NONLINEAR DYNAMICS,
MATHEMATICAL
BIOLOGY, AND
SOCIAL SCIENCE
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NONLINEAR DYN
ICS,
MATHEMATICAL
BIOLOGY, AND
SOCIAL SCIENCE
Joshua M. Epsteh
Senior Fefl~w,Economic SmJies Program,
The Brooking$ Instirution, and
Member, External Faculty, Santa Fe Institute
Lecmrrt: Notes Vo
S m t a Fe Ilns~trute
Studies in the Sciences of Complexity
Adason-Wesley h b l i s b g C ~ m p m yh, e .
me Adv
B w k &W
Reading, Massachusetts Menlo Park, Callfornia N m York
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Santa Fe Institute Editorial Board
December 1996
b n d a K. Butler-Villa, Chair
Director of Publications, Santa Fe Institute
Prof. W. Brian Arthrrr
Citibank Professor, Santa Fe Institute
Dr. David K. Campbell
Chair, Department of Physics, University of Illinois
Dr. George A. Cowan
Visiting Scientist, Santa Fe Institute and Senior Fellow Emeritus, Los Alarnos
National Laboratory
Prof. Marcus W. Feldman
Director, Institute for Population & Resource Studies, Stadord University
Prof. Murray Gell-Mann
Division of Physics & Astronomy, California Institute of Technology
Dr. Ellen Goldberg
President, Santa Fe Institute
Prof, George J. Gumerman
Center for Archmlogical Investigations, Southern Illinois University
Prof. John H. Holland
Department of Psychology, University of Michigan
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Dr. Stuart A. Kauffhan
Professor, Santa Fe Institute
Dr. Edwad A. Knapp
Visiting Scientist, Santa Fe Institute
Prof. Warold Morowitz
Robinson Professor, George Mason University
Dr. Alan S. Perelson
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Laboratory
Theoretical Division, Los A l ~ o National
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Department of Physics, University of Texas
Santa Fe tnstitlrte
Studies in the Seienwiit af Camptexw
Proceedings Volumes
Mitors
L). finess
A, S, Peresbon
A. S. Perelson
G, D. DooXen et d.
Title
Emerging Syntkaw in Science, 1987
Theoreticd Immunobg~y,Part One, 1988
Theoreticd h m u n o E o ~ Part
,
Two, 1988
Lattice Gm Methods far PsurtiizE DiRerentiaJ
XV
Equations, 1989
V
P. W. Andersan, K. Arrow, The Economy as an Evoking Complex System,
1988
& D. Pines
Artificid Life: Proeeedirmgs of an Interdiscipiinay
Vf
C. G. Langton
Worhhop on the Synthesis and Simulation
of Living Syskms, 1988
Computers and DNA, 1989
V11 G. I. Bell, & T, G. Marr
Complexity, Entropy, a d the Physics of
V111 W. H. Zurek
Information, f 990
Molecular Evolution, on Rugged Lan&ert-pe&;:
IX
A.S.Peretson&Proteins, RNA and the Immme System, 1990
S. A, KauRman
Artifidd Life 11, 1991
C. G*Langtorr et d,
X
The Evolution of Humm Languagw, 1992
XI
J . A, Hawkins &;
M. GeU-Mann
XII M, Casdsli St 5. Eubank Nonlinear Modeling m$ Forw~tsting,f 992
,
Principles of Orgmisation in O r g a ~ s m s 1992
XlIf J, E. Midtentha1 &
A. B. Bakin
The Double Atrction Market: Institutions,
XIV D. fiiednnan & J. Rust
Theories, and Evidence, 1993
Time Seria Prediction: Forecasting the f i t u r e
XV A. S. Weigend 8d
and tfnderstmdr'ng the Past, 1994
N. A. Gershenfeld
Tlnberstmding Complexity in f be
XVI G. Gumerman &
Prehistoric Southwmt, X994
M. Gell-Mann
Artifieid Life IIX, 1994
XVYZ C. G. Langton
Auditory Dispfiay, 1994
XVXII C. Kramer
Complexity: Metaphors, Modelsi,
XIX G. Gowan, D. Pin=,
and Reality, 199.1:
& D, Mettzer
The Mathematics of Generalization, 1995
XX
D. H, Wolpert
SpatieTemporizX Patterns in
XXI P. E. Clsurfis &
Nonequilibrium Complex Systems, 1995
P, PalEy-M&oray
The Mind, The Brain, and
XXIT B. I\i"lorowida&
Complex Adaptive Systems, 1995
J. L. Singer
Madurational Windatvs and Adult
XXIlX B, Julesz &
Corticd P l a t i c i t ~ 1995
~,
X. Kov&s
Economic Uncertainty a d Human Behavior
XXIV J. A. Tainter &
in the Prehistoric Southwest, 1995
B, B, Tainter
XXV J. Rundle, B. n r c o t t e , 8t Reduction and Predictability Ctf Natural
Diswters, 1996
W. Klein
Adaptive Xndividuah in Evolving Popda-t;iom:
XXVZ R. K. Belew 8r,
Models and Algorit
M. MitcheEl
l
X
11
I11
VoI.
I
I1
III
IV
V
v1
Lectures Volumes
Editar
D. L. Stein
E. Jen
L. N d e I & R, L. Stein
L. N d e l & D. L. Stein
L. Nadel & D. L, Stein
L, N d e l 8t D, L, Stein
Lecture Nates Valums
Author
J. Hertz, A. Kro&, &
R, Palmer
G.. Weisbuck
W* D, Stein lk F, J. Vaela
J, M, Epstein
B. F. Wijhout, L- Ndelt &
I). L. Stein
Vol.
a
Referexsee Volumes
Author
A. Wuensche gt M, Lmser
Title
Lectures in the Scieacw of Complexity, 1989
1989 Lecturm in Complex System, 1990
1990 Lecturw in Complex Systems, 1991
1991 Lrrcturw in Complex Systems, 1992
1992 Lectura in GornpIex Systems, 1393
1993 Lecturm in Complex Systems, 1995
Titf e
fntroduction to the Theory af Neural
Computation, 1990
Complex Systems Dynamics, 1990
ThinEng About Biology, 1993
Naniinear Rynmics, Mathematical Efiiolo~~
and Social Science, 1997
Pattern Formation in the Physicd and
BioIogical Sciences, l997
Title
The Global Dynamia of Cellular Automata:
Atdration Fields of OneDimensional GeUular
Automist;a, 1992
Dedicated in loving memory to my father
Joseph Epstein
(1917-1993)
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Contents
LECTURE 1:
On The Mathematical Biolog of Arms Rmers, Wms, and
Revolutiom
LECTURE 2:
An Adaptive Dynamic Model of Combat
LECTURE 3:
Imperfect Collective Security and Arms Race Dynamics:
Why a Little Cooperation Can Make a Big Difference
LECTURE 4:
RevoEutions, Epidemics, and Ecosystem: Some
DynaMlical Analogies
LECTURE 5:
A Theoretical Perspective on The Spread of Drugs
LECTURE 6:
An. Intraducdioa. to N~lnline-it~
Dyn
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ntroduction
This book is based on 131 serim of lwtures I gave at the 1992 Santa Fe Institute
Cornpiex Systems Summer School, and on my Princeton University ""Complex Systems, Simple Models" course, ollFered in aademie years 1991-92 through. 1993-94.
A god of my texhing, and of this bocsk, is to impart; the mathematical tools and,
as important;, the isnpulse to build simple models of eomplm procwses falling outside the artificial confines of the established fields. Many Estscinating and important
problems cry out for rigorous interdisciplinary study. And recent advances in seientific cornpuging have m&e the construction and "experimental" "udy of dpamical
systems remarbbly emy. The stage, in short, is s t far new synghetic wark, indmd
for tz, new dbcipline or, perhaps, tramdiscipline.
Thrm themes run through these j e c t u r ~ ,The first is that simple modeh can
illuminate essenrtial dynamics of complex, and crucially important, social systems.
The second is that mathematical biology oEers a powerhf, and hither2;o underexplaited, perspective on both interstate and intrastate social dynamics. The third
theme is the unifying power of xut&bennatics, and specificdIy, of nonlinear dynamical
sy~tennsthwry; formal andctgles be-een memirrgly disparate sociai and biofogie d phenomena are highlighted. One overarching aim is to help stimulate some
thing of a reconstruction in mathematical social science, relming-in some ernes
2
Nonlinear Dynamics, Mathematical Biology, and Social Science
abandoning-the predominant ~tssumptionof perfectly informed utility maximization, and exploring social dynamics horn such perspectives as epidemiology and
ecosptem science.
OVERVIEW OF THE LECTURES
There are six lectures. The first is entitled, "On the M&hematica Biolow of A r m
b e e s , Wars, and Revolutions." Here, some af the book" recurrent themes are
first souabed. It is demonstrated, X believe for the first time, that the most hmous equations in the mathexnatksl thmries of war (Lanchesterk @equations)and
a r m races (Mehardson's equalions) are both speeializations of the famous L o t b
Volterra ecosystem model. The essay introduces the related idea that explosive
mlutions-might be modeled as epidemics, using yet
processes of civil violenc
other parametrizl2tions of Loth-Volterra. To me, it is surprhing and interesting
that the Loth-Volterra ecosystem equations have artgthzng to say about wtzx, a r m
races, or revolutions. Of course, to claim that t h e ~ esimple equations say everything on, such complex topics would be foolish. And, in subsequent fectures, I move
beyond them,
Lecture 2 delves hrthc?.f into the mathematical theory of combat. Drawing on
mathematical biology and, of coume, history, this lecture oEers whrzt; f believe to
be a fundamental critique of the dominant approach, bmed on the equaekons of
F, W. Lanchester. Lanchester Theory, by which P mean the origind equations and
their csnt;emporary extensions, produca anomaIous results, mathematically precluding important absemed behaviors, l i b the trizding of spaee for time, Them deep
problem arise baause the belligerents, W idealized in the theory, are wmpletely
non-adaptive. War is, I argue, precisely a proems of codaptation, though the contestants bear a much closer regemblanee to Ashby" seat than to Homo economicus.
My awn Adaptive Dmamic Model tries to capture this as simply as possible, in the
process overcoming the anomalies of Lanchester Thmry.
The "nonlinear dynamics of hope" iis illustrated in lecture 3. By way of introduction, many applications of nonlineakt. dynamics show that complex systems
can be poked at the brink of dis&er; smaEl perturbatism in crucial variables can
produce casede @&inetionsin ecosystem, devmtating epidemics in human populatioas, or ozone holes in the atmosphere, d l unhappy events. Sensitive dependence
is b d news. Perhaps id is salient that we edl the area ""cat~%rophe
thmry'bnd
not, for insita~ce,"mirmle thmry." Well, this lecture invites us to consider the
Aipside of the nonlinear coin: can we identify cmes in which the right, smdl local
perturbzttion, can produce countezin%uitfiveexplosions of happy eve~ts?1think so.
As one example, there is an idea cdled '%collective seeuri~'"(65) that is receiving wide attention, Imagine three countries, A, B, iand 6 , Perfect CS would
then operatf? as follows: If A attach B, C allocsa2la all force to B; if B a t t w h C,
A aocates all farce to C; and so an. The general rule is simply that *e odd man
aut instanfrlg allomtes all J o ~ to
e the attackd pady.
NW, many aedemic political scientists and &absmen d i s ~ s any
s form of collective security because this ps;feet form is implausibly altrubtic. But, as nonlinear
dynamichts, we ask: What about a. tiny bit, of mllective ~ c u r i t y a, highly diluted
farm of altruism? The bcture shows that in m m race modeh susciently nonlinear
to produce really vola-~ilr;dynamics, highly difukd, or imperfect, collective ~ c u r i t y
regimes eaa damp the explosive omillatiow atld induce convergence ta stable q u i fibria below inikial wmament leveh. Put differently, the injwkion of tiny degres of
altruism can profoundly calm the athemise voXatib dynamics, The benefits of the
system are very great and, becaum of the wnlinearity; the level of risk to individual
participants is wry low. Here, mnsitive dependence is good news!
Whether the next lecture bears good news or bad depends, X suppose, on o ~ e ' s
political leanings. It examines the analogy b e t w ~ nepidemics (for which. a welldeveloped mathematical theory exists) and processes of expbsive social chanp,
such as revolutiom (for d i e h no comparable body of mathemat;ical theory exists),
Are revolutiom "lih" epidemics? If one t h i n k of the revolutianav idea as the
infection, the revolutionaricirs as the infee-tives, the public bealth audhoritim ins the
pawer elite, m d social indoctrination as inoculation, then an analogy begins to
take shape. E t is develop& ixt $he fourth lecture, with, 1 hope, some novel political
inhrpretations*The a n d a w to epidemics, which are nonlinear dhreshold procemes,
may help explain how small changes in; political conditions-marginal diminutions
in central authority-can catalyze explosive sociali tramformations, much to the
surprise of eli* and revolutionarim d i k ! The model also sugmts the existence of
social bifurcation points at which regression abruptly changm from being stabilizing
to being destabilizing and inflaming revolutionary sediment.
The &&h lmture combines arms race and epidenriolom perspwtives in building
a simple model of the sprearl of drug addiction in an, idealized cornmunit;): revealing
basic, and pahaps counterintuitive, relationsfiips bemeen legalization, pricm, and
crime. The analysis suggats once again the relevance to social science of seemingly
remote fields like mat;l.lema$ical epidemiobg and ecosystem science, and id trirts
to illustrate how simple models are built, and explored using methods of nonlinear
analysis,
These methods are the topic of lecture 6,Entitled ""An Introduction to Nonlinear Dynamical Systems," it consists entirely of mathematics, The ainn is to offer
a concentrated course in. the qualitative tlt-imry of nonlirrea autonomous diEerential systems, beginning with linearized stability analysis &and movirrg efficiem_t;ly
through Lyapunov functions, limit cycles, the Poinear&Bendixson and Hopf Bifurcation Theorem, Painear4 maps, various negative tests, and on to Index Thwry
arrd the celebrated Poinearh-Hapf Theorem from difterent,ial topology. 1 see this as
a coherent body of mathematics, much OE which i8 quite beatudifut, m d powerful
when applied to social dynamics.
I hmten to point out that lecture 6 is not dmigned as a mathematical founclation for the other le&urw, Not all the techniques developed there are ii~pplied
4
Nonlinear Dynamics, Mathematical Biology, and Social Science
in other lectures. Index Theory, for example, is developed for the sheer joy of it,
not because I use it elsewhere, though its surprising applications to mathematical
ecolom and economics are noted. Xn turn, not all techniques used in other lectures
are covered in lecture 6. Lectures 2 and 3, far instance, hvolve digerenc
quations, which are not treated in lecture 6. Lecture 6, then, is
a frestanding essay oEering a particular dewlogment of nonliaear dpamical systems, from linearized stability analysis through the PoincarkHopf Xndex Theorem
via, results of long&anding mathematical interest, such as Hilbert's 16th Problem
and Brouwerk s h e d Point Theorem.lxJ
One final point regarding the lectures should be made. While I oEten use history
or empirical studies to argue for the qualitative plausibility of a model, no new data
b m s are assembled or statistical tmts performed. As oRen occurs in science, thmry
may ultimately inspire the collection of data and the performance af tests. But these
lectures are purely theoretical, the goal being do demanstrate to social, physical,
and natural scienti~tsthat simple mathematical models can provide inszghl into a
wide range of complex social pracemes and th& mathem&ical biolah~yand nonfinear
dynamical systems t hmry in particular offer the social theorist powerful conceptual
and analytic tools,
THE LARGER INTELLECTUAL MNDSCAPE
These, of course, are not the only methods available for the study of social phemmena. And, in my Princeton course, I gave equal time? to the agent-based modeling techniques employed in Grovring ArtificialSocieties: Social f i e a c e Rom tne
Bogom Up, co-authored by myself and Robert k e l l . Both nonlinear dynamical
sydems aad agent-bwed models deseme a place in any "comgle~tyc ~ r r i c u ~ u m ~ "
But the former techniques are the ones employed here. For agent-bued models of
social syslerns, see Epstein and k t e l l (1996) and the g w i n g literature cited there.
ACKNOWLEDGMENTS
A number of colleagua ~ n institutions
d
dwewe special thank. For careful reviews
of the manuscript or portions thermf*for stimulating discussions, encouragement, or
flf%heIstare msumw familiarity with vector csleulus, linear diEerentiaf equations, eigenvalu+
eigeavector methods, phase plane analyak, and c&rain elements of camplm variables, real analysis,
and pa&ial differential quations. Partions of the other Imlurm &also m u m e exposure to certain
of thme topies.
f210ninsight, ars against pre-didion, as a goal of m~deging,see Hirsch (1984).
advim, I thank Robe& Axelrod, Robert &ell, Bruee G. Blair, John Casti, Malcolm
DeBevoise, George Dawns, Samuel Bavid Epsdein, Marcus W. Feldmm, Dunean
Foley, Murraty Gell-Mann, Attee Jmkson, Jean-Pierre Langlais, Steven McCarrofl,
Etaine C . MerJulty, Goafried Mayer-Kress, BenoiL Morel, Lee Segel, Car1 Sinnon,
Daniel Stein, Arthur S. Wightman, and H. Pefion Young. I thank the Princehn
University Council on Science and Technoloa for funding, and the Woodrow Wilsan
School for hosting, my courE, f am grateful to the Brooking@fnstitution for its
support and especially to John B. Steinbruner far the climate of unfettered inquiry
in which this research W= eorrdueted, I thank Daniel Stein for organizing the Santa
Fe Institute Complex Systems Summer School, I offer dwp t h a n k also to my farmer
SF1 and Princetan students. For expert wistance in preparing the manuscript,
I thank Risha Brandon. I am grateful to Ronda K, Butter-Villa for editing the
manuscript, and do Dells L, Ulibarri far production msistranee,
Finally; far their love and support, X thank: my wife Melissa, our daughter Axlna
M&ilda, my mother Ltrcy, and my brother Sam.
The views expressed in this book are those of the author and should not be
aseribed to the persons or organizstt.ions aeknowIedged above.
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LECTURE 1
On The Mathematical Biology of Arms Races,
Wars, and Revolutions
In this openbg lecture, I will attempt a uni@ing owrview of cerl;ain social
phenomena-war, arms raeing, and revolution-from the perspective of mathematical biology, a field which, in my view, must ult;im&ely subsume the social scienees.[31
Uafortunia;Lely, few social scientists we exposed to mathema;tieal bioloa, specifically the dynamied sy~kemspmspective pianer& by Alked, htb, Vita "Vofterra,
and athers, In turn, few m&hemil,tical biologhts have considered the application of
mathematicd bia.ofowt o problems of human soeiety.f41
Particularly in are= of interstate and intrastate conflict is there a need to
explore formal a n a l o @to
~ biological system. On the topic of animal bebavior and
human warfare, the anthropologist %chard Wrangham obmrves,
1 3 ) ~ hpempmtive
e
t&en here, however, is quite distinct from that t&en by Wward 0, WiLon, in
his book Soeiobzolog%r(1980). Specifically, X do not disc= the role of gena in the eantrol of h u m n
social behavior. h t h e r , the wgument is that mwro social behaviofs such m was, rewlulion, arm
races, and the s p r d of clrug nnsy conform well ta muations af mathematical biob
f%Y
ltnd epidemiolow in p&icutas, h r h a p s ""socioeeolo@" would be a suitable nBme for this bvel of
analysis.
W or a notable mmption,
Gam1Xi-Sforza and Feldman f 1981). See also the innovative and
undemtudied -works, b h e v s k y (1947) and h h ~ v s k y(1949).
Nonlinear Dynamics, Matnthemiliti~alBiolagy, and Social Science
"The social organization of thousands of animals is now kxtown in considerable detail. Most animals live in open group with Auid membership.
Neverthekss there are hundreds of mammals and birds that form semiclosed groups, and in which long-term intergroup relationships are therefore found, T h e e intergroup re1ett;ionsbips are knows well. In general they
v a y fmrn benignly tolerant to intensely competitive at territorial bordexs.
The striking and remafkabb discovery of the last decade is that only two
species other than hum- have been found in which b r d i n g males exhibit
systematic stalking, raiding, woundilng and killing of members of neighbor-.
ing groups. They are the chimpanzee (Pan troglodytes) and the gorilla (Pan
gom'lla beringei) (Wrangham, 19%). In both sgeeies a grow may have p s
riods of extended hastaity witb a particular neighboring group and, ia the
only two long-term studies of chimpanzees, attach by daminant against
subordinate eommurtiti~appeared rmponsible for the extinction of the
latter.
"Chimpanzws axrd gorillas are the species most closely rela;t;ed to humans,
~
diverged earliest
so close eh& it is still unclear which af the t h r species
(Cbchon & Chiarelli, 1983). The fact that these thrw species s h x e a pattern of intergroup aggression that is otherwim unhown speak clearly for
the imporl;ance of a biological component; in human warfare'"(VVrangham,
1988, p.78).
Although man has engagd in arms rmixlg, waring, and other forms of organized violence for a1E of recorded history, we haw comparatively little in the way
of formal thwry. Mathematical b i o l o ~may proGde guidance in developing such
a thmry m(arrgham writes, "Given that biology h in the process of developirrg a
unified theory of animal behavior, th& Iiurnan behavior in general can be expected
to be understood better as a result of biological theories, and that two of our closest evo1uLionary relatives show human patterns of intergroup aggression, there is a
strong case for attempeing to bring biolow into the andysk of warfiare. At present,
wwld like to sm mare efT~Ipt;,specifically
there art: few effort;s in this direction." @"l
more mathematical eEort, in this direction and hape to stimulate some interest
annong you, To convince you that there might coxreeivably be some '"unified field
tftmry" worcth. pursuing, I: w a d to shztre mme observations with you. To set them
up, a little backpound is required.
The Eundamexltd equatiom in, $he mathematical theory of arms races are the
so-call~dRiebardson equations, named for %heBridish applied mathematician and
soeiat scientist Lewis ~e Richardsan, who first published them in 1939.161 The fundamental equations in the mathematical theory of combat (warfare itself, as against;
peacetime a r m racing) were publishcl in 1916 by Rederick Wililiarn Lanchmter.t71
~ r a n & a m(1988, p.18).
[6]~ichm&an( 1939) and (1960).
[']SW ifranchmter (1916). For a cantenrporq diaussion with referencm,
Epstein (1986).
The formal thmry of ixrt;erstate conflict, to the extent there is one, rests on these
w i n pillms, if you will. Meanwhile, the clwic equations of mathematical x o l o ~
are the Loth-Volterra equatiom.
In fight of the remarks above, I find the following fact i~triguing:The Richadson and Lanehmter mdels of humm conflict are, mathematicaXly, specializatians
of the Loth-Valterra ecosystem equations.
Before proceeding, X must make one point unmistabbly char. I do not claim
that any of these modeh is really ""right" in. a physicist" sense. They are ilfumirrating abstrwdions, f think it w w Picwso who said, <'Art is ~l lie that helps us
see the truth." So it is with these simple models. They continue to form the conceptual foundations of their rmpeetive fields. They are universally taught; mature
practioners, knowing full-well the models~pproximatenature, noxlethelss entrust
to them the formation of the student" most basic intuitions. And this because, like
idealizations in other sciences-idealizadions that are ultimately "wrong" -they efficiently capture qualitative behaviors of overarching imt;erest. That these ecosystem
and, say, arms rme equ;zdionts should look ad all alike is unexpected. That, on elosc3r
inspection, they are virtutdly ide&ica! is, to me, really quite int;csrc?lstixlg.Let me go
a bit further.
Under yet other parameter set;lings, the Loth-Volterra equations yield standard modeis of epidemics. And, in other lectures, I will argue that social revolutiouss
and illicit drugs ma~ywe11 s p r e d in a strictly analogous way or-at; the very leastthat an epidernietlogical perspective on such social praeessrss is promising. Once
more, the poixrt; h simply that socid science might learn a lot from mathematical
biology and, conceivably, might inherit some of its apparexlt unity
Let me now introduce the LotbVolterra equations and show how the classic
arms raee and war models fall out as special cases, Then, I will aplore the analogy
between revolutions and epidemics, In, subsequent lectures, we wil move beyond
THE LOTM-VOLTERRA WORLD
The Loth-Volterra equations are as follows:
In discussing these equations, I will freely invoke nonlinear dyniamicd systems terminology present4 in leeture 6.W n r n i n g now to system (1.11,zi(L) iS the s p e c k
4 population at time t; the a" and r's are real parameters,
isl~nderthe rime, ""qdratic madeX," 'equimlent equat;ions and a; number of specializalionsincluding combat variants-we discus& in Beltrami (1987).
10
Manfinear Dynamics, Mathematical Biolagy, and Social Science
If all a@'sequal E r a and 7-1,Q > 0, we have unbsunded exponentia1-secalled
Maltbusian-grohh- Since, ultimately; there are limits, for instance, environmental.
casryhg capacities, the t e r m all,a22 > 0 are preceded by a negative sign. Then, in
the languwe of lecture 6, the species are self-inhibiting. Lewing rl and rz pwitive
and still assuming a12 = a21 .= 0, this mrtnnption yields a Iogirstic approach for
ewh species t;o the positive: p h e plaae quilibrium
a node: sink.
Now, life really gets inbrwting only when species intermt, and this invoivw
%hecross t e r m a12 m d azl.
Leiaving everfihing else as is, let us now msctme alz,azl > 0. X-n that case our
d be in a relationship of mutualhm, or reciprocal activation; the
spwim are ~ & to
population level of one f&s back poSilively on the growth rate of the other, Bees
and flwers-polfinators and go&natm, if you will-provide examplXcks. There u e
many others.
S e t t i il
~ = $2 0, the inLerior equilibrium coadiLions are
5~-;
Of courm, these are alsa the equilibrium conditions for the linear syystenr:
But t h b is exwtjty the fmous R i c h a r h n model of an arms rwe! The more bees,
the more flawers, and vice versa. It" the same in (1.31, but not quite as idyllic.
The more weaponry my abwrsary has, the more f wmt, and vice versa, up ta some
r ecalogical-limit or csrying capwily
Richardson's basic idea is that a state" arms race behavior depends on three
overriding fwtors: the perceived external threat, the economic burden of military
competition, and the magnitude of grievancm against the other parw I discuss
these at greater l e n s h in lecture 3. Sufice it to say here that Q ,rz > 0 repremnt
fundament;al grievmces; ars, a21 > O are the reciprocd wtivation coeEcients (the
rates at;which emh arsenal grows in rapame to the other); and all, azz are the selfinhibiting, or damping, t e r m which Eefiardmn identified with eeonornic fatigue,
M&hematical biologists have long ssked how mutualistic papulatiom avoid exploding in what Robert May called an "orgy of mutual benefaction."[%l i h w h e , we
can a ~ what
k
mwhanism d m p s the upward action-reaction military dynamic r e p
raented in the Richardson model. In each case, self-inhibitory eEects must somehow
dominate reciprocal activation effects if a stable species equilibrium-r
military
"blance of powerB-is to emerge. Stability analysis bears this out.
The positive (or
Clearly, we can write (1.2) in matrix form r Az = 0, s E
interior) equilibrium of system (1.1) and the sole equilibrium of (1.3) is therefore
given by Z = -A-%. For each model, the stability of can be evaluated by the
methods of lecture 6,
By a simpfe translation, the mehardson equations (1.3) are globally asymp
totically stable at Z if and only if = Ay is globally asymptotically stable E& the
origin, w h r e y = z - 2. Rom lecture 6, we have the well-hown stabilty criterion
+
Now, Richardson's economic fatigue means
all, a22
> 0. So, we have
And we will have Det A 3 Q precisely when alla22 > alzaz~,which is to say that
inhibition (allazz)outweighs activation (alzazl),confirming our intuition.
One c m demonstrate[lQlthat the eigenvdua of the Jacobian of (I.l) at f have
negative real parts (indeed, are negative reds) when the s m e condition is met.
An isocline andysis is also revealing. W recall that an isocline is a c u r v e h e r e
a finewhere one side" rate of groMh is zero; clearly, an equilibrium is a, point
where isoclines int;ersect. Erom (1.2), the baclinw ape given by:
all
411(21) = -X1
1312
a2 1
(zl) = -21
a22
bfi2
T1 (the s
l - isocline) ,
-a12
7-2
+(the zz - isocline) .
a22
For local stability of the equilibrium Z, we require the configuration of figure
1.1, But, this occurs only if the slope of c;bl exewds the slope of (Bzl which is to say
allfa12 > az1la22, or
a11a22
> a2xa12 .
Our intuition is agt.zin confirmed: stability requira self-inkbitidion t d exceed reciprocal activation in this sense.
12
Nonlinear Dynamics, Mathematical Biology, and Social Science
The main. point, however, is that the classic Loth-Volterra model of mutualistic
species interaction embeds, in i b equilibrium behirlvior, the elmsic Mchardson arms
rwe modd.
In the models above, of course, the ""penotypes'Vo not change. in fact, ecosystem
dynamics mlect against certain phenotypm. Roughly speakng, phenotmic frequencies and papulailion. 1eveh have interdependent trajectories, This is very clem, for
exwple, in irnmunalow, where antigens and antibodies coevofve in a s+calIed
"'biologicd arms rwe." But, of course, real a r m races work this way, too. Ballistic
m k i l w beget antribdlistie missile defemes, which beget various evasion and defense
supprmsion teclrnologiw. The mwhine gun makm cavalry obsalete, giGng r h to
the ""ion horsen-the tank-which begets antitaxlk weapom, which beget special
armor, and so on, Michwl hbinson's analogy between moth-bat coevolutiorr and
the coevolution of World War XZ air war twtics is apposite,
"Moths and their prdators are in an arms raee that started millions of years
before the Wright brothers made the Brmden rdds possible. Butterflia exploit the day, but their 'sisters2he moths dominate the insectsbhare of the
night skies, Few vertebrates conquered night flying. Only a small frwtion
of bird species, mostly owls and goatsuckers, made the transition. Bats, of
course, made it their realm. Many species of bats are skilled 'moth-ers':
they pursue them at speed &&erdettecting them with their highly attuned
echolocatian system. Some maths, however, have developed kars' capable
of detecting the bat" ultrasonic cries. When they hear a bat coming, the
moths take evmive actian, including dropping below the bat's traek. The
pstrallels of the responE of Allied bombers to the r d a r used by the Germans in Wrorld War 11 are interesting. If we visualize the bombers as the
moths, and radars on the ground and in the night-fighter aircraft;as bats (a
reversal of sizes), the situation is similar. Bombers used rearward-listening
radar t o detect enemy night; fighters. When they detected a figheer, they
took evwive action. But heavy bombers, heavily l d e n , were not wry maneuverable. They couldn't dodge about quite as well as moths. S m e pilots
tried to drop their aircraEt into a precipitous dive. Moths also do this; it is
ewy h r tbenn to fold their wings and drop. The next; stage in the nightbattle escalation is predictable. The night fighter's radiar was eventudy
tuxled to detect the bomber" fifigter-detector, and thus the bomber itself*
Bats have not yet tuned in on mothsbars.
""Bombers also used tecfr;tnologiealdisruption. Night fighters came to be
guided to bombers by long-distance radars on the ground. The fighters
started winning. But nothing remains static. The ground radars could be
jarnmed by various kinds of r d i o noise. The technologicd battle swung
the other way. Then the fighters wquired radar. Much like a bat, a fighter
e m i t t d and listened to radm signals of its own. Thwe, too, proved to be
susceptible to countermeasures, however. The RAF could jam the fighters-adar or 'clutter' it with strips of aluminurn foil. Each bomber in %
formation dropped one thousand-strip bundle per minute, so that huge
clouds of foil foiled the radar. Amazingly, there may be a similar counterweapon among moths. Some rnoths can produce ultrasonic sounds that fall
within the bats' audio frequency The moths' voice boxes are paired, one
on each side af the thorm; double voiws must be particularly confusing.
Alien sounds in their waveband could confound the bats, exactly in thc?
same wrtly the foil confounded the f ghters.
""The next steps in the bat-versus-moth war may simply be awaiting discovesy by some bright r-earcher; &er aft, we did not know a lot about
echolocation in bats until &er World War II. My guess would be that the
detectar will get more complex to m e t the defenses. Thb may already have
happened; bats specidizing in moths with ears may have moved to a higher
I4
Nanfineatr Dynamics, Mat,them&tlticaiBiology, and Ssciat Science
frequency sound outside the mot;hsYhearing range!" (&binson, 1992, pp.
77-79).
Quite clewIy, levels of asnnamsnt (in the international system) a ~ tevebs
d
of
papulaeion (in an ecosystem) interact;, W in the ZothVo1t;erra and Ridardson
models, but phenotwes themsc3lves are stfw changing. In, biolog, there is a mathe
In socid science, there isn't. There probably could
matical Lhwry of coevol~tioa.[~l)
be, so I simply mention it as a prom&ng &redion.
NW, let us shift, p a r s froin the rnutualistic/arrns rme variant of (1.1). Speeifically, instead of wuming that a12 and a21 are positive, sassume that they are
nt3gat;ive.
Rearranging slightly?the equations (1.1) eake the form
where ki E ( r i / a i i ) > O is the carrying capacity of the environment for each species.
These equadions were published in 1934 by the gre& Russian mathematical bbiologist
G. F. Gause in his boak The Stmggke for Existen=. Indeed, he termed nlz and a21
"LcoeEeients of the struggle for dstenee." im1
Now, exmining (1.6), each species wutd @hibitlogistic gro*h t a its respective carrying capacity but fix these Interaction-truggl
erms- XneEuding them,
(1.6) gives a picture of uniform mixing of the populations z l and zz, with contacts
proportion4 to the product slsz,Now, however, since the interweion coeEcienQ
and species 2 at rate asr. Quite
are negative, ea& contwt kills spmies 1 at rate
clearly; a parallel tto combat is suaested, But more is true.
TR fact, unbebownst to Gaum, (li.6) is an exmt form of the famous-and to
this day uttiquilous-Lancfrestc3r model of warfaretim"l
The tramition fom arms race to war, then, might be seen as a transition from
the e w of al2,azl > O ta tbe erzse of alz,a2l < 0. In the fatiter context, the wUknown biological "principal of competitive exclusion" simply maps to the military
principle that;, usutxollyt one side wins and the other side loses. Both these competitive
exclusion behaviors reflect the mathematical faGt that the interior (zl, 22 > 0) equilibrium of (1.6) is a saddle. The stable equilibrium in the mutualistic-peacetime
me was a node. To the extent thew models are correct, then, we can
say (pacem Poincar6) that war is topologically differ& from peace; the outbreak
of war is a bihrcation from node to saddle.
Thus far we have been exploring a mathematical biolou of interstate relations;
what; about intrwtate dynamics? 1s there a Loth-Volterra perspective on rwolution, far instance? And, to what biological process might such socid dynamics
correspond?
REVOLUTIONS AND EPfDEAnfCS
Consider the fallowing specialization of (1.1):
Then (1.1) bwames
kl = -(3125122,
52
= a12z1x2,
which is the simplest conceivable epidemic modet. Now, ratficsr than armament
levels, $1 represents the level of susceptibles, and licz the level of infectives, while the
parameter a12 is the infection rate, expressing the contagiousness of the infection.
Ideal homogenmus mking, once mare, is assumed. If population is constant at PO,
then z1 = Pi - 22 and we obtain
our familiar friend the logistic diEerential equation. Here, rcz = O is an unstable
equilibrium; the slightwt introduction of infectives, and the disewe whips through
the whole of society.
A trad&ional t x t i c for combating the spread of a disease is removal of infectives.
Sometimes, nature does the removing, as with fatal diseaes; afierr, society removes
infmtives from circulation by quarantine. The simpfest possible mumption is that
removal is proportional to the size of the infective pool, yielding the bllowing variant
of (1.1):
with r2 > 0. This is the famous Kermwk-McKendrick (19-27) threshokl epidemic
model,il4J seedled because it exhibits the following behavior,
t l 4 l ~ e r m w kand McKendrick (1927). For a contemporary sLaternent, sew Walkman (1974).
16
Nonlinear Dynamics, Mathematical Bioiogy, and Soda! Science
By definition, there is an epidemic outbreak only if k2 > 0. But this is to say
- r222 > 0, or
1"2
s1>-.
(1.11)
alzllslsz
a12
The initial susceptible level ~ ~ ( must
0 ) exced the threshold p z r z / a l z ,soncl*
times called the relative removal rate, for an epidemic to break out. The fact that
epidemics are threshold phenomena has impofiant implications for public health
policy and, X will argue below, for social science.
The public health implication, which was very controversial when first diseovered, is that less than universd vacination is required to prevent epidemics, By
the thrmhold criterion (l.ll), the fraction immunized need only be big enough that
the unirnmunized &action-t he actual susceptible pool-be below the threshold p.
"Herd immunity," in short, need not require immunization of the entire herd, Far
instance, diphtheria and scarlet Emr require 80-percent immunization to produce
herd immunity;(l" lethcote and Yorke argue that "a vwcine could be very effective in cantroning gonorrhert.. .for a vaccine that gives an average immunity of 6
months, the calcufat;ions suggest that random immuniz;l.tion of 112 of the general
population each year vvould cause pnorrhett to disappear."i161
Mathematical epidemic models are discussed more fully in lecture 4,With the
above as background, let us now consider the analogy between epidemics (for which.
a rich mathematical theory exists) and promses af explosive social chmge, such m
revolutions (for which no comparable body of mathematical theory exists). Again,
a more careful and deliberate development is @ven in lecture 4. Here, we simply
oger the main idea. It will facilitate exposition to re-label the variables in (1.10). If
S ( t ) and i ( t ) represent the susceptible and infective pools at time t and if r and y
are the infection and removal rates, the basic model is:
with epidemic threshold
.%I
The basic mapping from epidemic to revolutionary dynamies is direct, The
infection or dimme is, of course, the rewlutianary idea. The iafectives I ( t ) are
individuals who are actively engaged in articulating the revolutionary vision and
in winning over ('"infecting") the susceptible class S(t), eompxised of those who are
receptiw to the revolutionary idea but who are not infective (not aetively engaged
in transmitting the disease to others). Removal is most naturally interpreted as the
polit;ieal imprisonment of infectiva by the elite ('the public health authority'").
melatein-~msshet(1988, p. 255).
i1B1~&h~otc3and York@(1980, p. 41).
flSj
Mmy familim t w t i c ~of totalitarian rule can be ~ e asnmrasura to minimhe r
(the effective contact rate between infectives and susceptibles) or maximize y (the
rate of political remavd). Prms cewor~fiipand other res2;rictions on, free speech.
rduce T , while increases in the, rate of domestie spying (to Idc?&i& infeetiva) and
of imprisonment without trid increase 7.
Smmetricstjtliy, familiar revolutiomary taet iw-ucb
W the publication of underground literature, or ""smizdat*%wk to inereme r. Similarly, Mm%directive
that revolutianarim must ''swim like fish in the sea," mafxing themselvm indistinguishable (to authorities) &orn the surrounding susceptible population, is intended
ts reduce 7,
GORBACHEV, DeTOQUEVILLE, AND THE THRESHOLD
Interpreting the threshold &&ion (1.13), if the number of susceptibls is, in fact,
quik close to p, then even a sligh2; reduction (voluntary or not) in central authority
c m pwti society over the epidemic thrmhold, producing an explosive overthrow
of the existing order. To take %heexample of Gorbacfiev, the policy of GIasxrost
obviously produced a sharp inerewe in r , while the relmation of political repre-ssian
(e.g., the weakening of the KGB, the release of prominent political prisoners, and %he
dismantling of Stalin"s Gulag system) constituted a reduction in y.Combined, these
meabsures widen;t;ly depressd p to a level below So, and the ikevolutioxls of 1989''
unfolded, Perhaps DeToqueville intuited the tf-treshold relation (1.13), describing
this phenomenon, when he remarked that "KberaIizatiort is the most dificult of
political arts."
As3 a final element in the analogy, systematic social indoctrin&ioxr can produce
herd immunity to potentially revolutionary idea. We even s w "booster shots" administered at regular interwIs-May 1 in Moscow; July 4 in Americ
occasions the order-sustaining; mfihs ("The USSR is a clwsless ~vorkers$it,rdise";
"Everyane born in America has the same appo&unit.ies in lifen")re ritually celebrated.
Now, as 1 said behre, all these analogic are doubtlessly terribly crude, X certainly do nod claim either that any of the models are right or that the dmamical
analogies m o n g them are ex
Yet, the very faet that a single ecosystem modelould specialize to equations that even caricature,
the Lath-Volterra equationt
however crudely, such basic and important social processes as arms rwing, warring;,
and rebelling is, f believe, very injeerestixlg and serva to reinforce the larger point
with which X began: social sciertce is ultimately a subfidd af biology*
18
Nonlinear Dynamics, Matthamatt;ticafBiology, and Social Scisnce
Finally, let me conclude with an, admisssion. X was surprised when I began to no"ciee
these colnxsecLions. But why should we be surprised? In certain non-Wedern cultures, where our specia is seen as ""a part; of nature," where gods---like the s p h b can be part man and p a t lion, "all these connections be-mn ecosystems a d socid
systems might appear quite unremarbble. But in Wesit;ern cultures shaped by the
Old Testament, where God creates only man-not the khes, birds, and bushesin his own image, man is swn as ""apstrt from nature*" And, accordingly1 we are
surprised when our modeh of fis
worse yet, of viruses-turn out to be i&erest;ing models of man. Perhaps we are true Daminians more in our heads than in
our hearts. Creatures af habit, we me captive to a transmitted and slowly evolving
cultwe, But, of course, this too is "ody natural*"
LECTURE 2
An Adaptive Dynamic Mode of Combat
In this lecture Z, would like to give m introduction to some simple mathematical models of combat, including my own Adaptive Dynamic Model. Here, we are
concerned with the course of war, ritf;her than the arms rww or criss that may
precipitate war. Before discussing specifics, it may be we11 to consider the basic
question: What are appropriate goals for a mathematical theory of combat at this
poinit?
First and foremost, we need to be humble. Wxfare is complex, Outcomes may
depend, perhaps quite sensitively, on technological, behavioral, environmental, and
other factors that are very hard to rneasure before the fact. Exact prediction is
really beyond our g a p .
But, that" not so terrible. Theoretical biologists concerned with morpbogenesis-the development of pat;tern-are, in some cases, situated similarly. For the particular lmpard, we certainly cannot predict the exact size and distribution of spots.
But, certain clwsfj~
of partial differential equaeions-rewtion-diEusiort equationswill generate generic animal coat patterns of the refemnt sort. So, we feel thizt this
is the rigfit body of mathematics ta be exploring. The same sort of point holds
for epidemiologists. Few would claim to be able to predict the exact onset point
or severiw of an epidemic. Theoreticians swk simple models that will generate a
reasonable menu of core qualitative behaviors: threshold eruptions, persistence at
20
Nonlinear Dynamics, Mathamaticaf Biology, and So~iaiScience
endemic levels, recurrence in cycles, perhaps chaotic dynamics. The aim is to produce transparent, parsimonious models that will generate the core menu of gmss
pralitative sgstem beha~ors.This, id s e m s to me, is the sort of claim one would
want to m&@ for a mathematical theory of combat.
NW, in elarssical mechanics, the crucial variableu are mass, position, and tirne,
In classical economics, they are price and quantity: War, traditionally, is about terrihry and, unfortunately, death, or mutual attrition. A rapectable model, at the
very lemt, should oEer a plausible pieture of the rdationship beween the fund*
mental processes of attrition and withdrawl (i.e., territoriaf, sacrifice), I will discuss
attrition first.
The big pioner in this general area was Reclerid William Lmchester ( 1868-1945),
The eclectic English engineer made contributions to diverse fields, including autaMe is best remembered for his
motive design a d the thmry of
equatioas of war, appmopriately dubbed the Lancfi&er equations. First set h r t h
in his 1916 work, Aircrafi in Wa.rfare,these have a mriety of form, the most;
renowned of which is called-for reasons that will be given shortly-the Lanchester
""square" m ~ d e l , I With
~ ~ I no air power and no reinforcements, the Lanchester sytlare
quations are
of
ers of "Blue" and "Red" wmbata
EXere, B(t) and R(t) are the
which is an idedized fire sourc
d 6,r > 0 are their respective firing
SS
per shot. Qualitatively, these equations say something intuitively very appealing,
indeed, seductive: The atf~tlionrate of each belligerent- i s propodional to the size
of the adversary. In the phwe plane, the origin is obviously the only equilibrium of
(2.1) and the Jaeobian of (2.1) a-t Z is
[l71~mef.lesler
(1956).
[l81~w
Lanchwter (1916). The same model was appwently deveEop& independently by the RUBsian M. Osipov (1915).
The eigenvalues are clearly f G.Hence, the origin is a saddle, though the positive
quadrant is all we care about. The system (2.1) is, of course, soluble exactly. With
B(O) == B@and R(O) =. &,
with various trajectories for R and B over time. Depending on the parameters (b,r )
and the initid values (h,
h),
either side can gtast ahead and lost;, or start; behind
and win, W Es observd historicdly~l~
The mast celebrated reult of the thwry is the so-cafled L a n e h ~ k rSquare
Law, which is obtain& easily. Ram (2.11, we have
Separating variables and integrating from the terminal values (R(t),B(t))to the
higher initial values,
we obtain the state equation
Of course, stalemate occurs when B(t) = R(t) = 0, which yields the Lanchester
Square Law:
b ~=
: r@ or
This equation is very important. It says t b t , to stalemate an adversary t b r e t h w
as numerous, it does not suffice to be Lhre timm as effwtive; you must be nine
times as egwtive! This prwurned heavy dvantage of .numbers is deply embedded in virtually all Pentawn modeb. For d e e d s , it supported the official dire
ssments of the canven$ionaf balmce in Central Europe, giving enormous. weight
1291Xndd,the numerically smaller force was the victor in such
Auterjitz (1805);
of fi~ntiem(1914);
Antiet- (1862); R d e r i c h b w g (1862); Chanceflorsville (1M3
;the battle of Kursk
the fall of Ranee (1840); the invmion of R m i a (Operation Bar
(1943); the North K o r m &=ion (1950); the Sinai (1M7); the Gotm Heights (1967 and 1973);
and the Falklmds (19821, to name a few.
22
Montinear Dynamics, Mathematical Biology, and Social Science
to sheer Soviet numbers and placing a huge premium on western technological
supremacy. That, of course, had budgetary imp1icr;rtions. But, the presumption of
ovemheinting Soviet conventional superioritzy also shaped the development of sacafled the;zter-nuchar weapons and producecf a widesgreacf assumption that their
early ernployrnent w u l d be inevitable, which drove the Soviets to seek prmmptive
offensive capabilities, and sa on, in an expensive and dangerous military coevolution
(SW the preceding lecture) .
The whole dynamic, while driven by myriad politicd and military-industrial
intcrats on all sides, was certainly s u p p o ~ e dby Canchester" innocent-looking
linear differential equations (2.1). But, the linearity itself implicitly asumes things
that; are implausible on refiection and it mathematically precludes phenomena that,
in fxt,are observed empirically Moreover, anyone expctsed to mathematical.biology
would have f'crund the Laxxchester vaiant (2.1) to be suspect immediately,
DENSIN
The equations, once again, are
IDthis framework, increttsing density is a pure benefit, If the Red force R grows, a
greater volume of fire is focused on the Blue force B, and in (2.61, the Blue attrition
rate grows proportionally. At the =me time, however, no penalty is imposd on
Red in (2.7) wher~,in fact, if the battlefield is crowded with Reds, the Bfue target
acquisition problem is eased and Red's attrition rate should grow.
In warfare, each side is at once both predator and pmy. Inerewing density is a
benefit for an. army as predator, but it is a cost for that same army BS prey. The
Lanchester square system captures the predation benefit but completeXy ignores the
prey cost of densiw, The latter, moreover, is farniliar to us all. For instance, if a
hunter fires his gun into a sky black with ducks, he is bound to bring down a few.
Yet if a single duck is Aying overhead, it takes extraordinary accurwy to shoot it
down. Ebr duck, considered as prey5 demity carries costs,
And, as any ecologist would expect, $be eBect is in&ed obmwed. Quoting
Herbert Weiss, "the phenomenon of lossw increasing with force committed \;V=
observed by Riehard H, Petersan at the Army Ballistic Research laboratories in
about 1950, in a. study of tank battles. It was again obsemed by Willard and the
present author [Webs] has noted its appearance in the Battle of Britain data."(20]
The work referrd to is D, Willardk statistical study of f 508 !and b;zt;tles.(211
To his credit, Lanchester rtctually agered a swond, nonfinew variant of these
equations, which is much more plausible in this ecological light. Here,
In parentheses are the Ganchester square terms reflecting the '"redation benefit1"
of ctensity, but they are now multiplied by a term (the prey force 1eveI) reflecting
""prey costs," as it were. The Red atlrition rate in (2.8) slows as the Red population
goes to zero, reflecting the fact that, as the prey density falls, the predator" search
("foraingW")requirements for the next kill increase. Equivalently, b d ' s attrition
rate grows if, like the d u c h in the a n a l o ~ its
! density growt;, In summary, a density
cost is prment tX3 balance the density benefit reflected in the parentheshed tern,
If we now form the casualty-exchange ratio
separate variables, and integrate as before, we obtain the state equ&ion
and the stalemate requirement
r& = bB@.
W ,as agaiast the Lanehester Square Law,it does suffice to be three (rather than
nine) timw W good to staremate an dversary three times as numerous.
AMBUSH AN0 ASYMMETRY
h r t h e r , mymmetrica~,variants of the bmic jtanchester equations have been bevised. Far example, the so-called ambush variant imputes the "qqu~relaw" fire
concentration c q w i t y to one side (the ambushers) but denies it to the other (the
ambushem). Here,
24
Nonlinear Dynamics, Mathematical Bistqy, and Sacial Science
so that
Now wguming a fight tm the finish (R(1).= B(1) =r 0 ) and equal firing effectiveness
(r = h), a Blue force of B. can stalemate a Red force numbering B:-a
can hold off een $howand. It" TherxnopoEm.
hundred
REINFORCEMENT
Thus far the discmsion has concentrat& on the dp~miersof engaged forces, Ofden,
bowever, there is some flow of r i j i ~ f ~ r c e r nto
e ~the
t ~ combat zone proper, But, there
are limits t a the number of forces one can pack into a given area-there are "force
to space" constraints. One might therefore think of the combat: zone as having a
carrying cwacity and, wcordingly, posit logistic reinforcement. Attaching such a
term to the Lanchester nonlinear attrition, model prodlitem
where a,p, K, and L are positive constants. As obsewed in the preceding lecture,
this is ezac.11~G w e k ((1835) famous model of competition bebeen two sweies,
itself a form of the general Loth-VoXterra ecosystem equations.
Equ&ions (2.10) admit four basic cases, corresponding to different ""war h i s b
rim," Thae are shown in the p h a portrai$s in figure 2.2.
FIGURE 2.1 Phaw p o r t r a ~for hnch-erfiaulse
Msdel
(c)
Source: Bassd on Clark (l996), p. 194).
C w s (a) md (b) sre clear irzstancm of %hebiological ""principle of competitive exclusion,"" or mihtary principle that one or the other side usually wins. Case
(c) shows the horrific stable nod the "permanent war" "at neither side wins.
Finally, we haw case ( d ) , a saddle equilibrium. Any perturbation (off the stable
manifold) sends the trajectory to a Red or Blue triumph. There is, however, the
interesting and important region below both lerocli~es.Each side fwls enmuraged
in this zone; reirrforeement rat= exced attrition. rat= so the forces are growing.
But, for instance, as the trajectory crosses the b = 0 imclioe, matters start to sour
for Blue; goes negative while Red forces continue to grow. Expectations of Blue
defeat; may set in, Blue morale may collapse, and, as a result, the Blue force can
'%reakVh n g before! it; is physically annibifated. Indwd, the general phe~omenooof
B
''bbreakpoints'?~ common.
26
Nonlinear Dynamics, Mathematical Bialogg and Swial Science
BREAKPOINTS
Ziterd fights to the finish are aetuafly rare. Normally, there is same lewl of attrition
at which one bc3Uigerent "crach." S~upposeBlue brealrs if B(t) = P& and Red
breaks if R(t) == p&, with B < p,@ 5 Z and p not necessarily equal to p. Clearly,
breakpoints divide phwe space into four zones, m shown in figure 2.2.
In Zone 111, each ~ i d excwds
e
i t s breakpoint, so %bereis comb&. Red wins if a,
traject0l.y crosgm from Zone 111 to Zone 11. All" quiet in Zone I, and so forth,
F1GURE 2.2 Breakpoints
Substituting the ddernate conditions, EZ(t) = @Baand R(t) = p&
illustration, the Lanchesster square state equation (2.4) yiekts
which implies the (wiLh breakpohts) &alemate condition
inLo, for
GENERALIZED EXCHANGE RATIO
As discussed in Epstein[221thesevariants are all special cases of the genclral system
The corresponding casualty-exchange ratio is
where c-values are simply reals in the closed interm1 [O,11.
Clearly, from (2.10, cl is Blue's preclfalion benefit ham increasing densitJr while,
from (2*12),q is Blue's prey ~ o soft increasing density. Hence the exponent c l - q
might be thought af as the net predatzon benefit of i n c ~ a s i n gdensity, which is
net; fire concentration capacity in Lmchesterk sense. The Red exponent c3 - ez is
analogously interpretd.Therefore, let us define
Xb
-- Blue's net predation benefit
X, = Red's net predation benefit
= el - q
=: c3
-
,
.
Then
Again separating vasiables and intt;egrat;ing from terminal to (higher) initial values,
With skatemate defined as B(1) = R(t) = 0, we obtain the stailfematecondition
which specializes to at1 the ernes discussed earlitir (e.g., X b = X, = 1 implies square
law), and many mare,
stein (2985 and 1990).
28
Nsntinaar Dynamics, Mathematical Biotsgy, and Smial Science
Equation (2.13) is the algebraic form of the exchange ratio p(t), used in my own
Adaptive Dynamic &del. On sepanation of variabla and integralion, it also yieids
the memure of net military advantage used in the n o d ~ e a ra r m rrzee modds of
lecture 3.824
Of course, mere casualty-exclrange ratios do not mcessarily determine m u a l
outeamm, Even dehcfers with favorable exchange r&ios in engagennertt;~may run
out of room or run out of time @.g., pqular support; may collal>se bc?forcl the
attwkerk breakpoint is reached). Duratio~m d territory-spacc3 and tim
loom every bit as l a r s as physical attrition in. determi~ngoutcomes. And this
brings us to the topic of ~novement.
MOVEMENT
Historically, war has been about territory: On a map of the modern world, the
j a a e d borders are o&en simply the places where battle lines finally came to rest, X t is
imt;ermtingto compare thee with the straight borders arrived at more esntractrrdly,
geaceftrlly-say9 the borders between Nebrash and Kanszts or betwwn the U.S.
and Canada, This is the remon that mount;i%jtxlranges are such common borders:
they w r e natural lines of military defeme. The Alps, Hirnalaya, Pyrenes, and
Caucases; are exampfm. The same obviously holds f a major bodies of water, like
the English chaanel, and rivers, like the Yalu. In sho&, poll.l;ical borders refiect
military technolow. In any event, movement irs a central aspect of war. And, ;zs I
argued at the outset, a pl~twiblemodel should capture the basic connection between
the Euadamental procams: attrition and movement.
Lanehmter himself had nothing to s;zy about this and offered no model of
movement, Contemporary extensions of Lanchmter all handle it in ictssentialtlly $he
same way: they posit that the velocity of tfi, front-that is, the r&e of defensive
withdrawal-is some function of the force ratio. 50, if
these modeh posit a withdrawal rate, a velocity, W ( %with
)
W(lf =. 0; W'(%)> O
if > l and eventudly W M ( z<
) Q, imglying some w p p t o t e . (The direetion of
movement is alwayg ""fomard" for the 1arger force.) One prrbfkbed example1241 is
la31~orfurLher d k w i o n of the X", see Epstein (1N O ) .
g2"1 ~ a u f m n n(1983, p. 214).
Lecture 2
29
The basic setup, then, is this: farcm grind each &her up via the attrition equations;
the force ratio changes accordingly; and, as a function of that changing fbree ratio,
the front's velociity changm, a;s depkted in the ffw diagram of figure 2.3.
FIGURE 2.3 Flow Diagram for the Standard Modst
The frsmewrk is very neat indeed, The only problem is that any comb& model
wiLh this bnsie structure is fundamenkdly implausible, and for one bwic rewo~:
movement of the h n t 4 e f e n s i w withdrawal-is anornalau~!For a glven pair of
attacking and defending forces, the Gourse of &trition on the defender" side, ;ls
edeulat;ed in this framework, is mactly the same whether ht.? withdraws or not. The
course of attrikion on the attmkerk side is also unchmgd whether the defeader
withdraws or not. In short, defensive withdrawal neither benefits the defender nor
penalizes the attxker. So, why in the world would the defender ever withdraw?
Thc: framework i h l f mathematically eliminates m y ralionale, or inmndive, for the
purports to represent. Movement is infiuenced by
very behavior-withdrawal-it
attritian, but not convtersdy The movement of the front (witbdrawaf) is not fed
back into the ongoing attrition process, when the entire point of witMrawal w m
presumably to affect that; process-in the prototypical crtse, the point is to reduce
one" attrition. Surely, it ki eontrdietory to assume some benefit in, withdrawal
(otherwise, why would anyone withdraw?) and then to reAect no benefit whatsoever
in the ongoing attrition cdculatbm. Yet, a11 the contemporary LanGh&er variants
of which I am aware suEer this inconsisteney.~~~]
In turn, because defensive withdrawal cannot slow the defender's attrition (or,
for that matter, the attacker's), the sacrifice of territory cannot prolong the war.
And so, the most hrndamental taetic in military history-the trading of space for
is mathematically precluded. But, this tactic saved Russia from Napoleon
and, later, from Hitler. A plawible model should ce&ninly permit it.
IZ511t is intermting to note that the battle of Iwo Jima-m island, where movement of the kont
the only case (to my knowledge) in which there is any statistical
corrapondence betwmn evenb as they unfolded and iu; hypothmizd by the Lanehater quations.
Even if the 8t;tatisLicd fit were gm&,there muid be RO basis for eMrapolation to cases where
by imuacient data. On this issue,
~ubstantiaImovement is pmibfe. And, in fact, the fit is
SW EpsLein (1985).
was tsl but impwibl-is
30
Nontinear Dynamics, Mathsmatical Biology, and Social Science
So, how do I fix it-how do X build in a feedback from movement to attrition?
As simply as possible. The key parameters are the "equilibrium" attrition r a t e ,
a r d ~and a , ~ .The first, adr, is defined m the ddly attrition rate the defender is
willing to suf"flerin order to hold territory. The second, a,T, is defined as the daily
attrition rate the a t t w k r is willing to su&r in order to take? territory: X assume
0 < c r d ~ , r x , l . < 1.
War, in addition to being a contest of technologies, is a, contest of wills. So it
is not outlandish to posit basic levels of pain (attrition rates) that each side comes
willing to suger to achieve its aims on the ground. If "ce defender" attrition rate
is less than or equal to ad^, he rem%.insin. plwe. Xf his attrition. rate excmds this
""pin threshold,'"e
withdraws, irr an egord to rwtore attrition rates to tolerable
levels, an eEort that may fail dismally depending on the adaptations of the attacker,
a similar creature. If the attacker's attrition rate exceeds tolerable levels, he cuts
the pace at which he prosecutes the war; if his attrition rate is below the level be
is prepared to sufir, he increases his prosecution rete.f24
X
t is the interplay of the two adqtiue systems, meh searching for ifs equilib~um,
that pmduees the observed dynamics, t;he actual movement that occzlrs and the achal
att&t.ion suflered b$ each side. Indeed, in its most basic Ebrm, withdrawal might be
thought of 4ns an attrition-regulatirrg servomechanisnn. The pain thresholds c r d ~and
a , play
~ the roles of homeostittic t;trgc?Ls,in other words. The ixlCrodtxetiian of these
thresholds struck m w a n d stilt strikes m as the most direct mathematical way do
pernit defensive withdrawal to 8ff:ect attrition and, thus, to permit the trading of
space for time, Their introduction &so generates the fertile analom between armies
and a broad array of goal-oriented, feedback-control (cybernetic) system.
Before delving into the mathematics, one passible xn;tsconceplion. about these
""pain" thresholds should be addressed. 1 do not claim, nor does my model imply,
that battlefield commanders are necessarily awaw aE the numerical values of a s d ~
and a,r. Humans in the sixteenth century were not '"ware" 'h& they were sweating and shivering depending on the error: "body temperature minus 98.6 degrees,
Fahrenheit ." But the homeostatic behavior was there nonetheless.
OVERVIEW OF THE MODEL
Let me now turn to the Adaptive Dynamic Model itself. The fufE atpparatus includes air power as we11 m air and ground reinforcements, faci;ors I will not discuss
f 2 " ~ o rearlim versions see Epstein (1985,1990).
[271~hw
parameters reprment daily rates of attrition, not total or cumulative attrition levels,
discuss& above in conneckion with breakpoints.
ilrs
31
~ecture2
ltzere.f281The model is a system of delay equr%lionswhere the unit of tirne is usually
inkrpmted as the day. If A(t) and D(t) are the altackr's and defender's ground
forces sufvivhg at the sta& of the t t h day and a,(t - 1) is the attacker's attrition
rate over the preceding day; we have the wcounting idemtity
The attaderk force on n e s d a y is his force an Monday, minus total losses Monday.
Likewke, it must be true that
D ( t ) = D(t
- 1) - (Defender's losses on day(t - 1 ) ) .
What are these logs=? Well, if we define the csually-aehange ratio as
p(t
-- 1)
Attackers Lost on day t - Z
Defender8 Lost; an day t - 1
the defender" losses must be
since the numerattor is the at;txkers b t on (1 - 1). Thus,
accounting identity
W
have the second
Obviously, mce we attmh specific functional forms to cr,(t) and p ( t f , we no longer
have aecaunting identidies; we have a model. Above tve discuss& p(t) and a r g u d
that a plausibie and relatively general functional form is
where X, Ad E [O, I] are parameters. The red action--&l feedback from movement
to attrilian-is inside cr,(t). Here is where the interplay of adaptive belligerents
tmfalds, As men"F-ioned,this iaterplq is betwwn the attacke~"'~
prosecution rate
(reflecting the pwe E& which he chooses to press the izttaek) and the defender's
sewomechanisms, in @Recta
witfidrauoval rate, both of which are 5tttrition'reguIitti~;tg
The defender is, in same respects, simpXer. We discuss him first;.
Nonlinear Dynamics, Mathematical Biology, and Swial Science
ADAPTIVE WITHDRAWAL, AND PROSECUTION
The defender" withdrawal rate for day E is wsumed to depend on the difirence
between his actual and his equilibrium attrition rate for the preceding day, day
(t --- 1). The functional form of that dependence should satis@ sonre basic requirc3menies:
1. As the actual attriLion rate for day ft - 1) wpraaehes 1, the withdrwd rate
for day t should approach the mmimum fesible ddly rate, W,,,.
2. If the actual dtrition rate for d a y (1 - 1) is greater than the equilibfium rate
ad^, the withdrawal rate for day t should be greater thaxt for day (1 - 1).
3. If the a u a l attrition rate for day (t - 1) is less than or equal to the equilibrium.
rai,-t;e ad^, then %hewithdrawal rate for day t is zero.
X
t may nod be correct;, but the simplest functional form I can think of that satisfies
these requirements is
W ( t )=
Q
W ( t - 1)
+
if ~ (- 1)t
( a d (t - l ) - a n ) othemise ,
<a;i~
where
While in particular c w s , there may be departures, exeegtions,@gfand so forth, as a
first-order idealization, the notion that, ceteg..ispribus, the aim of withdrawal is to
reduce one" attrition rate swms fairly compelling. It also en,jioys a cept;ain biological
plausibility. If the beat, is too gre&, we p k our hand from the fire; Ashby" seat
a bwic mechanism of defense for all species. One
comes t o mind. Surely, flight i~
of the more famous experimenb in this c o n n ~ t b nW% conducted by our friend
G a u s and is known as his "&or beetle'' experiment. He began with two bwtle
species competing in aa environment of flour. He found comp&itiw exclusion to
be operaive; left alone, one species consistently exterminated the other. But, when
Gause insert& small Ien&b of glass tubing in60 the Aour, the weaker species was
able to retreat into the tubing, establish refuges, and sum
hey could "trade
space for time,'' as it were. So can the defenders in the Adaptive Dynamic Model.
As we will see, they may choose to forego that option. But, a remonable model
should not preclude it.
nrming ta the attacker, the model assumes that the pace at which he prews
the attack, his prosecution rate for day t , which we denote PP(t),depends on the
digerenee between. his wtual and Ms equilibrium attrition rates for the preceding
day, day (t - 1). The functional form of that &pendence should satisfy some basic
requiremexlds:
1. As the attacker" aactud attrition rate for day (t - 1) apgrowha 1, the pros*
cutian. rate for day t should approach. zero.
2. If the actual attrition r&e for day (1 - 5) is greater than (lem than) the equi, prosecutian r;tLe for day t should be Less than (greater
librium r&e a , ~ the
than) for day (t - 41).
3. If the actual attrition rate for day (t - 1)equah the target, or equilibrium, rate,
then there is no change in the prosecution rate.
It may not be correct, but the simpkesk functional farm X can think of that satisfies
these requirements isPO1
As 1 said earlier, it is the interplrxly of thwe aclaptive agents that shapes the
dynamics; $hey are link& in the formula for &,(g), the attacker" attrition rate for
day t. This Euxlctiond form should satisfy some bmie requirements:
1. Ceter*isparibus, the higher is the attwker's proswutian rate, the higher shouXd
be his attrition rate;
2. Cetercis pamfius, the higher is the defender" withdrawal rate, the lower should
be the attacker" attrition rate.
3. As the defender" sitMrawal approaches full Aight (W(t) -+ Wm,,), the attxker % attrition rate should approwh zero.
It may not be correct, but the simplest fiunetional farm I can think of that satisfies
Once the initial conditions and parameter value are specified*these equations
produce the dynamics. And, as noted above, it is the coadaptation of these agents,
ewh searching for its equitibrium, that determines the actual moveme& that occurs
and the actual &trition that is suEered by each side.
In a nutsheU, the attwker makes an openiag ""hid" on the pace of war, the rate
at which his own forces are consumed (of course, he clzn set his rate at zero by not
attacking). We may want to press the attaek at an e ~ r e m e I yhigh paee and may
be wriljing to suEer extremdy high attrition rates, if-for operational, strategic, ar
E3Of~ am pgraLeEul to Mike Sobel for pointing out to me that itthe functional form for PCt) that f
originally publish& in Eptein (1985) m$ualfy fails rquirement 2, Sukaquent to our dkcwion,
I notic4 that it aho fails; L,
34
Nonlinear Dynamics, Mathematical Biology, and Social Scisncs
political reworn-a quick decision is p a r a m o ~ n t Via
. ~ ~the
~ csualLy-exchmge ratio
(defenders killed per attacker killed), tbis imposes an &trition rate on the defender.
The latter may elect to hold his position and accept this attacker-dictated rate, or
he may c h o w to rduce his attrition rate by withdrawing ad a certdn speed.
The mathematical mechanism whereby the defender" withdrawal reduces his
attrition is not obvious. From (2.16), the attacker's attrition rate over day t , a,(t),
praduces, via the inverw exchange ratio I l p , a defensive &&ition r&e aver day t ,
ad(t).If this exceds the defender" movement threshold a d r , then on the next day
the defender withdraws at R, rate W ( t;t- 1).This action reduces (th& is, feeds bwk
negatively an) the attacker's sttritian rate a,(t 1). In turn, this decreme in the
atlackerb attrition rate produces (again via I l p ) zb reduction in the defender" attril),whose size rcjllative ta ad^ determines the rate of any subsequent
tion rate ad(t-Iwitkrdrawal. ff ad(t 1) is less than ad^, no subsequent withdrawal occurs. The
front then remains in place unless and until the attacker-by attempting to force
the combat at his chosen pac imposm on the defender an dtridion rate exceeding
his withdrawal threshold, and so on. Qae might think of the defender a an adaptive
system, with withdrawal rates as an adtritjion-regulating sewornahanisnn. 1321
All the while, the attacker, too, is adapting; the grosecut;ion rate B(t) is his
sewomechmism. Just as there is some t k ~ s h o l da r d ~beyond which the defender
will withdraw, so %beattacker possesses an "equitibrium" attrition rate a , ~ If
, on
day (t- I) he recards m dtrition rate exceeding a , ~ the
, attackcer reduces the pace
at which he prosecutes the combat. If he records an attrition rate lower than a,T,
he accelera- by raising P ( t ) . The magnitude of these changes in P(t) agprowh
zero if the attwker" attrition rate appromhe a , ~ the
, equilbriunn rate, Each side's
anlaptation may damp or amplify, penalize or reward, the adaptation of the other.
The adaptations are perhaps more sophisticated than. m e t s the eye. Specifically, a primitive type of learning can occur. Suppme that OR Monday?the defender's
T some amoun%X. In response, the desrttrition rate exewds his threshold I X ~ by
fender withdraw at a. rate W(1)on m-day. Suppose, however, that-beeause hi?;
own attrition rate on Monday tvw below his thrmhold aaT-the a . t t ~ k e increizses
r
his prosecution r&e on nesday and that, as a result, the &fender% attrition rate
on mesdaty again excwds his threshold by the s m e amount ;Ye Only a defender
+
+
I3l1~s
an operstional matter, a quick dwision can circumvent fogisticai problem that could prove
telling in a prolonged war. Strategicafly, the attwker may seek a dmision before the defense has a
ckiance t o mobilize superior industxy, superior reinforcemen_ts, or superior allies. An attmker with
unreliable alies of his own may seek a quick win Eat they be@n t o d e f ~ tAn
, a t t d e r may aXso
chaos;@t o prm the a t t s k a t a ferocious pace I;o smure a decision before the d e f e ~ d e r knuclear
options can be m e c u t d . A clmic stratem of ~ t s t wFacing enemiw on multiple Eronls has been
to win quickly through offensive wtions on one front and then switch forcm t o the second.
X have d i s c o v ~ r dthe combat modef of b h e w k y
~ 3 2 ~ ~ initiaffy
i n m pubfishing t h quations,
~
(1947). Although it diEem from my model in numerous ways, U h e v s k y ' s model dotls posit that*
cete7.1;~pdvibw, defemive withdrawal should r d u c e the attrition rat= of both attacker and defender, and-using a diBerent mathemalkalion-it incorporates; the idea af a defensive withdrawal
thrmhold.
unable to learn would withdrm at: W ( t )again, since that rate alreatly failed to sdve
his problem. A more deeply adaptive defender would withdraw at a rate greizLer
thaa W (t);in the Adaptive Dynmic Madef, he does. To me, tbis makes a certain
amount of biologicd sense. If walking slowly away ffom a swarm of attacking bees
does not reduce the sting rate, we try jogging. If jogging doesn" reduce the sting
rate, we run, and so on, until WC?are running W fmt as we can (Wmax).Of course, in
the bee cme we actuauy are free to pick something clam ta Wm,, as a first "trial retreat rate" k c m s e we are not concerned with territorial sacrifice. Analogous paints
apply to the attwker and his learning behavior in adjlusting his prosecution rate,
P(t), sts we will illustrate in the simulations helm.
By setting the two fundamental thresholds c v d r a d a , in
~ various wzcys, the
model will generate a remnable spectrum of war types-bellotopes-from
the war
of entrenched defense, 8 fa, Verdun, to guerrilla war. S will discuss the four extreme
settings and then present; same sinnulations.
CASE 1 :a , ~ 1, The British at the S o m m (1916) oger perhaps the great example
of an attacker with no apparent pain threshold. Considering the extraxtrdinary pain
involved, we can ask with. Jack Beatty, "What made them do it?"
"'Xt.' WW to march, in, an orderly way; rank by rank, column by column, to
their death. That is wfr& 20,006 British soldiers did on July 1, most of them
falling between 7:30 and 8:30 A.M., the taste of tea a d bxon still fresh
an their Lips. They got out of their trenches and marched to their death, or
to mme other h r m of mutilation.. .. Methodiealll~y;these German] gunners
raked the British formations. Methodically new formations set out, were
shot dawn in no-man%-land,were replmed by other formations, and so on,
turn and turn about, through the long day" (Beatty, 1986, pp. 112-f14).
Long indeed. Here, perhaps, is a cszse of c r , ~m 1. Along similar lines, one Lhilzks
of the fateful Argonne Forest ofXensive of 1918 and, in particufar, of Pershingk order
to ""push ahead &th,sut; rqard to losses and without regard to the exposed condition
of the Aanks." "rely, for Pershing, a , vvas
~ ebse to 1. And, S Beatty notes, ""Iis
no wonder that the cemetery at bmagn~Sous-M~ntfaueon,
deep in the Argonne,
is the largest American military cemetery in Europe, containing the remains of
14,246 soldiers.'' P3f
36
Nonlinear Dynamks, Mathemarea1 Biology, and Smial Science
CASE 2:a d T
1. The defemive analogue of the British at the Sornme is undoubtedly the Rench, at Verdw, a h in 191Gnot a wad year, as Beat@ recounts:
"The hench. rotated seven tenths of their army though the meat grinder of
Verdurr. A colonel" order to his regiment givw the death-heav Bavor of the
b&tle: You have a mision of sacrifice,. .. On the day they wmt to, they
will mizssacre you to the last man, and it is y w r duty to fall.' The losses
on both sides were appalling-perhaps a million and a quarter casualties in
all. (The ossuainl at Verdun is fuH of the hones of the 150,000 unidendified
and unburied mrpes.) In short, Verdun was a demographic eat&rophe
for fiance. Yet, bllowing P4taink famous order, "4s ne passeront pas!' the
Rench Army held Verdun for the ten months of the battl
courage m d endurance hut not of victory. The standoff of Verdun, in the
words of Alistair Horne, "-was the indecisive battle in an indecisive war; the
unnecssary battle in an unnecasary war; the battle $bat had no victors
in a war t;hat had no victors"""(Be;ttty, p. 117).
Perhaps this is the terrible stable node I spoke of ab
l.
the sink of all sinks and,
1would ugue, a cme of adr
GASE 3: c r d ~ 0. Diametricdly opposed to the Reneh at Verdun are guerrilla
defenders; their wiZ;hdrawal thrahold c r d ~is close to zero. In guerrilla wars, like
Vietnazxl, larger "superior" hrees seeking direct engagements find themselves frustrated by defenders who withdraw---"vanish into the brush"-at the slightest attrition, the extreme case of trading space for time. Indeed, the entire strategy of
the guerrilla-his only real hop is precisely to prolong indecisive hostilities until domestie support for the war disintegrates, as it did for the United States in
Vietnam.
GASE I:a , ~ 0. The fou&h and final ""pure" variant is the case where the attacker's equilibrium rate a , is
~ close to zero. The naturd example here is the
"The clmsic c m is where an attacker is attempting a
secalled " ~ n operation."
g
concentrated breakthrough in some sector of the battfe front, He wants do prevent
the defender from shifting forces A m neighboring sectors to reinforce the breakthrough sector. Standard procdure for the attacker is to '"inn," or "Ex," these
neighboring defensive forces by applflng some prwure, but not emugh to incur
serious losses.
By specializing these two parameters, ad^ and a z , ~ , the model will produce the
""pure" forms, shown in table 2.1, m well as myriad mked eases.
TABLE 2-1 Adadwe Dynamic Modat
Defender's Threshold
&tacker's Threshold
ad^
%BT
Qualitative Range
CL^
--+
I
e v d ~-4 0
a ,+
~f
Q,T
-+O
RTench War (Verdun)
Guerrilla War
The Sornme
F g n g Operations
For illustrative purposes, f oEer two simulations repreenthg mked caes. The numerical setkings are given in table 2.2. In the first, I pasit a ferocious attachr, with
an equilibrium aterition rate of &,F = 0.6. The defender" withdrawal threshold
attrition rate is et Ltt; Q ~ T= 0.3, respwtably stalwart;, Though not shown in figure 2.4, the forces are inlrtidly equal (at hdf a million). WhnL coadaptive storx
then, is this picture telling?
The attacker's opening ""bid" on the pace of war, his opening praseeuticrrr rate,
is P(I) = 0.1. At this low level, the resulting att;ritia~rate far the attwker is well
below the 0.6 levet he is, in fact, prepared do suf;fer.And so, as s h w a , he begins
~
0.4 (on day 3 ) before
raising his prosecution rate. But, this must, climb t ; amund
id produces a defensive attrition rate above the defender's thrahold of ad^ -- 0.3,
which induces withdrwd.lM"l&h curves then rise to day 6. Xn this phase, the
defender's withdrawals (partid disengagexnents) are t h w ~ t i n gthe attacker" eeAEart
to attain his ""ideaI"" attrition rate of a , ~
--- 0.6, so the attwker prosecutes with
Inerewing vigor, which eEofes induce successive withdrawals at increaing rates.
t341~hecomputer has simply e a n n s t d the dots in thwe picturw.
38
Nonlinear Dynamics, Mathematical Biology, and Social Science
TABLE 2 2 Numeri~alSettings for Figures
2.4 and 2.5.
Setting
Figure 2.4
Figure 2.5
V
Dash indicates "same as in figure 2.4."
FIGURE 2.4 High Unequal Thresholds
atbcker
II defender
Brosecutian Rate
0.8
Wia&awal Rab
S
6
0.6
4
0.4
2
0.2
0.0
Q
Lecture 2
Now, ail the while in this sirnuladion, the casualty-exchange ratio (attmkers
killed per defender killed on day t ) has been constant at, a rate favoring the defender.
And, by d a y 6, he has whittled d w n the attackr to such an extent that, even at
high prosecution, the attacker cannot exact defensive attrition sufficient to induce
withdrawstf-o,
withdrawal, stops, the defender ihdts, on day 7.
Xn egeet, the attacker "slams into" the now stationary defender on. that day?
producing a t t ~ k eattrition
r
we1 in excess of the attxker's talerance a,T, '"ouch,"in
other words. The attacker reacts to this extraordinary pain by cutting his prosecution rate sharply on day 8-too shwply, it turns out. He has overshot, as evidenced
by his subsequent increases in P(t) which ultimately levels off at around P($)= 0.6.
A rather biEerent history is portrayed in figure 2.5. The prosecution rate deereas= moxlotonicalty, while the withdrawal rate rises and falls twice, In this case,
the attacker's equilibrium, and defender's threshold, attrition rates are set equal at
a , =~: ad^ =r 0.1, considerably lower than in the preceding cam. initial. force levels
are as before.
FIGURE 2.5 Low Equal Thresholds
B
1
3
5
7
9
Time
13
attachr
defender
Nonlinear Dynamics, Mathematical Biology, and Smial Science
Here, the attacker's opening prosecution rate ezeee& his equilibrium rate:
P(1) =; 0.2. This openiElg rate imposw on the defender an attrition rate that excmds
his withdrawal thrahold. Over the first six days, bath sides are above tolerance;
the defender wiLbdraws at a grming (though diminhhing margind) rate, while the
attacker deerewes his prosecution rate.
These coadinptdiom (plus a casualdy-exchange ratio favoring the defender)
gradudIy depres the defender" attrition rate to a level below his withdrawal
thrmhold; so, on day 1, be halts. Though the attwker is steadily reducing his
prosecution r&e, the weight of his h p a c t on the stationary defender is ssuEciently
painful to drive the latter &am his position, once more until, on day 10, the front
stabilizm, The attack norretheless persists, tbsugh at a declining level of ferocity,
P(tf -
SUMMARY
which X mean the original equations and their contempobehaviorat dimensicms of combat are ignored, Mere opposrary &exlsiong-thme
ing numbers and tahnicd firing effectiveness completely determine the dynamics:
there is na adaptation. En the Adaptive Dynarnic Model, the parmeters c r d ~and
a , allow
~ one to reawe the diEerent ways in which given fisrces can behave, As we
have seen, with it given form, a;n attacker may prosecute the ogensive at a ferocious
pace, virtually unrsponsive to losses, The British at the Sonnrne in 1916 come to
mind. Or, an attacker may operate the same forces at a more restrained pace, as
in f i n g operat;ions, A high d u e of c r , ~will prodwe the former
of attaclser;
a low value of a,T will genera@the laeter.
Similarly, the tmtical defender may be mare or less stalwart in holding his posieven slight attrition
tions. Guerrilla defenders may withdraw-""disappear"-when
is sufired. For such twtical defenders, the withdrawd-threshQlb attrition rate r r d ~
is close to zero. At Verdun, by contrast, no attrition rate ww high enough to dblodge the defenders from their entrenched positions*Pl?taink famous order-----'"h s e
Eectivc;ly set ad^ equal to one,
These st;riZl;egicand buman realities w e captured, however crudely, in the Adapdive Dynamic Model. And they are captured by a mechanism that; permits movement to deck a%ritiaxr, a feedback that h not possible in any version of Lanchester's
equations. So, I f=I some confidence in claiming t;hat my equations present a less'
crude cslricature of combat dynamics.. But, given the connplesty of the process,
that is all I claim.
fa Lanchester Theory-by
me
LECTURE 3
Imperfect CO ective Security and Arms Race
Dynamics: Why a Little Cooperation Can
Make a Big Difference
This lecture uses simple mathematical models t o explore the relationship betwmn security regimes and arms raee d ~ m i c s . f 3 5The
)
main focus is a regime
known EIS collective security; which is receiving wide attention. Little of the attention is mathematical, however, and, to my knowledge, none of id involves dynamical
systems. One aim of this paper, then, is to formalize col1ect;ive mcuridy in a dynamical systems context, whieh will allow us t o extract some unexpected results, This
formalization, of course, requires a rigorous definition of collective security. To wit:
Imagine three countries 3, y, and x. Perfect collective security would then operate
as follows: If rr: attacks g, z allocizees all force t o y; if y a t t m h z , ;t. afloeiates all
force t o z; and so on, The general rule is simply that the odd man out instantly allocates all force to the attacked pady. In more biological-or sociobiological-terms,
perfect collective security is a form of reciproca1 allrui~m.1~~1
f 3 5 1 ~ regime is a ""st of implicit or explicit principltjs, norms, rules, and decision-making procedurw
around which actors3expectations converge in a given area of international relations." the Krasner
(1983, p, 2)- On tha emergence of norm generally, see Axfjlrod (1986).
la61%eeWilson (1975; 1978, chap. 71; Gould ancl Gauld (1989, pp. 244-46); and Smith (1989, pp.
l67+9>.
Nonlinear Dynamics, Mathematicat Bioiog)r, and Social Science
NW, a heated debate surrounds this idea.P71 The debate turns on the question, "How much cooperationIJ8f is possible?" Reptics argue that substmtial levels.
of cooper&ion are not passible, and therefore that eolleetive security c m be dismiss&. Proponents counter that a substantial degrw af cooperation is possible
and, therefore, that collective ~curit;yis worth pursuing. Notice, however, that
both positions assume substantial leveb of cooperation to be neeessam for collective security to be worthwhile. What about this centrd assumption;? What about a
Little bit of eolfective security; is there merit in a highly diluted form? As nonlinear
dynamieists-wutely aware that small perturbations can have huge egects-we are
intrigued by the question.
And in fact, a central conclusion of this analysis is precisely that collective securiw regiakes-ven
in highly diluted forms-can exert remarkably powerful stabilizing egects; in arms rwe models sufkicie~tlynonlinear to produce really volatile
dynamics, highly imperfect eolfective security regimes can damp the explosive oscillations a d induce mnvergenee to stable equilibria below initid armament levels,
y
of altruism can profoundly clam the
Put digerently, the injection of t i ~ degrees
othemise volatile dynamics. The benefits of participating in the system are very
great and, because of the nonlinearity? the required level of commitment from individud participants i s very low,
fn addition, one might assume that the more volatile a system is, the less value
there will be in a given, low, level af colkctiw security But, caun%erintuitively
within the clam of models examined here, precisely the reverse is true! These results
are examples af what I call ""Le nonlinear dynamics of hope," aand would appear
f research on. altruism in a
to iwite a rmrientation r, at least, an extensio
variety of fields.
ALTRUISM: WOW LITTLE IS ENOUGH?
Specifically, the present analysis demonstratw that, in some dynamical systems,
excmdingly low levels of indizl.iduat altruism produce high levels of collective harmony. Everydbing depends on the intervening dynamics, and these, id w m s to me,
are XmgtlXy missing kern the debate. Far ixrstance, in a memorable p h r w , Edward
Wifson writw, "the genes hold culturr: on a leash" "(9) (emphwk added), But, what
he actually argues is that the genes hold indiwzdzcals on a Eemh. The present; analysis suggests that, even if the individualsqemh is very short, the culture's Ierzsh
might be v e v long. hdeed, the analyslis swm to open an emotionally appeding
niche: perhaps we can be optidstic about the prospect of social harmony while
vigorous debate on the generai tapic has appewd in Maswd's jowal, Intmatisnal SeMewsheirner (1994/95), and Kupchan and Kupchan (1995). 53% also
Kupchm and Kupchan (1991). For a eollwtion of thoughtful analym, see Downs (1994).
fJ81~hroughouit,
E will use the term ""cooperation'band "r~ciprocalaltrukm," as just definrtd,
eu~tyX
.n pardiculzrtr,
interchangeably;
i391~ilson(1978, p. 167).
retaining a certain degrm of skepticism. toward the prospect of indi~duabaitrzlism.
In short, the issue is not simply haw much individual altruism is possible; but, for
social harmony, how little is enough?
ORGANtaTtOM AND METHOBOLQGV
The discussion is organized as follms. Fir& by way of introductian, the simplest
twecountry form of Lewis Rye Xtichardsonk classic purely eornpetitive arms race
model is prewnted.14QI That model is then generalized slightly and expanded to
encompass three competiWrs, the minimum number necessasy to exaxniae colective security.f"j In, ztddition to Richardsonian competition and collective security,
I will examine a regime characterized by the presence of a world policeman, or
""glob~eop."l~~f
In this moctel, a force, C , located outside the competitive three%&msptem is held at the ready to swoop in to support any attack4 p&y. The
object is to compare these three regimes formally. 1do so first ~ u m i n that;
g the uacierlying arms race dyrramics are linear, as in Ridarkon's mode1.[4YThen I = s u m
nonline= arms race dynamics of a specific sort. X would hope that the full nonlinear
model introduced here will contribute to the theoretical arms race 1il;erature in its
own right.
Now, I make no attempt to test any model. Nor do I claim that any of these
models is ""rght," Rather, the models are coarse I e m s under which we compare
the regimes. To be specific, the generd procedure would be ars follows, Take some
r, if you prefer, anarchic-arms race competition. Call that
model of Hobbesi
model
(I use the Ii~earRiGhardscrn model).lqa1Then construct (see beiomr) that
model's collective security variant, ~ fAnd,
. for expository purposes, suppose col= M1(0),then
lective security damps the competition in the sense that if
~f(t<
) Ml ( t ) for all positive t.1451 NOW?take a second Hobbesian model (I use
a nonlinear Richardson model), Mz, construct its collective security variant,
and compare the dynanzies. Agdn, suppose that collective secwity damps the competition, Co&inue in this way. If this compamtiue result recurs without exception
M?)),then one may be justified in concluding
over a huge set of model pairs, {(Mi,
that collective security exercises a systematically deprmsive eEeed on competlltive
MF(o)
MF,
[$Of Richardson
(2960).
f41jIdo not treat the n-=tor, or globally inclusive, cme here,
lazl1n principle, globoeop could be a conso&iurn of powers. I thank Brian fillins for this name.
(d3f~ssurning
linearity, globocop is not an i n t e r ~ t i n gvariation on Riehardwn. In fact, it .ts liilear
Richardaon with griewnce & r m translated by a fixed amount, Hence, little is said about gbbacap
in P& 1. Indeed, since its eEwts are, from a qualitative mathemtieal standpoint, pre%Ly8tra;ightfomard even in nonIinear cmw, globocop is includd for completeness but will receive relatively
little attention.
[44~~echnica1fy,
the model is f i n e .
gasj~ospell this out compl&eXy; we are positing M1 ( t )
(Mrl ( t ) ,M12 (L), . . . , M l n (L)) aind
~ f( t')ZE
(t), n/lg(t),
,. , ,
(t)).Then M? ( t )< M1 ( t ) iff M$ ( t )< MI ~c( l )for every k.
( ~ g
ME
1144
Nonlinear Dynamics, Mathematical Biology, and Social Science
dynamics. This discussion suggests that may be the erne, though more of this
""sructura1 sensitivity analysir;;" would be neded before confidence is obtained.l*f"l
from a methodological standpoint, it is also important to distinguish this analysis from &her treatments of the issue. Tn particular, I am not examining the stability of collective seeurity from a game theoretic standpoint,t47fNo claim is made
as to the likelihood of compliance with, or defection from, the system, b t h e r , I il.m
trying to contribute a dynamical systems perspective to the theoretical literature,
=king, with all else fixed, what the dynamic effect of purely '5nstitutionil.I))
haage? The results bear on. the game theoretic literature in=
as compliance depends on p a y o E ~ , fThe
~ ~ jpayoffs associated with compliancethis
analysis suggmts-may be surprisingly high, and individual behavior may change as
a result, Or, to couch it more prosaically, maybe if leaders appreciated the pote&ial
payoff of ewn limited eolteetive securily; they would be more interested in braad
compljance. Indeed, the andysis would appear to raise starkly the question where
do draw the line betwwn altruism and self-f-ixrlerest in the international system.
Findly, I make no attempt to evaluate the practicality of implementing collecLive security in Europe, the Middle East, or any other pafiiculm region, However,
if collective swurity in practi
uld behave at all like the idealized regime erramined here, then its institutio
n in highly diluted forms-might well be wort;h
substantial effort, To begin at the begiming, let us revisit Richardson3sodginal
model,
f4q~ltimately,
an elegant way to p r o e d would be ta chaxterize mathematically the entire cl=
of formal a r m race modefs under which collmtive mcurity (or globocog) would show a deprwive
eRat, and then examine whetha modeh falling outside that c f m are a t all plausible. If not,
we might wish to concIu& that, by virtue of its membership in that ccla~,the "right';'"modelilj indieate the same depressive effect far eoflwtive swurity (or globocop), abnd
ipiio~.afaeto, that the eBmt is quite real, The basic: id=, again, would be to say mmething reasons;bb
about compamtive dynmics, wiUtout claiming to know the 'kright'brm r e model, ReIatdIy,
given in the
it is worth stating explicitly that no mnsitivity analysis on the pwameter VB~~UE?S
Appendix ir; conduct& here. It ia of eonsiderabb interest that there &sG paameter mttings at
which a sharp sensitivity to rule regime is evident. A separate study would examine the robustnms
of this result under a wide range of parameter s&t.ings.
1 4 7 1 ~ eNiou and O r d ~ h o o k(1991). Afso reXemnt are Axelrod (1984, 1987).
i 4 8 j ~ h elocation of mked strategy equilibria-ad of evolutionarily stable strategies in bimatrix
games4epends on pay08 magnitudcts, not just orderings. The s p d of canvergence to any stable
quilibria also depends on payoff xnagnitudm.
PART l. LINEAR MODELS
THE CMSSIC RECHARDSON MODEL
The fdlowing differential equations constitute %chardson% bbasie model, with a:
and y the wtars:
The basic idea is that ac state" a r m race behavior depends on three overriding
fwtors: the perceived efiernal threat, the economic burden of military comp&ition, and the magnitude of grievancm against the other party. Each merits a brief
discussion.
The constants, and g,,are muclfly i&erpre"?t.d simply as grievances. h d ,an this
reding, a core mmsage of Riehardson" sodel is that there can be no permanent
disarmament without the rmolution of mderitfing poEticd grit3vaxrcw. Even. if d b
armament is total (i.e., z(t) = y ( t ) = O), arms racing will reemerge if g, or g, is
greater than zero. For instance, take ($.l),and assume z(t)and y(t) are both zero. If
g, > 0, then, dnce the gro&h rate k(t) equals g, , that rate, too, must; mcmd zero.
Hence z(t) be@ns %agrow, which, via (3.2), stimulatw a. l l ~ ; ~ reilction
o ~ h in y(t),
which f e h back to further sLimulste z(t;), and the race is on. Or, as Richardson
himself put it, "mutual disarmament without satisfmtion is not permanent,f"491
It m e w t a me th& g, and g3, c m be inderpretd somwhat more braaxily. States
compete militilarily not only because there are grievancm, but aXso becaum they lack
confidence th& grievances can be resotved without resort to arms. The emergence of
inseitutions offering high confidence that diEerencr;s could be resctlved nonviolently
might permit disarmamend despke outstanding pievance~.So, I think of the t e r m ,
g, and g,,as capturing both the underlying grievancm and the level of confidence
that grievances can be rmolved nanmilitarily. If confidence ranges from zero to one,
then each g could be intergfeted as: (grievance).(l-confidence), for =ample. It; is
far horn cbar how one wauld memure g, or g, under either interpretation, Luekib,
their measurement is not necessary far our purposa.
f " l ~ i c h w h n (1950, p. X?).
46
Nonlinear Dynamics, Mathematical Biology, and Social Science
THE ""ECQNQNFIC FATIGUE" E R M
Leaving the grievance k r m aside, the rate at which z grows, k , is proportional
to the perceived exkemd threat y, a i d vice versa for the growth rate g. Wit;hout
same damping term, this is a pure posieive feedback system that simpiy. blows up.
Richasdssn posited economic fatigue terms ( - a p and -b2g) that damp the process.
Ckasly, so long as the fatigue meEcknts at and b2 me greater than zero, the process
me negative, then -at and -& are positive,
is dmped. Importantly, if a1 m d
meaning $h& military gr&h rates inerewe the larger is the miXitary establishment.
In such cases, 3; and y may both grow even in the absence of underlying grievances or
any perceived threat. This would be aut;ocatafyt;icgroMh in the "military-irrdustriaf
camplcx," The "fatigue" mefficienta might; be though of as embocSyinl;;the net eivilmilitary, or '"guns m s u s butter" balance in society*If the terms are nqative, then
there is ante-catalytic dlitary expansion, or "milititarism" "for short.
THE EXTERNAL THREAT TERMS
Finally, the terms blz and azy incarparate perceived externd threats, obvious camf will generafize this model
ponents of any plausibfe model sf arms race behavior,f5@f
below, acfding a third ga&y and htradtteing wrtiain nonfinearitiw, among other
thine. But clearly, Richmdsank lliiaear model has some basic appeal, parsirnoniously relating arms rme behavior to grievances, perceived external threats, arrd
intern& economic fatiwe.
The simple anaEytics of the famous model deem@a concise review. Defining
the basic Rchardsan model is simply
Isalfn reality, thme t e r m are probably nonlinear, since the military" power t o shape t m t perceptions itself may w g with the size af the militau rixstablbhment, sugejciting t e r m of the form
az f z)g and b1 (g)%. See More1 (1WI).
[ E i t f ~ hmodet
e
providt3s s nice framework for imxterpreting grom ehangw in, for instance, Soviet
behavior, Thinking of the Soviets as ~ a u n t r yS, one might argue that the end of the cold w x
manifwts the facts that Garbmbwk a2 < B r s h n e v k s.2, Eorbmhev's g < <rezhnev% 9, and
Corbwhev" sal 3 ffrahnev's a l . The last of tlz
nt of the economic
r a e w w a s perhaps crucial;.But,
s h w l y and, suddenly, a r m race d y n m i c s were tzhanged.
A positive equilibrium 3, if it e e t s , is @ven tay
Stability is independe& of g; that is, is a stable equilibrium of (3.3) if and only if
the origin is a stable equilibrium of t = Ax, where s = X-3. Pis covered in lecture 6,
that requirement is met if TrA < 0 and Ded A > 0. For a' and bz positive, the trace
condition is obviously s;ztisfied by A, and the determinant is positive if the product
af economic fatigue terms (al&) exceeds the product of reciprocal activation terms
(bIa2).As ohsewed in lecture 1, the gbbal equilibrium of the Richardson model is
the interior equilibrium of the mutudistic Loth-Volterra ecosystem model,
Now, simple linear model in hand, we wish to examine the tramition from this
Hobbe~ianworld to a collective security regime first and, secondarily, to a world
characterized by the presence of a world policeman, or "globacop."
As noted at the outset, there has been no attempt to frame the comparison
in ciyxramical systems terms. An immediate issue, then, is how to operationalize
mllwtive security and globoeop. Recdl $hat un&r a perfect (as wajinsk diluted)
t h r e p a r t y collective security regime, if z attacks y, z imtantliy csntributw all
forces to y. If x a t t m b z, y imtantly allocates all forces t a z, and so forth. The
odd man out imtantly allocaks d l force to the a t t w k d pa&y. Under globocog,
a hrce aE h e d size, C , from outside the tjnreewtor system is instantly allocated
to any attacked pa&y* Various imperfections will be d i s e u s ~ dbelow. But prfiect
collective security, and globocop, are operationalized in figure 3.1, as variatiow on
an underlying linear Richardsorr model,
There are clearfy some digerences beween these system and (3.1) and (3.2).
Fir&, rizther than LW actors, there %re thres?, the minimum number necessary
to examine colkective security. Second, them are systems of diff'erence, rather khan
diEerential, equations, For notational simplici%y,h z ZE sg+l-zt ,and unsubseripted
st&e variables on the right-hmd side8 reprrttjent values at time t. R a m this point an,
we will work in discrete time, mainly bwause major decisions on national armament
levels are taken in discrete tirne (e.g,, anaualiy).1521
a m i n g now to more substantive i~sues,notice that the Rieha~rdsonmodel is
reformulated slightly in that actors respond not simply to adversary m i l i t ~ ylevels,
as in (3.1) and (3.2), but to the gaps betwen their own levels and those of patctlnt;ial
adwrsaries; country s reacts to (y 3;) in the first equation rather than to y as
before. Here, with Richardmn, we "suppose that what moves a government to arm is
no%the absolute magnitude of &her nationsbrrrtamerrts but the diRerence between
its own and theirs," lB31
-
fszlln fmt,the discrete nonlinear dynamics (below) me considerably richer than the anaXogorrs
continuous dynamics.
f54
~ i c h a d s o n(19130,p. 35).
48
Nonlinear dynamic^;, Mathematical BiolowIand Soc=ialScience
FtGURE 3.1 Linear Models: The Three Basic ~ariantspq
LINEAR RICHARESSONIANCOMPETITION.
LINEAR AND PERFECT WLLECTiVE SECURIW*
113:= -alz
+ azfy - (a-+z)) + as(s - (z-t- y)) -+ g,
ay = bl(Z - (9+ 4 )- bz3/ + b 3 ( ~- (Y + X)) 4- gy
b z = el(z - ( z
L?Lz
=I
--a15
+ g)) + cz(g
.--
( z -t- S)) - Q Z
+ gZ
+ az(g - (z-t- C)) + ag(z- (Z + C ) )+ g,
+
+
+
Ay = b l ( ~ (P C ) )- ~ Z Y b3(z - (g C ) )f gy
h z = CI(Z - (Z + C ) )+%(g - ( Z +C))-C@ + g z
Under Rehardson, when z evaluates the external threat from g, he computm
y - 2. But under perfect collective security, be can count on x's cmdiiuted support,
So, be computes y - ( X t-. 2). In turn, when a: evaluat;es the poteneial threat posed
by 2, he campuks z - 1; under Richard~on,but z - (a + y) undfzr pedect wllwtive
securilty, Likewise for all exterrzd thre& terms in the modd. Even this very simple
formulation suggests that life under couective wurity would: diger from life under
Richardson in nonobvious way^. For instance, put yourslf in. s% position and ask:
am T better of%or m r s e sff if y k arsenal grows? In a Riehwdsonian worEd, the
amwer is clear: you are worse off. Under coltee%ivesecuri$ypby cantrat, the amwer
is not clear since y is a thr& in one context but is an dly in nnother (namdy, if z
attacks). Whether s ultimately risw or falIs with an inereme in y depends on the
quantity a2 - a3, wkeb might be positive, negative, or zero.
Relative to linear Rchardsan, however, linear perfect ecrklective ~ c u r i d yhas a
systtematie~tIlydepr~i-rre
eEect on the competition. While this rersttlt is intuitive, a
h m a l proof (below) wilt, in fa%, lead to eouderintuitive results. In particular, it is
the magnztude of "the collective security eRect7'-not its sign-which is unexpeded,
padieularly in the nonlinear variants below. But, having found a simple way to
r reciprocal zlaflruism-into a bwie arms race model, let
import collective securi
us d e l y proofs and structural variations for some elementary simulations, Though
i54f~lthoughquimlent matrix farrnulations will be u d below, I =chew m a t r i m b e r ~for two
r e w n s . Fkst, for the uainitiatd, the bwie di@erencmbetween the regimw c o m e through mare
cle8rly with this natation. Seeand, ruzd more important, the reilationshiw beWmn the Iinetx
versions and the commponding nonlineas wia-tions below will be mueh clewer in the notation,
informal, t h e can help us develop a ""f1I"' for how collective security may effect
dynamics in the perfect linear ease,
For ilEwtr&tivepurpose, then, a B= Case simulation of the linear Rehardson
model is given in figure 3-2. All actors are set at 1OOO units initially, and all reaction
coefficients, economic M i m e caeEci&s, aad grievances are assumed posieivr?. All
mumptiorrs are given in the Appendix.
On those msumptioxrs, the evolution is shown in figure 3.2. The defense Ievets
all increase, leveling off to some equilibrium, 5 E R:, given by 2 = (1 M ) - l g 9
for appropriat.ety defined M and
FIGURE 3.2 Linear Riehardson
Befense Levels
2000
1800
1600
Now, leaving d l initid defense levels and other numbers bed, what is the eEect
of the purely instit~tianaltramition to cdlecti~escuriw? How doe8 the chaage
in rule mgimt: alter dynamics? In*&
of gra&bl we have deertine, W shown in
figure 3.3.
fas11n m&dx notation, all thrm linear models in figure 3.1 share the generill fom, zt+l = M z l +g.
Ely definition, an ffquiUbTium, Z*~atisfiwE = M2 i: $.g SO, where it exists (i.e., where I - M is
nonsingluar), 2 = ((I - M ) - l g .
50
Nontinear Dynamics, Mathematical Biology, and Social Science
FIGURE 3.3 Linear Collective Sseurity
Defense Levels
1200
In faet, i f all eoe%"ieientsare positive, both syst;ems will a"citain equilibrium (more
on this below),
FIGURE 3.4 Wichardson ALctocatalytic
Defense Levels
60001
Time
51
Lecture?3
So powerful is the collective security effect, however, that even cases of autocatalytic arms growth ( a l ,bz, CS all negative), or "militarism," can be reversed. Figure 3.4 shorn an autocatalfiic (negative economic fatigue) c ~ under
e
Riehafdson.
FIGURE 3.5 Collective Se~urityAutmataiytic
Defense Levels
z2001
And, in figure 3.5, we have exactly the same numerical wttings, but run under
perfect collective security.
Tfrae results are systemalic. Indeed, if we denote the Richardsonim, collective;
security, and globocop levels at time t by zp,zFS, and zfC, then
Theorem. If z$ = sgS = zfC z 0, then for all positive t ,
%,R > zfC. He%, for Z, y E R", > 9
> for all i.
> zFS and
52
Nonlinear Dynamics, Mathematical Biolowt and Social Science
Proof. Cmtirrg the above qstexrzs in m&rk form, let
Far future reference, it, is importmt to note that If z > 0, then Bz
t;hese3 definitions, it is a matter af trivial algebra to show that
> Cl.[5" 1Wil;h
~fsf = zfCpthen globocop (3.6) i s simply Richadson (3.4) with grievsnces reduced
by the constant vector CV > 0; the effect is obviously depressive. To prove that
collective securi'cy is strictly depressive, we establish a simple lernxna.
Lemma. If, for some time t^, sf > zfS, then zf > zYS for all t > i.
Proof,
zLl = ~ s f + b
since ~ s >f 0
> ( A - ~ ) s +f b
> ( A - B ) Z ? ~+ b since zf > 5y by hypothesis
r==
2F5
L+ l
by ( 3 . 5 ) .
a, a proof that zp > sfS for all positive t will be in h a d once we
show that zf >'?X
But this is simple. Since zg = zfS, call them both zo > 0.
Then,
z p - s f S = [ A X ~ + ~-] [ ( A - B ) Z ~ +=~ B] Z >
~ 0,
[561~a
ensure physical rmlhnr (z> 01,vur? must stipulate that a1 -i
a2
-3- a3
61 3I- c3 < 2. X thank Jean-Pierre Langlois for this o k w a t k n .
and
< 2, bX f b2 + 63 < 2,
aad we are through. C3
Now, as noted earlier, these rmults obtain. so long ins BJ: > O for z > 0.
The specific B matrix above can be altered considerably while leaving the strictly
depressive effect of collective seeurit;?r ixldwt. Connwtionist b r m i n o b g wilt prove
natural for discussing imperfect collective si~tcuriw
THE CQNNECTIONISM OF COLLECTIVE SECURITY
This perspective emerges f m closer scrutiny of the B matrix. The ijth entfy, bij,
reprmxttrz the level of altruism that pafty j shows party i. If we let the s p b o l
"'re --r y" represeat the altruism z shows y (i.e., it is
> Q ) , then, eonceptaally,
the B matrk is
a
y--+Z
z--rz
z+y
0
z-+y
4;+2
y-+x
0
Gr;l_phieally;this would correspond to the "(altruism web'hhowrz in figure 3.6, Pairwise, iaff altruism is reciprocated; arrows run in both directions. If z a t t w b z,y
aflocaterz force to z and vice versa if s atdwh y, and so on. When this is the c s e ,
we will say th& the collective securiw system is m&mallg connected*
FIGURE 3.6 Maximally Connected Altruism Web
AXE off-diagonal elements of the B-matrh are strictly positive; the sign structure is
then
The strength of any c~aneetion(in cont;rwt to the connection pattern) is: the! red
number, b,, which can assume values in 10, l]. So, in these terms, perfect collective acurity entdfs m d m u r n connectivity and, maimurn connection stre~@h,In
turn, imperfect co11ective security regimes result from: reductions in conneckivilty,
rductions in connection strength, or both.
54
Nonlinear Dynamics, MathematicalBiology, and Soeial Science
MAXIMAL CONNEGTIVIW WITH DILUTED STRENGTH
It is obvious that, if Bs > O then
> 0 for a y real y E (0,I). This is
the most transparent cme of imperfat or ""dluted" wllectiw security, Mmimal
connectivity-reciprocal altruism-prevaiis~, but, irrsted of sending all of one's
forces to the aid of the attackd party, one sends a fraetion y. In fact, the strictly
depressive effect is preserved if every oE-diagonal entry in the IS-matrix is a different yij E (0,l). Everyone is better off even if the reciprocal altruism is, in this
sense, discriminatory.
MtNlMAL COMMECT1VEW WITH DILUTED STRENGTH
More intriguing, however, the altruism need not be; reciprocal to leave all parties
strictly bfjtter off. Specifically, altruism matrices far more sparse than B will fulfil1
our drictly depressive requirement, Bz r 0. Indeed, it is necmsary only that each
row contain a single positive entry. So, for instance, any matrix with the following
s i p strtleture will do,
Graphically, this would correspond to the cyelie LLaltruism
web" in fifigure 3.7,
FIGURE 3.7 Cyclic Altruism Web
Here, z is unilaterally dtruistic to y; y is unilaterally altruistic to z; and x is
unilaterally altruistic to z. Everyone is better off, but there is no mzprocat altmism,
Imtead of "mu scratch my bwk and 1'11 scratch yours," the appeal i s " p u scratch
my back, and 1'11 scratch Samks,and Sam will scratch yours." 1 c111 this "qyslic
~ l t r u i ~ r nThe
. " ~direction
~ ~ ~ of the cycle Is reversed if B has the sign patttern shown
below.
[ w ~ ~ b v i o u sthis
~ y , is a form of d i l u t d altruism, with some yid'S equal t o zero. But, as it has a
different f i ~ v aand,
t
beeause the positzon of the zeros mtsttters, E give it a separate name,
It is, in f a , not necessaq thsrt these altrukaz, graphs, or '%eh," h closed.
For instance, any B mitttrk with the followi~gsim pattern will satis& our strictly
degremive, Bz r 0, requirement.
But, its graph is open, as shown in figure 3.8,
FIGURE 3.8 Open Altruism Web
Everyone is strictly better off if z and are reeipracal altruists and g is unilaterally
ait;rui&ic t o z , eveat if z is. dtruisdic to no one!
In summaryl for the linear models &ove, there are b ~ i e d l ytwo sengm in which
collective s w u r i e can be imperfect and stiitl exert a strictly d~j~remive
eff:ect on
dynamics. A1tru;ISm can be perfmtXy reeiprocd but diiukd in stren@h. It can also
be imperfectly reciprocal (as in fipre 3.8), even unreciprocated (M in figure 3.7). As
we will see, it may in faet be both highly diluted and imperketly reciprocal a d still
have a profoundly depressive effect. It is in precisely the systems that concern us
most-the volatile! systems-that such highly imperfect eoflective security regimes
can have dramatic stabilizhg e E ~ t sSuch
.
dynamics, however, really arbe only in
nonlinear systems. Let us turn to thwr?varimts,
PART II, NONLINEAR MODELS
Nonlinearitties may enter the model in numerous ways. One way is through the
balance assmsmexlt, s r e ~ e r n a threat;,
l
terms, Moder n military e~ta'blishmendsdo
m t memure military b a l ~ n c
ernal theats-by simple subtractions of the
form y - z, EM in the above models. b t h e r , they o&en usc; methods that;, at some
level or other, embed mutual attrition models implfing that net military advantqe
i s a diflerence of levels raised to powers. Where does this come &am? BasicaUy,
konn the attrition stalemate conditions of generalized Lanchester equations which
were discussed in the preceding lwture. Allow me to derive this quickly.
56
Nonfinear Dynamics, Mathematical Biology, and Swiat Science
GENERALIZED AmRITiION STALEMATE
Let R($) and B(t) be Red and Blue forces at time t , and let r and b (real numbers
between zero and one) represent their effectiveness per unit. With constants cl
throu& c4(0 c < l), the most general Canchester attrition system is
<
These relations imply that the casualty-exchange ratio (Reds killed per Blue killed)
h1581
dR = b
dB r R c 3 - e ~
k3cf-c4
'
Assuming, for simplicity" sake, that q - q =. q - Q = X, let us separtlte variables
and integrate from terminal to (higher) initial values.
With stalemate defined as R(t) =; B(t) -- Q, we obtain the stalemate condition:
If the lebhasrd side is greater than zero, Blue "wins." If it is less than zero,
actd "wins,'"~ssuxning equal firing efictiveness b = r, and defining P = X + 1, the
measure of net military advantage implicit in all Ganehester-bmed attrition models
which ia exwtly the form employed in the nonlimew Richardson, collective mcurity,
and globocop variants display.ed in figure 3.9.
[?S81~or
a fuller dkcwion of Lhk gexlerdizcld exchange ratio and its interpr&ation, see Epstcfin
(1993, and 1990, pp, 85-93), and Ifleture 2 above.
FIGURE 3.9 Nonlinear Variants
NONLINEAR AND VARIABLE COLLECTIVE SEGURIW.
With inexeasing p, whieh w w innplicitly set at unity in the linear models &m,
the volatility of the system grows. %call that in figure 3.3, with = X, thr: Richardson model produced msnotonie e;ro&h to some quilibrium. wth a15 else? as in that,
run, but with P = 1.7, the dynmics are rdically altered, Figure 3.10 gives the re-sult. By any definition, an acute crisis develops. This, strictty speaking, is not chants,
but it is wild. 1591 System-wide strategic manic-depression and inererasingly explosive
oscillations develop.
tss;"l~or
a p r a b e definition of ehms, SM Deaney f 11389, p. 50). On ctzaotk dynamics ia a m race
madeb, see Mityer-Kr- (1991).
58
Nontinear Dynamks, Mathsmatksf Biolwy, and Social Science
FIGURE 3.10 Nonlinear Richardson 1
DeEense Levels
Time
Now, the striking fwt is that highly imperfect collective security regimes of
various mrts can contain this crisis. First, let us =same m a i m a l conxrectivity
(reciprocal altruism) but dilutd strenoh.
MAXIMUM CONNECTIVITY WITH L W STRENGTH
The dilution. is captured by y in the colle-ctive mcurity equation^ of figure 3,9
above. If y =: 1, perfect collective security. is restored, with 108 percent of tbird
party forces going instantly to the defender. Xf y =; 0.5, half the forces are so &llocatd, and so forth. Here, of course, we are inter&ed in the weakest for- of
eollmtive security that are of illterest; and surprisingly, with dl other numbers set
as in the exglcrsive case of figure 3.10, a onepercent solution of collective security
(y
0.01) contains the blaze. Crisis management occurs, as s h o m in figure
3. Ila, which displays ~ ( t y)(G),, and z ( t ) superimposecrf. Figure 3.11b gives a threedimensional phase portrait of the same dynamics.
-
FIGURE. 3.11 (a) Nonlinear Collective Security 1; (b) Nonlinear Gallmive Swuriv 1,
Phase Portrail
30
(a)
Defense Levels
A crisis iaded develops, but it is contained, and the system then calms down Lo
an equilibrium below the inieial state. This coiorfitl transient behwior-which
we
60
Nonlinear Dynamics, Mathematical Biology; and Social Science
interpret as crisis containment ccurs at the ""buundizry" betwen explosive oscillation and monotone disarmament. That; is, if we begin with y == Q and the
explosive oscillations of Ggure 3.10, and smoothly i n c r e w y, we will arrive at a
value beyond which all oscilfations will be damped away and all t r a j = t o r i ~simply d ~ r e z t s emonoto~icallyto equilibrium. Some ml?ly find even these transient
oscillations worrisome., Particularly in the early phwe, when the oscillations are
increasing in amplitude, the expectation that one" securie (Le., superiority) is
transitory and will soon give way t o inferiority may generate prmxnptive (strike
while you're &he&) pressures. But even thme trmsient oscillations can be elimi,
twepercend
nated outright at remarhbly low y vdues, C e t e ~ ps a ~ b y~=~.02-a
commitment-haa that effect.
Now, w b a f s going on here; how M e such low leve2s of cooperation producing
such dramatic eRects? The key is the parameter P. The higher is P, the more
volirttile is the Richardsartian, system. But, in collective s ~ u r i t ythat
~ same P is
applied to a sum-the Richardsonian defender plus his collifsetive mcurity ally. The
more powerhlly are low levels af eotlective ecttrity amplified (by ,L3 > I), the
more will a little coXlective security enhance stabili*. The greater the underlying
volatility (high p), the greater the stabilizing eEect of a given level (7) of collective
security; One might aasume that the more volatile the system is, the less value
there will be in a low level of collective scurity. But, prechely the reverse is true!
By the same token, if we go to a low @---aless vBiatife unregulated Riehardm
nian world----idtakes a much stronger tramition t o collective weurity to significantly
the same taken, there exisb some r e d 7 for which the transient phase will haw any spwified
duration. I thank h b m t Axtell for this thought;.
IG1lrf"hedependence on P of eoXIective security'~depremive eEmt can be analyzed by examining
~ colleelive ecurity,
the partial derivative of, say z's growth rate A z with respwt to y, the d e g r of
for vwious ps, It suffica Lo consider the term
from the Ase equation of the nontinear coilwti.re smurily model above. Considering this twm only
(since the as term wiH bebrave anitlagouirly), compute
Ciwty, if p =f 1, then yk ddtsprmdve eEet by) varies linearliy with z , But, a t the otber
extreme of p -.-- 2, the product of a: and z, and the sgvalre of z, enter in. Then,
alter trajectories. To wit, figure 3.12 shows our nonlinear Richardsonian case, but
with p .= 1.2 rather than P = 1.7 from the crish case.
FIGURE 3.12 Nonlinear Richardsan 2
The same level of collective security (y = 0.01) that contained the crisis at j3 = 1.7
has virtually no effect here, as shown in figure 3.13.
FIGURE 3.13 Nonlinear Collective Security 2 (Low y)
Nonlinear Dynamics, Mathematkaf Biology, and Social Scienw
Recall that a one-percent collective security commitment (7= 0.01) was s u s eient to contain explasive oscillations and p m d w redu&ions in the P = 1.7 ease.
Here, to merely produce constancy (a weaker requirement than reduction) requires
a 7 of 0.10, a kenfold inerem. This case is shown in fipre 3.14.
FIGURE 3.14 Nonlinear Collective Sa~urity3 (Higher 7)
Defemse Levels
20
Let us rctstore the value p
security.
60
H:
100
Time
1.7 and witmine another form of imperfect callective
LOW CONNECTlVlTY WITH HIGHER (BUT STILL VERY LOW) STRENGTH
Specifically, let us now relax the assumption of maximal connectivity and posit a
low connectivity configuration like the cyclic altruism web of figure 3.7. If we leave
the connection strength at 7 = 0.01 as in figure 3.11, crisis containment will not
occur; we will see the explosive dynamics of figure 3.10. So, connection strength (7)
must be raised to compensate for the reduced connectivity. How much greater than
0.01 must the connection strength, y,be in order to contain the explosion and force
convergence to an equitibrium below initial levels'! The value of y = 0.02 produces
the dynamks shown in figures 3.15a, and 3.15b.
FIGURE 3.15 fa) Cyclic AItruism with y = 0.02 fb) Cyclic Altruism vvith y = 0.02, 3 0
Phase Portrait
(a)
Defense L e v e l s
20
80
Time
Again, what is striking is that this 7 is still very low; a two-percent collective
security commitment under a sparse eonneelion pattern still suficm,
A comparabiy low value of 7 = 0.025 also suffices to contain the explosion
and force convergence to a low equilibrium under the open connection pattern of
figure 3.8. Dynamics in that case are shown in figurm 3.16a ancf b.
64
Nonlinear Bynamlcs, Mathematical Biology, and Social Science
FIGURE 3.W
Po~ralt
((a)
Open Web with y = 0.025 (b) Open Web with 7 = 0.025, 30 Phase
Defense Levels
Clearly, collective seetlri.t;y can be highly imperfect in a spwtmsn of ways and
still exercise a powerfully depressive eEecf,oa dynamics.
Lecture 3
65
Thus far, I have said dmost nothing abo& globaeop. Under g l O b o ~ ~when
p , evaiwting the threat; of external aggre~ion,ail =tars w u m e a defensive reinforcement of
C. In the fineaz carje, globocop is equivalent to mchard~on.with reduced grievances
and is, fof that reason, quite uninteresting mathematically: In the nonlinear ease,
our exponent p reenters the picture, amplifying the globocop-augment;ed defenders
substantially for, agdn, v i t e modest C-values, if P is high. At the @-valueof 1.7,
under which a onepercent ~Xutionof collective security coditin& %hecrisis of figure 3.10, a onepercent globocop of C = 10 does perfectly well, too. The simulation.
is given in figura 3.17a and b.
FIGURE 3.17 (a) Nontinsar Globotzop; (b) Nontinear Elobocop, 3D Phase Portrait
(a)
Defense Levels
66
Nonllnear Dynamics, Nathem;lltieal Biology, and Social Science
A general nonlinear model tha%speciahzes to all the above variants and, nstably, to rnkw of gfoboeop and coll~tivesecurity is given below.t621
FIGURE 3.W General Nonlinear Model
AZ
= -als
+P
bx
Ay = -- (zP- ( y + y ~ + ~ ) P
P
STRUCTURAL SENSiTtVtW ANALYSIS AND CONCLUSIONS
In the usual sensitivity analysis, one uws a h e d mathematical model and examin@ $fie mnsitiviw of that modef" outputs to varia;tions in modd inputs. Here, we
have fixed inputs (e.g., z(0) dO) == z(0) = 1000) and exmined the sensitivity
of output to variaLiom in the model. W started with the linear Richardson model
and we perturbed it into a linear collective security variant. m e n we constructed a
nonlinea Richardmn model and pelrturbed that into a nonlinear collective security
variant. In bath cases, we found cofleetive secudty to have a powerfulfy depressive
egect on the competition. f cerLainly da not claim that either underlying Richardson; model is correct in an empirical ar predictive sense. If, however, the ""coUective
security eEeet" is .isyst;c?maticallydegressim across the entire set of plausible u ~ d e r lying Richtardsonian and non-Ricbrdsonim models, it is fair to regard the effect as
quite robust, Thus far, of course, the r ~ u l t are
s only suggwtive. But, wbhin the
clws of models examiad here, the following points apply*
With all numbers-propensitiw to ztrrn in response to perceived imbalances,
grievancm, and so forth-bed, modmt institutional changes alone can drastically
)
in hightg imperfeet f o m s , i s is ppowerfil dealter dynamiez;. Collective s e c u ~ t g even
pressant, And the d q ~ s s i v e(or '"stabilizin9") efleet i s the greater the more volatile
(in the P - S ~ R is
S ~UEe underEyi7lbg stmctum. I h e uentured no opinion whatever
on the femibiitity of implementing collective security anyhere. But, based on thh
analysis, it would be unfortunate indeed if imperfect collective security arrange
ments were dismissed simply on the ground that perfect ones may-indeed, probably would-be difficult to instiitute.fm31Highly imperfect forms of collective security
=I
~ " ~ ~ u ~ eand
h aKupeban
n
appear to be suamting some sort of mix betwem, in effmt, globocop (a
coneert, ~0mpri8dof "a smll group of m j o r powers") md collective wurity. But it is unehas to
me where on tfx%?
spmtrurn b&wmn t h e rqimf?stheir "concert;-bwdcoXleeLive security" regime
lies. See Kupehn srnd Kupcfian (1995).
["l81mong other things, the purely togisLica1 demands of a working co1I~xLivesexuriw re?@mecould
be nontrkM.
Lecture 3
may have profound effects and demrve serious study. Depending on the: magnitude
of C, globocop shows sirnilar powerhl propertiw. Indeed, an the bmis of this andysis, na especidly clear choice bebeen collective security a d globaeop can. be made.
Obviously, there may be political arguments pointing to one over the other, but not
It would be interesting to examine wh&her, within the even
rnathem&ical
d
arms rwe models, the same depressive
wider class of stochmtic a ~ gametheoretic
effects of eollectivc?security and globocap are suggested.[65t
f4"[
F O ~instance, collmtive murity rdws the class-ie "free rider" woblern. See Olson (1965). On
the other hand, globocop hzls overtontxl of wrXd government and rnacy be men as entailing an
intolerable swrifice in national sovereignty.
i6"see Downs and Rocke (1990). Here, utility functions include a term proportion& to the gap in
arm, so a coXEective securigy variant could easily be construetd and exarnind from this standpoint;. The same can be said of the model present4 in F~mwtand Mayer-Krm f 1991, pp. 166-85).
T
APPENDIX
TABLE 3.1 Numerical Assumptions for Lsdure 3 Figures
Figure ~umber'
Parameter
3.2
a1
a3
0.003
0.0085
0.0051
0.0068
as
fil
b2
b3
cl
c2
c3
p
y
C
g,
g,
$z
3.3
3.4
3.5
3.10 3.11 3.12 3,13 3.14 3.15
3.16 3.17
-0.003 -0.003
-0.W2-0,002
0.0034
0,0051
0.0085
0.001
-0.001 -0.Q01
1.0
1.7
0.0
1.0
1.O
0.0
20.0
5.0
3.0
0.002
1.7 1.2
0.01
1.2 1.2 1.7 2.7 1.7
0.01 0.10 0.02 0.025
10.0
A blank indicates the same value as in column 1 (figure 3.2).
LECTURE 4
S,and Ecosystems:
The p r a d i n g iedurm have concerned iaters.tate arm ri;tchg (mutudkra) and
(competi$ion). Let us now turn attention to btrastate procwes. This iecture canczc3rm rwaIu$iaw. The next wncerm the sgred of drugs, C1ine;ing do our
Valbrra-like "mand mifid thwry.It) the processm of interat in t h m fectura is the
ion sf same "simd" though a popuiatian, epidemic-lib prow s ~ in, short. Epidemics proper are facinating-md obviously very importmtt h k s . You would ce&Gnly enjoy Wiltiam McNeill" sanderhl book, Ptawes and
Peoples, which conmrns the role of ixlfectiow dkmes in, human M&ary.@@]
Since
t h m Iectwm procmd &am %hea n d 0 0 to epidemia, perbaps an. btroduetory word
or twa on dynmicd adagim per se is in order,
w;zr
ANALOGIES
Any two procmses whose matthematicd daeriptions have the s w e hnctionaf farm,
and whose ~ t a t ew i a b l a and peurmekrs cast be put in o ~ e b a n corrwpondence,
e
70
Nonlinear Dynamics, Mathematical Biology, and Swiat Science
are said to be dynamical analogies. It is a sta&ling fact &h&a huge variety of
seemingly unrelated procmsa are analogous in this sense. For example, the same
equation that describes a damped harmonic oscillator, such as a pendulum with
kiction, also describes an oscillating electric circuit:
"all th& is required is to relhel the sta-te variables and p ~ a m e t e r sinvolved. Thus, the state variable representing the displacement of the mechanical system becomes the electric4 charge of the electrical system; the
ve1ocit;y becomes the current; the mass of %heparticle becomw the induetmce, meehanieal force bmomes EMF, etc. With similar reinterpret at ion^^
the same dynamical equation can be regarded as diwcribing rotational sys1978, p,
tems, goustic systems, hydraulic systems, and so on" "sen,
54).[671
knother example is the analogy b e m e n e~ectrastiat;ic aetraction under
Coulomb" LW and gravkationd attraction. under Nez;vt;oxx%Law, The magnitude
of each force is proportional to the produet of the two chargesjmasses, inverwly proportional to the square of the distance separating them, and directed along the tine
joining them. As another instance, Kelvin's circulation theorem in fluid mechanics
is identical in its mathematical form to Farday" LW iin electrodynamics, Both r e
late, via Stokes' Theorem, the flux of a vector fiefd to the circulation (or current) in
a boundary such as a conducting loog.p81 Countless further examples could be provided. The plnysical diversity of diffusive proeeaes satis&ing the ''he&" equation,
or oscillatory proeemes satismng the "wave'bequation, is virtually boundless.
But dynacmicc=afanalogies are more than beautiful testaments to the uniEying
power of mathematics: they are wef"Llk. In particular, ""AndoGe~are useful for
analysis in unexplored fields, By means of anitfogies an unfamiliar system may be
compaed with one that is better known. The relations and wticrns are more el2sily
visudized, the mathematics more readily applied and the analflical mlutions mare
rtsadily obtained in the familiar system.'5f@gl
Ainaloa in this sense has played ta powerkf role in the development of science,
engheering? and also social science, B notabb example of the 1Lzt;ter being Samuelson" sapplication. to economics of classical m&mum principles of physics. In one
eolorfuE diseussisn, for irrstaace, he argueri $In& "if you look a t the morrapolistic firm
as an example of a mdmuxrz system, you can mnnect up its structuraf,rdations
with those that prevail for an entropy-maximizing thermodynamic sptem. Pressure
and volume, and b r that matter absolute temperature and eneropy, have to ewh
other the s m e conjugate or dualist relation that the wage rate has to labor or the
land rent has to wres of land." %muehon, provides an elegant diagram that, in
~"~EMFis declromotive forw.
fssfSW, for example, M w d e n and mamba (1976, p, 338).
is910bn (1958,p. iv).
his w r d s , does "buble duty, depicting the ecommic relatianships W well as the
t h e r m ~ d y n m i cones."
Mutrray Gell-Mann has written on the application of nanlinear dynamics to
various s y s k m , including socid systems. In his words,
"Many of these applications are highly speculative. h&hermore, much
of the thmretical work is still at the level of hathematical metaphor."
But, X think this sieuation shaufd cause us to respond with enkhusiwm to
the challenge of trying to turn these metapfnarical conneetiom into real
scientific: explanatisns" "elk Mann, 1988, p. 4).
It is in. this highly speculative, metaphorical spirit that f proceed in the next two
bctures. This essay examines the malogy between epiciemim (for which a welldevelopid mathematical thmry e ~ s t s and
)
processw of ~sxpbsivesocial change,
such as revafutions (far which no comparable body of mathematicd tltwry exist;s). Are revolutions '"ike'kepidemics? More precisely, is id useful to think of these
process= a analogaus? Connections to predator-prey systems are also explored
and the spatia-tempord generdizatiomts of t h a e revolution('epidemic/predator-prey
motrEeb-reaction-diEusion equations-me examined* The realm of rewtiamdigusion equizeions is a natural one to explore. Such equations are central to the
mathematical theory of pattern. fomaLtion, and id is the evolution, propagation,
and stability of soeial patterns that is, ultimately, our concern, We begin with the
sinlple analogy between revolutions and infectious d i c ; e ~ e s . l ~ ~ ]
REVOLUTTONS AS EPIDEMICS
The particular a i m of revofutionary wtion, af caurse, vary widely from case to
case. In one instance, the revolutionafies"goaf may be %heoverthrow of monarchy;
in. another, it may be the installittian of theocrwy; in yet a third, it may be the establirshment of democrwy Given this enormous mriation in objectives, the thought
that there might be an underlying:structure common to all revolutions is an idriguing one. By a ""cmmon structure," "I course mean a mathematical madet whose
dynamics-at feast at wme crude level-are mimicked by revofutionaq processs
regardgas, that is,
in gmeraf, rqardlws of their political ""substaxlce," as it we
~ 7 0 ~ ~ a m u e f(2972,
m n pp. 8-9).
f 7 l f ~ l t i 1the
e connections betwmn revoliutiona, e p i d e ~ a md
,
~ o l o g i c dsystem prwnted here
have not, to my knowldge, bec3.n pr-nted e f ~ h c l r ethe
, thought th& the spred of id= might
be anafogom to the spred of d
h= bmn explorctd. A small literature sprang up in the 196Qks,
foitlawring the publication in I957 of the =mina1 work, Bailey (1957). For s wad overview with
referenet3s, we Dietz (1967). See a h hppaport (1974, pp. 47-59).
72
Nonlinear Dynamics, Mathematical Biology, and Social Science
of what the revolution ""is about." h considering this notion, I take, as one tantalizing poi& of departure, the mathematical thmry of epidemics; these processes
exhibit common dynamical structures despite obvious digermes among cornmunicable diseases. Measles, mumps, and smallpox are clearly digerent diseases; yet
their pwpagational dynamics may be indistinguishable from a mathematical standpoint. l72f Although the points of corrmpondence bewwn epidemics and revolutionr;
will be quicMy cvidont;, it will prove uscifirl to delay specific analogizing u&il a simple epidemic model is praented,
A BASIC EPIDEMIC MODEL
The epidemiolo@&"sproblem, as Paul Waltman puts it, "bto dwcribe the s p r e d
of an infection within a population. h a canonical example one thinks of a small
group of individuals who have a communicabfe infection being inserted into a large
population of individuals capable of ke(zdhing' the d i s e ~ eThen
an attempt is made
.
to describe the spread of the infection in the larger group.""3] In the simple model
first developed by Kermmk and McKendrick,E7" the population is assumed to be
constant and divided into three disjoint clmses:
S(t): the sweeptible claw comprised of individuals who, though, not infective,
are capable of becoming infective;
I@): the infective class, comprised of individuals capable of transmitting the
disewc: to others; and
R(t): the removeb class, consisting of those who have had the d i m e and
are dead, ar who have recovered and are permanently immune, or are
isolated until recwery and pernnanexlC immunitty occur.
The following rules are assumed to govern the s p r e d of the diseme:
(i)
(ii)
(iii)
The populatioa is conslat over the time intervai of interest. Births,
dedlt;h~from catus= other than the disese in question, immigration,
and emigration are all ignored.
The r&e of change of the suswptible class is proportional to the product
of the number of susceptfblm S ( t ) and the number of infectiws I(t).
Individuds are
Eram the infectious d m s at a rate proportional
t a I(t).
Rule (i) is a straightfomard simplifying assumption whose re1a;rat;ion is discussed
below. Rule (ii) reprwnts the assumption that the transfer of individuals from the
i 7 2 j ~ e eHetheote (1976, p, 336). Sec? aho Hethcottt- (1989, pp. 11s44).
f731~altmim
(1974, p. 1).
1741~ermwkand McKendrick (1927). See Murrsy (1989, pp. 611-18).
susceptible class into the infwtious pal p r o c d at a rate proportiond to the number of eontms bewmn infectivm m$ suseeptibiw, That the eontmt rate shott3d
be proportional to the product of the class sbes X and S implies unifmm m
of the twa groups and in&mtanwus contraction of the d i s e upon
~
exposure (Zakncy and incubation periods me both zero). In egect, the law of mass aceion is
msumed to apply. h W d t m m not=, ""This is remrrabje if the populiztion consists
of students in a school whrasr? changhg clmses, attending a.tk1etic events, etc. mix
the population." Impodmtly for our purposes, he eontinuw, "It would not be true
in, an environment where saeia-economic fwtors have a major inf3luence on cont a c l ; ~ , " fPinaliy,
~~l
rule (iii) implies that all infectiva have the e r n e probability of
removal (recovery, death, or isalation). The rnodel does not wcount for the length
of time m individual has b e n Infective.
Acmpting thwe definilio~ssad rules, and treating the aver& popu1a;t;ion as
a eorteinuum, the flow of individuds &om the s u ~ e p t i b l eto the infective to the
removed class is described by the foltaAng system of nonlinear differeneial q u a
with initid conditions S(0) = > 0, I(Q) == l. > O and R(O) = 0.
The comtmts r m d 7 are called the infection rate and the removal rate, and
p = ylr is termed the relative mmovd rate.
THE THRESHOLD CONDITION
Now, under what con&tions will m epidemic occur in this mod47 To say that
an epidemic occurs is to say that the infectious class grows or, equivalently; that
> 0, which h m (4.1) implia that rSI - y l > O or, simply, tbzvt.
[ 7 " 5 1 ~ d t(1914,
m ~ p, 2).
[76]~mam
the A m is from susceplible (S)to infmdive (If to r e m w d ( R ) ,this is term4 an SIlR
rmodk?l.ff the inf&iuw p h w is fo'otlawd,not by remod (e.g., irnmuniky), but by rentry into the
sweptible pool, tm SIS model would be call& for. "Is general, SIR madeh axe appropdatrt for
such ~ 3 8m ~ l wmumps,
,
md smdlpm, while SIS m&eh itre appropriate for
such as meniagitis, phgue, and venered d i
some bmteriaf agent d
such W d w i a and sleping siekn-." Hdhcot;e (1976, g. 336). See &m Hetheode
(1989). The wrnerstone af the nnrrlhemalical epidamiolu~literature mmains Bailey (1957). See
h Baile3y (1975). A cormprehemive mntennporw t a t k Anderson and May (1991).
74
O\lanlintsar Dynamics, Mathernatk-calBiology, and Social Seienee
This is a basic result.iT7jFor an epidemic to occur, the number of susceptibles
must e x c ~ dthe threshold level -the
relative removal rate defined above,
POLITICAL INTERPRETATION
The bwie a~fS\lfom
to revolutimi~fydynamics is dirwt. The infection, nr diwase, is
af course the revolutionary idea. The infwtives Ift) are individuds who are xletively
engaged in articulating the revolutionary vision and winning over ("infecting") the
susceptible class S(t), comprised of those who are receptive to the revolutionary
idea, but who are not infective (not netively engaged in transmittiag the diseme
to others). Rernwal is mast naturally interpreted ijls the political imprisonment of
infectives-----R(t)is the ""Gulag" "population, the set of unfortunate revolutionaries
who have been captured and isolated frorn the susceptible populatim.f781
Many familiar tactics of todditarian rule cttn be seen EM measures to minimize
r (the eEective canLaet rate b e w e n inkctives and susceptibles) or m&mize 7
(the rate of political rcjmoval). Press cemarship and the systematic inculcation of
counl;errevolutionafy beliefs reduce r , while increases in the rate of domestic spying
(to idedify infcxtives) and of imprisonment w i t h u t trial increase y.
Spmetrically, familiizf revolutionary tactics-such as the publication of underground literature, or " s ~ m i z d a t " ~ e etok increae r. Similarly, Mm% directive
that revolutionaries must ""sixn like fish in the sea," making themselves iadistinguishable (to authorities) from the surrounding susceptible population, is: inteded
to reduce y.
GORBACHEV, DeTOQUEVILLE, AND SENSITIVITY TO INITIAL
CONDCTIQNS
Interpmting relation (4.2) =mewhat differently, if the number of suseeptibltss, &,
is in fact quite close to p, then even modest reduct'rans (voluntary or not) in central authority can push society over the epidemic threshold, producing an explosive
overthrow of the edsting social order. To take the example of Gorbwhev, the policy
of Glasnost obviously produced a sharp increadie in. r , while the relaation of potitieal repression (e.g., $he weakening of the KGB, the releae aE prominent political
prisoners, the dismantling of Stalin's Gulw @@ern) constituted a reduetion In y.
Combined, these mewures evidently depresmd p to a level below So, and the ""rvolutions of 1989" unfolded. Perhaps DeToqueville int;uited reldion (4.21, describing
f771~bviously,
the system (4.1) has a gre&t many further mathematical properlim sf interwt. For
a d i ~ m i o now
, Braun f 1983, pp. 456-73).
IT81ln tfib disewion, we ignore exmutiona
Lecture 4
75
tbis sensidivi/t;y to istial conditions, when he remark4 that '"iberalization is the
most diEeuEt of pafjiLieal arts."
TRAVELING WAVES
In the discussion thus far, the sqtatial dimension has only been implicit. In fact,
epidemics spread wross geographical a r e a over time. And one generally t h i n k
of revolutions spreading as well. Specifically, we o&en invokcl, the t e r m i n o f o ~of
waves, Recently, we saw "a wave of democratic revolutions" sweep acmss Eastern
Europe, Perhsbps this sort of language swnrs natural for a reaan: if one generalizm
model (4.1) to explicitly include the spatial dipusion of infectiws, traveling w m s
do indmd emerge. And this process, of course, has a political interpretation.
The one-dimensional spakio-temporal gen~ralizationof (4.1) is:
An infective spatial diffusion term, Da21/@x2,has been introduced into the second
equation, which bears some resemblance to the clmsicaf heat equation, It == BI,,,
where D is %betherma
r, in this cwe, the political-""dEusivityB of the medium.
The presence of the paredhesized term makes the equation a so-called reactiondiEusion rdation,
Now, as set h r t h in lecture 6, one posits traveling wave solutions to (4.3) of
the form
S(z, t ) == S(z), I ( x , tj = I ( x ) , z = z.- - &,
(4.4)
where c is the wave speed. The boundary conditions S ( m ) = 1,S(-m) = O,I ( m ) =
I f - m ) = O must also be met. B y p ~ s i n gmat;hema%iedspecifies that are well presented elsewfiere,[7" the basic conclusions are, first, that no epidemic wave prop&
< y/r. This, of course, is the basic threshold condition from model (4.1).
gates if
What is new, however, is that if that threshold level of susceptibility is exceded,
an epidenaiclrevolutionary wavefr;onl propagates. And its speed of propagation, c,
is given by
c = 2[D(rSB- y)]'/2 .
(4.5)
Basic cortnterrevofulionary tactics aim not o n b to minimize r (the rizte at
Ftakich contact produces a transmission) and metximize 7 (the removal rate), but
to minimize D as well. Physical curfews, ratrietiom on free w~embly,internal
lT9f~e@
Murray (1989, pp. 66143), BritLon (1986, pp. 51-71), and the discurnion in lecture 6 of
dbk volume.
76
Nontinear Dynamics, Mathematical Biology; and Social Sciems
pmsport requiremexrt;~,a g a ~ h e i din all its forms, are m e w to '"atchifyt" and
limit the ""politic& diEusivityB of, the sacid medium. A high density of ipternal
police reduces the diffixsion af revolutionaries just as a high demity of wet insmts
or Dr&ards the spread of a brush fire.@qfndwd, the mosti obvious ""fiebre*,"
minimizers, are the borders bewwn e o u n t r i ~which
,
physicdly enforce certaLin. of
the ideological '"patches" "viding h~mpanity*[~~J
VITAL, DYNAMICS AND THE EWLUTION OF IC)ISCONTENT
Thus far in the dbcussbn, the total popuiatbn has bmn assumed eanstant. Proeases of profound social change may unfold owr periods in which birth md deathwhat epidemiologhts cd1 ''vital dynamicsn-play a rate. The introduction of thwe
factors expands the range af gomible revoIutionary/epidemic trajectories?. Indwd,
the ixlt;roduction of some rudimentary vital dynamics connects our disetrssion, perbaps surprisingly, to the field of mathematical e c o l ~ o ~
Fbr expasitmy ewe, let us recall the basic epidemic model (4.1). Keeping matters simpk, now introduce a MaXthusian birth rate i&o the susceptible papulation,
If p > Q is the birth rate, we obtain the sys;t;em
Students of ma;t;hematicaI,ecology will recognize this as prmimly the Lath
VolLerra predator-prey modc31. The prey (susceptibtes) would increm exponentidly
if not for the pred&ors (infeetives), who m u l d die off exponent;ially *bout their
""foodsource,'"he prey, Qnee a birth term is introducd, the infwtives and suscep
tiblm may be w n as forming an ecosystem*
As shown in figure 4.1, the orbits of (4.6) are closed cuwes in the SI p h a e
plane; soluti<ms are undarnped
184sm Murray (1989, pp. 375-76)f 8 Xbait
i~ strate~
to collf;r~li
the s p r d af rabim in foxr?s,for =ample, is to thin the susceptible
fox population to a level belaw the eriticd thrmhold demity acrm some swath. lying in &ant of
the d ~ ~ e i epidemic
n g
wave; to create a 6re:c?bre&,as it were. Far an andflic approxinnstion for
the minimum width of a rabies control break, as sueh b m i m are knwn, see Mumay (1989,pp.
681-89).
fS2fftdamp& mriants rmulb if pS is rmh& simply by p. See Elailti?y (1975,pp, 135-37).
Predators
Prey
The political interpretation is straightforward: extending model (4.1) to include rudimentary vital dynamics ('het births"), as in (4.61, society may experience periodic cycles of rewlutionary discontent (the infection) analogous to cycles
GharwLeristie of basic model ecaystem, and rt3current epidedcs.[=l InteratinglLy;
Gaodwin's madet of tht? clitss strug1.e b e w e n workers and capitalist;^ is of e x ~ t l y
the s m e f"orm.f8"1
In the physical sciencw, it is often of great interest to identify entities that
are ceased by the dpamical system-funetions that are comtant along sy-m
trajectories. As discussed in lecture 6, such functions are termed Hamiltonians of
fsqOn cyclical behavior in - e g i d e ~ cand waI~&calmdeb [sspwificisily,the inhrw%eclr d a r might
(1989, pp. 192-211) and H u a g sfld MerriII (1989).
[&4j~oadwin.
(1967). The mdei is3 e!ewky d k u d in brenz (1989).
78
Nonfinear Dynamics, Mathematical Biology, and Social Science
the system. Interestingly, this model admits a Hamiltonian formulation. Under the
transformation p = In S, q = In l,system (4.6) becornesI85J
Separating variables and integrating the equation
dp
p-reQ
dq
rep
-5-
-7'
we obtain
reP-7p+reQ-pq=c,
where c is a constant determined by initial conditions. The left-hand side, constant
along trajectories, defines a Hamiltonian:
It is easy to verify that (4.7) is, equivalently, Hamilton's equations:
A socid interpretation of this energy-like Bamiltonian-a
constant of
r~lutio~asy/counte~~revolutionitry
motion-wouldb-intrigdg.
This purely o s c i l l a - n e u t r a l stable-behavior of system (4.6) breaks down
when the assumption of Malthusian growth is abandoned. Indeed, the model is
structurally unstable in the language af lecture 6. Assuming some carrying capacity
for the susceptibles, a more plausible system would be
where K is the carrying capacity..
[ 8 s l ~Verhulst (1990,pp. 21-22). For further generalization, and an elegant discussion of HadE.
tonians in this cantext, see Samuelson (1971).
Once more, the int~oduGtionof' spalial diffusion terms renders (4.10) into a
reatet ion-diEwian system. f n one spatial dimension,
And, as in the e m of the generalized epidemic model (4.3), traveling wavefront
soltrdions mist. Assuming for consistencyb sake that it is again the infeetives th&
diffuse, let D1/Dz = 0. Then, under the usual boundary ~onditions,[~~1
a minimum
wavefront ~ e l a c i ht i~n~t @ven by
An inl;er&ing digerenee betwwn (4*11)and (4.31, however, is that this wavefront
solution can be approach4 in. an. oscillatory or monotoxlic fwhion depending on
parmeter vaalues.fB7J
THE REMOVERS
The modcsls thus far prmented plrtee no upper bound eta R(C);thme is an unlimited
removal czlpaeily. One refinement is the imposition of same upper bound on Rlt).
Second, the counter-revolutionary elms (the elite and its gendarmerie), or "public
health authority," b not explicitly represented. Clearly, since the power elike is
doing the removing, it desemes a place in. the model. Let us denote this group by
&(C) and, for simplicity, assume it ta be constant at some initial level Eo. Then,
i f G denotes thq upper bound an removals, a slightly refined version of our hrst
(Kermaek-McKendrick) modet might take the hUowing form:
Nonlinear Dynamics, Mathematical B i o l ~ g
and
~ Sscial Science
The basic m m o d (repression) rate parmeter y has 'bwn supplanted by aEo,
where cu > O if R(t) < G and a = 0 o t h e m h . In other words, a ctquals a positiw
equals zero once the Gut%
constant so long as there is ""room in the Gul;sg,'"ut
is full (or, more generally, once the removal capacity of the State is reached).
We saw that, under the model f4.1), with no upper bound on removals, the
infection ultimately dies out, At the other extreme, if G = O (no removal capacity), the tramfer of susceptibles into the infective c l ~ ~ - t h &is, the sprt3axli of the
idectian-is in c;@ectgavernd by the sysgexn
dt
Since tfie total papulaeian P and the elite E are constants, we have S(2) -t
I ( t ) + E. = PO,or S ( t ) = [Po- Eo]- I ( t ) , and h n w e m wrik
a Bernoutli equation whose solution,
is precisely the Lo@stichnetion, This same hnetion, interestingly, has been found
to govera the digusion of ce1.lain befinoliogicd innovatiom-'%whnoJaw
epide~es."@88f
The clmfy related discrete Logktic map b,af course, the prototypical chmtie dyamical system. h d , in, fact, the question whether epidemics
ehiibit chastic behavior is under actiw st~dy.l8~1
Such canneetians, it seems to me,
are potentially quite intermling.
IMPERMANENT REMOVAL
Thus far we have wsurxrd any removals to be permanent; the Kermmk-McKendriek
flow is from S to I to R. In fact, even when the state" removal capwity k unbounded (a reasonable assumption in most practical cmtts), removal need no$ he
permanent. Removed individuals may eventually reater the susceptible pool-=
when we rwmerge from a stay in the haspitd, or the prison, a the case may be.
Extending
Models capturing this are, for abvious remons, termed SIRS modets,@Ql
ls8f see Mmsfield (1C)Cil).See also Cam11i-Sfon;a and Feldman (1981).
f891~ee,
for example, Olmn and SchaEer (1990). For ai rigorous mthernatical definition of the
much-abud tern "chws," m, for sample, D m n q (1989).
[ ~ X principlplr?,
R
one could return Erom prison to the infative, rather than suseplible, pool, producing an Sf Rl wdel.
Kermwk-McKendriek, we obtain. the system
where y = aEo from the previous (political) interpretation.lQ1lWe are most interested in the positive equilibrium. Clearly, from (4.18), $ = 7/Pf and we can write
Since, in this model, population is m & m t at N,we may witc? R = N
so that the system, in $1-space, becomes
-S -I ,
Denoting by F the vector field (fi,f2), the Jaeobian at any equilibrium Z = ( S ,f)
At the positive (or interior) equilibrium of interest, S = ?/P, so that P$ - y = O
Clearly; the traee is negative and the determinmt is positive; so, as reviewed in
lecture 6 , this quitibriunn is stable. With T the trace and D the determinanc of
D F ( f ) , its eigendum are:
Tbe resal p;art;s are negative, but we m y fitwe nonzero imcxg~h~wry
pms, and spiral
convergence to the equilibrium, as shown in figure 4.2.
~ b i analysis
s
pwdiek MeMein-Kabet (1988,pp. 2 4 H 9 )
82
Nonlinear Dynamks, Mathematical Biology, and Swiai Scieme
FIGURE 4.2 Spiral Sink,
Source: Based on Edetstein-Kashctt (1988, p. 248).
The political interpretatbn of spiral eonvergrtnee w u l d be periodic outbursts that
atre reprmsed to it steady level of endemic positive discontent,.
Now s u p p o s e m a g e d a d e n experiment-that you are a Mwhiavellian a c i d
enginwr and want to ensure that revolutionslry ideologiw pose ~o thre& to the
establishd order. One ins"trument is indoctrination. The parallel. $0 inocula;tion
may be instructive.
HERD IMMUNIW AND DECENTRALIZED TOTALIITARIANISM
Staying with the same model, we see from (4.20) that for f----the endemic levelto be positive we must have N - $ > O which is to say N p l y > 1. With May,
we define & =. N p / r ta be tht? intrinsic reproduct;ive rate crf the disem.fg"
represents the averii?ge number of secondary infections caumd by irrtrodueing a
single infeeted individual inta a host, population of suseept;ibles.'"g31 The epidemic
thrmhold condition is thea simply .& > 1.
Ig21~aS"
(1983). &J is dsa termed the ""reproductive number."
1 9 3 f ~ d e i s t e i n -(2988,
~ a ~ p. 241).
Suppose, however, that we can vaccinate some frxtion, P, of the population
(still eanstaa at N),The fraction immunized should be big enough that the unimmunized fraction (1- PIN is belaw the3 threshold*We will then have achieved "herd
immunity." Procwding, we require
from which the required inoculation level is given by
For variaus d i s e ~ e swe
, have table 4.1,
TABLE 4.1 Immunization Levels P Required far Herd Immunity: Various Diseasea
Inlwtion
Whooping cough
German memltzls
Chicken pox
Diphtheria
Scarlet fever
Mumps
Location and Time
-%l
Approximate
Value of P(%)
Developing countries, before
global campaim
England and Wales, 195+68;
U.S., various p I a e ~ 1910-30
,
England and Wales 1942-50;
Maryland, U.S., l908-lY
England and WaXes, 1919;
Wmt Germany
"CT.S., various places, 1913-21
and. 1943
U.S., various plwes, 19Tte-47
U. S., variaus places, 1910-20
U.S., various plwes, 1912-16
and 1943
Holland, 1960: U.S,, 1955
If we now recover our politic& analogues, we have the required level of social
indoctrination given by
An in~rea~se
in domestic police (&) or in their eEectiveness (a)will allow the elite
to do b s indoctrination, while an inerem in j9 (the infectivenws of revoludionary
84
Nonlinear Dynamb, Mathem&ical Biology, and Smial Science
i d e ~ will
)
lequire increaed prapagandktic effort, all of which maka a wrtak
amount of sense. A somewhat darker reading is invited by the rearrangement
If indoctrination is very high, even the most compelling revolutionary critique will
pose no threa;t to the mtablbfind order because too few will li&en. Once herd
immunity is whieved, the system is really on autopilol; a type of deeentoalized
to talita~anismis possible.
Regarding another crucial variable, reprwsion, there we many subt1&ia that one
might wkh to capture, For
ple, in some situations, reprmsion has a deterrent,
or "chitling," eEect, whib in other sliluations, it simply infjtmeshostiliw toward the
regim. Indeed, revolutionaries have been knawn to provoke reprwsion with precimly this infiammahry aim. Morwver, a @ven population may gip &am one &'response made" to the other in the wake 0f pa&iculm incidents, With these thoughb
in mind, consider the faElwing extension, which was pleopomd by Jean-Pierre
Langlak.
The extensian consists in adding a tern, &RI,to the preceding; model. hpression (removal) has a deterrent effect if S > 0. In that ewe there is a Aow out of
the idective, and into the susceptible, pool, The idea is that the awareness of removah (vmishinl;s) has the @Be&of drMng some idectivm out of the revolutianq
movement. If 6 < Q, repremion has precisely the opposite eEmt. Histx>ricaUy3both
rmpome mod= are obsemed.
Ig4I~ccordingto Hoover and K w ~ l m k i reprmion
,
r d u c d dimnt by the anti-Nai movemnt in
Germny (emIy 29403), the human rights mwement in the former USSR (19?&198Cls), a d the
demoeray movement in Burma (late 198k). Here, 6 W= positive. b p r m i o n had the revem eEwt
of increwkg d m n t by the mti-Vichy movemen6 in France ( w l y 294Os) and the anti-ap&heid
movement in South Africa (late 11$809). In them E
G W= negative, 1x1Hower ancl KmaImki
f 1W2) is a revim of the literature and mferenm.
hgarding thresholds, we see from (4.27) that the revolution spreads
when the follawing con&tiaa is met
(i > 0 )
With 6 = 0, are reeover the clmic threshold condition (4.21, w&h aEo rts .g., just a
in (4.13). Xf 6 r 0, however, more susceptiblm are nwded; repressian deters so it is
harder to initiate the revolution. If 6 < 0, fewer susceptibles are required; repression simply fam the flames- Xn sucb case^;., a ruling elite is best advised to refrain
horn all action. Equivalently, revolutionari~in these situations should bend every
ego& to provoke brutality. The parameter, 6, is what; dbtinguisbes "a revolutionay
situatbd"am
others.
We can study the d m a m i c ~of this model iix $1 spwe. With a bed total
population of IV and k e d elite of Eo, let k = N - Eo. Then R = k - 5 1.
Substituting this into (4.26) and (4.271, we obtain
..-
In the S1 plane, there are two equiIibriags51: (k,O ) , and
Far illustrative parameter vdues, the first is a saddle, the second (4.33) is a spiral
This, and
sink, and the Wa are connected by a heterodinic orbit (sm le&ure 6).rg6J
other LrSt;jmfrQria,are shorn in figure 4.3.
1nterpret;ingthe heteraclinic orbit politically, the equilibrium (k,0) represents a
world of idwlo@cal purity-there are no r~valutionarim,But, that society is "ripe
for revolution;'' (k,0) is a sa-ddle, and heace unstabfe. The slightest subversive agitaLion (I@> 0) and the system will run to the spiral a.t;traetorof endemic diseonterrt,
, our param&er 6.
whose e x u t location depends, eetelris p a ~ b v s cm
fQ51~he
s~ond
of t k m was obtained symbaliedly by IWathemtica. See Wolhrn (1991).
isCil~he
wlues emplay& are: r = 0.0.1,ib: =.. 100, v -- 0.45, & =. 0.02, a -- 0.04,sad & ;= 20,
86
Mantinear Dynamics, Mathematical Biology, and Sorziai Science
FIGURE 4.3 Hetaroclinic Social Orbit
EXTENSIONS
In point of faet, oT course, the class of" in&vidu& subscribing to the elite ideology
is not constant in size (at Eo).There is recruitment of susceptibles into the elite in
addition to recruitment of susceptibles into the revolutionary class-a "struggle for
the hearts and minds" of the susceptibles. And, equally impoftant;, there is ofien
d i r s t eonfiicit between. the power elite and the infwtives; there h o\llright civil war,
A more fully elaborated syStem might take the form of an ecology in which taro
warring (see lecture 2) and diffusing predators feed on a prey species. The resulting
reaction-diffusion sy&em muId be complex mathematically, and well worth study
It wodd also be interesting to attempt a formulation of such a society usiw cellular
automata (perhaps as a generalized Greenberg-Hastings modeliQ7l)or agents.t981
CONCLUDING THOUGHTS
Clearly, social dynamics of Fundamental interest can be generated by simple models
af the sort I have advanced (wry tentatively) above. Explosive upheavals; revalu-
< p), or that begin to
tio-ons that fizzle for faek of a remptive population (e.g.,
spread but are reversed and crushed by an elite; longer-term cycles (undamped or
Lecture 4
damped) of revolutionary action; even endemic levels of weid diseontenl, and trweling wavm of revolution are aH ewily produced in nonfineizr models. T h ~ modeh
e
have the additional attractions of reBecting sensitivity to initial conditions
recently in Emkrn Europ and of unifying in a few variables diverse
totalitaria rule and rewlutianary aetion. Findly, wen rather subtle social ""bhreation poixrds" emerge, where repression abruptly changes from being stabuizing
(S > 0) to being inAammatary (6 < Q).
Dynamical, andogies are of theoretical value precisely in their pawer to illuminate such nonlinearities, to parsimoniously suggest generd conditions under which
explosim, dissipative, cyclical, even cbmtie social dynamics are likely, and in so doing, to foeus empirical attention on the parameters and relatiomhips that, in fact,
matter most. Indeed, without some theoretical framework, or model, it is often.
quikite unclear wh& we should try to memure!
This page intentionally left blank
LECTURE 5
A Theoretics Perspective on The Spread o
Drugs
This lecture explores another social process of considerable interat, the spread of
drugs, and is divided into three parts, In Part 1, a simple dynamic mod& of a drug
epidemic in an ideaEzed community is built up ham bmie wurrtptiom concerning
the interaction of subpopufations-pttshers, police, and no$-yet-ddicted residents
of the communiz;y.[g@~
The model combines demerits oT the epidemic, ecosystem,
combat, md tzrms rwe modeh dkcusmd above. Quilibria of the rmalting dynamical 8yskem are located md c1wified using took of f i x l e ~ i z dstability analysis,
Dajwtories are pfottd for a set of initial conditions. In Part II, a spatid-reactiondiEusion-vwiant is prewnted, Then in Part 111, supply, demand, md.price considera;tiom we intrrrducd; a;sen%iaf,and perlralps caunterhtuitive, relationships
19910b~0wgytpmieuIar dynsnnies dt?pend on: p&ieular drugs. No part;ieular drug is mentiand
here. We imgine em idealized drug thag is totdly md imevemibly ddictive after some smI1, but
hard to prdiet, number of us=.
90
hlsnlinear Dynamics, Ma;thern;lticaf Biola~y,and Social Science
betwen legalization, price, and crime are revealed. And in this light, the role of
education is discumd.
PART I, A DRUG EPIDEMIC MODEL,
We begin with definitions and a brief discussion of variables and parameters. At
any time, the populiltion is msumed to be divided into four disjoint groups,
S(t): The aonaddicted and susceptible population.
I ( t ) : The population of addicts, all of whom are assumed, in this simple model,
to be pushers, The variable f is used because this group plays a role thak
is mathematically analogow to the infective group in epidemiologyf a
pilraflel we shdf expbit.
L(1.1: The law enbcernent, or policc;, force, whose sole function is msumed to be
the arrest and removal of pwhers.
R(t): The arrest& and removed, or imprisoned, population. For this simple
model, removal is msurned to be permanent,
In ddition to these variables, a number of parameters are involved.
p:
p:
y:
a:
b:
The rate at which a corrtwt beween a pusher a ~ da, susceptible produces
a new addict/pusber (price dqendence is discussed in Part XI1 below),
The natural grodhi rate in the susceptible pool, ss youths come of age, sa,y;
The r&e at whi& a contxt between a pusher and a cop raults in removal
of the former.
The rate at which an inereme in pushers incream the growth, rate in
police. This variable reflects social a1arrn.['~~1
The economic damping to which the police g r a e h rate is subject.
All parameters are nonnegative real numberg.
Let us SW if we cannot arrive at a plausible model by remning from first
principles, noting conxleetions to related phenomena as we go. Pedagogically, the
exerche may illuminate the type of resaning thi3Lt o&en goes i a h the csnstruetion
of modelis in mathennatieal b i o f o ~ a, field which, ukinnatefy, subsumes the social
seiencw,
flooi~wicatllytproblem get mare attention when they impinp on the elite than when they are
confind to the ghetto. In a more realistic modd, therefore, a would depend on the meio-economic
inLa which drug abum, andlor the trim w o e i a t d with it, had spred. Were a is a, comtant.
As the simplest conceivable model, then, let us imagine t h ~ there
t
is no population $ro&h and no police force. At every time t , the populztlion is constant at N
and is the sum of suseeptibtes S($)and pushers I(t). That is,
How do 5 and I evolve? Well, for a susceptible to become an addict/pusher, he
or she rnust first come ilnto contact with a gusher, Reeognbing that real societies
are hekrogenwtxs and patchy? let us nonetheless follow the practice of theoretied
epidemiology and ecology and, as a first cut, msurne hornogenmus mixing of pushers md susceptibtes. The number of coxlt;ats is then taken to be S I . Of course,
only some fraction p of contwts produces new addicts. One may think of ,B as
the "ust say DO" "rameter. If P = 6, every susceptible says n0 and there is no
growth in the ddicted, or "infected," pogool. Xf = 1, the^ every cant;aet produces a
nevv addiet/pusher, On these very primitive assumptions, then, the A m out of the
susceptible pool and into the &dieted pool is h l y described by the equations
This system ki none other than the most bmic epidemic model, termed an "SP'
model since the fiaw is strictfy fmrn susceptible to infective, Now, by virtue of (5.1),
we ma;y write S = N - I, axld (5-3)becomes
whose solution is the well-known equation of logistic gro+h. The acfdicted popula,
tion inerewes until it equals the entire population; the "epidemic'bhips through
the whole of mciety.
In faet, there are some brakes on this process. Hewing to our assumption that
the drug is iHegal, there is some rate at which puskerladdicts are removed from
general cireulation. As a first refinement on our model, let us imagine a fixed police
force of size Lo. AS in the pusha-susceptibfe sphere, "Eaw of nnas action" "dynamics
are nssumed. There is homogenaus mMng of pushers and police, so that contacts
proceed sts Lo1. And, per contact, the removal rate is y.The idea, then, is that, as
before, susceptibles flow into the addictedlpushing pool at rate PSI. But, pushers
Aow out of circulation and into the '%removed" dclrass at rate yLoI, Since ?Lo is just
a constant, call it Q. Then we have the model:
glf?
Nonlinear Dynamics, Mathematical Blolqy, and Saciai Science
Students of mathematical epidedolaw will recognize "c W the ciwie
Kermack-MeKerrdrick SIR e p i d e ~ cmodd. It is a threshold model in that suse e p t i b l ~must exceed some minbum level in order for the infected, or addicted,
class to grow, Thb is strdghtfomard. To say the d d c t e d clws grows h to say that
which is to say that PSI
- 01 > 6, or: that
The ratio
is o&en termed the relative remaml rate of tbc3 infxtian. It fs the
werything evenepidemic thrmhofd. m i l e the infwtion u l t h a k l y dies out-ince
tually flaws into the removed compartment-it deerewes manotonically oltlly if
S < o/P. Otherwise, it enjoys a period of growth-the epidemic ph
dyjing out, as shown in &@re 5.X, in which p = B/@,
FIGURE 5.1 An SIR Epidemic Model
Now, from ($.S), f5.6), md (5,7)?it is evident that papul&ion b still consea&, ~ i n m
Of courR, population is not generally corntat. There are sa-cdld vitd dmamics,
bi&h and death.
As the n e obvious
~
refi~eme&on our efementary model, then, let us assume
n& "birdhs'hr entrmts of pS into the swceptibh cohort, where p if the per capita
g r a d h rate. N e d l m t o sayf logbtic rather t h m Maltbusim g r o d h k mather
pamibility. But, kmping mat@rs m s h p l e M possible, we then obtain the model:
Notice that (5.9) and (5.10) are the classic Loth-Volterra predator-prey model,
wiLh pushers as prdatars and not-yet-addicled susceptibles as p r q Predaf;ors
would die out (at rate -cl) were there no prey to feed on (at rate PSI); and
prey would flourish (at rate PS) were they not consumed (at rate PSI) by p r d a
tors. Aside from the origin, this system has as its equilibrium the point (S,i ) =
(@/P,p/@),whieh is a ~ e n t e r . fAs
~ ~shown
~ l in figure 5.2, the populations oscillate;
the arbits are closed cumes in ehe Sir phast? pIaae.
Popufations
Predators
0.5
1
1.5
2
Prey
Above, we saw thaut in the SIR e p i d e ~ cmod&, the infected ar addi&d elms
ultimately go= to zero, as aIf addicts are remwedt. Here-, we have Loth-Volterfa
d y a a ~ e sin whieh predators and prey cycle around an equilibrium. C d d it be
that the complete modd wit1 combine t
amp& md oseiXtat~ry-tendencies
in same way2 We will return to this que&ion.
llollksdeveloped in lecture 6 , at the equilibrium, the eigenvalues of the Jaeobian of (5.9)-(5.10)
we i m a g h q . The quitibrium i;s nonhyperbolic and Iiaewizd stabifiv andysb dcm not apply.
%hequilibsiurn its a cent= or a hew; and b w u e the
But b e c a w the eigendua are ima&
system admits a Bmilts~lianformulation (mf&ure 41, Lhe wuilibrum. is B center or a saddle,
Hence, It h a mntm.
94
Nanlinsair Dynamics, Mathematical Biology, and Social Science
THE ARMS RACE COMPONENT
Above, we defined as the product ?.Lo where L. was some f i e d level of law
enforcement or police. But, the police force is not necessarily constant. So, let us
relax this assumption. What, to a first order, would determine the size of the police
force7 Well, if no one cares about the level of addiction in society, i ( t ) ,there will
not be any groMh. Thinking of the p a a m a e r a izs a coefieient of societitl &arm,
we might posit that, without any economic damping, the police force should grow
as a I , But, as in arms race modeling, id is remanable to wsume some eeanonnie
fatigue or damping, under which mtes of growth decf ine the Iarger is the military
establishment. ff the damping coeBeient is Et, then the police growth rate is given
by h = u l - bL, just as in the Richardson arms race model of lecture 3, and the
complete model is ais follaw~:
df
psr
dt
==:
-T ~ L ,
Notice that the term -?]L, in (5.13) is a pusha attrition rate reminiscent of a
Lanchwter combat model presented in lecture 2, so this dynamied system combines
elements of the epidemic, ecosystem, arms race, and combat models developed in
precediw ieeturw, Before engaging in a linearized stability maty.sis of this dynamicd system, let us briefly trme through the eEect if, horn some time, everyone says
no; t h d is, if p = 0. C/early, sin= P --- 0, the growth rirte in the ELd&ct;ed/gusher
pool in (5-13) is strictXy neg;ative; this entire group is eventuatly rmoved. That
being the ease, (5.15) reduees to
and the police force, too, "ithers away9"a rexonable qualitative result since the
apprehension of pushers is their sole ELxnction in this model. Xn the end, we have a
poliwless society of nonaddicts and a removed populat;ion of former pushrs. This
little thought experiment completed, let us bring to bear some more powerhl toots.
LfMEARlZED STABILITY ANALVStS
Assuming all parameters to be pasitive in (5.12)-(5.15), what are the nontrivial
equitibria of the system, the nonzero population levels where all derivatives are
zero? We really care only about S, I, and L, and a bit of algebra quickly leads to
the unique pasitive equilibrium:
Evaluated at this equilibrium (call it $1, the Jacubian matrix of (5.12)-(5.141,
which ecologists term the community matrix, is given by
The eigendues are solutions to the third-order ch~rl"acteristic
equation
Det(J@)
- Aid) = 0,
(5.18)
where id is the identity matrix. Expanding, this characteristic equation is
Equilibrium is stable if and only if the roots Xi of this equation have Re(Ai) < 0.
For a third-degree equation,
the RouthHurwitz necessary and sufficient eondition~l~~~l
for %(X) < 0 are
a1
>Ota3 > 0, and alaz -aa
>0.
(5.20)
The first two of these are obviously satisfied by (5.19)) and so is the third, since
Therefore, the positive equilibrium in (5.16) is stable. There is an endemic level of
addiction.
Earlier, I raised the question whether our simple made1 might somehow manifest
both the damped behaviar of the Kermwk-McKendrick SIR epidemic model and
fla2lSee,for example, M m y (1989,pp. 702-04), or May (1974,p. 196).
96
Nonlinear Dynamics, Mathsmllltkal Bislwy; and Soeial Seieme
the cyclical behavior of the LotkaVolterra predator-prey model, each of which is
a special case of (5.12)-(5.15). A canonical bebavior combining these would be a
spiral approach to our positive equilibrium. And this is precisely the behavior we
have, as shown in figure 5.3, which
- - offers
a small gallery of phase portraits. Here,
the equilibrium happens to be (S,I , L ) = (108,2, 12).11"l
FIGURE 5.3 Drug Modet Orbe and Sotutions.
Mote: Here z is our S, g is our X, and x is our .L.
A natural extension is to add spaee. As demonstrated earlier, t h k can be Wcomplished by appendinef;vmious dzusion terms to an underlying dynamic model,
yieldinpf a sa-called remtion-diffwion system. One such generalization is offered
next.
PART !I. DRUG WAR ON MAN STREET": A NONLfNEAR
REACTION-DIFFUSfON MODEL
The population is cornprig& of three subgroups, who~enumbers and spatial digtributioxls evoke over time. We ima@ne th& event8 unfold an on+dimensional
inkrval-it "'street." h
t us define S(;e,tf, l(%,t ) , and L(z,t ) as the susceptible,
infecdive, and law enfurcennent levels at; &rwt posilion ;e sbt time t.g1WJ Denotinf:
these functiom (of z and t ) simply as S, 6, and L, the generalized quations are as
follows:
bS
d25
== -PSI 4- p S 3- &ss-,
at
O%Z
Ig~oringd l diffusion and cross-dift"usion terns, the first two equat;ions are exwtly
as before, The third equation has been refind slghtty. The Ricfiardsoxlian damping term (-bL) is retained, but the first expression is now ESIL rather than the
previous aI. The idea, recall, is that the poliee force grows with the level of mcietal
alarm at the drug problem itselE This level of dmm is assumed to be a function
of arrests of which the (tax-payiag and police-buying) susceptibies are awue. Under our normal msumpti;an, the arrmts are proportional to PI;, and suxeptible
awareness grows with exposuxe to these arrests, hence further multiplication by S,
yielding the overall term [SPL,[1051These, then, are the reaction k-i~eticsin the
reation-diEusian system (5.21).
n m i n g to the diEwiw process@, the simplest is the susceptible c m . Here,
the term bss(a2S/ax2)is added, as in the models of the previous chapter, indicating that the susceptibles-while int;ersctim with other groups-diBust3, Analowus
and police
diffusion terms appear in the equations for infectives (611(a21/a22))
(6LL(a2L / a x 2 ) ) .All t h e ~ ediffwivities (6,) are positive. However, the infective and
police equations are more complex than this, In the paliee equation, there is dso
Pa4f ere, we will not drmk the remavd
(i.e., ammted) group c?xplicitly.
[ l D S 1 ~mume
e
again that the exposurw occur through homogenmus mixing, or mkinetics.
atian
98
Nonlinear Dynamics, Mathematical Biology, and Social Science
a cross-diffusion term (-61L(a21/a22)),
indicating that police diffuse toward infective eanwntrisrtioions; they @&gage
in " " c i m + t ~ s , "if you will. In turn, infectiws
(i.e., pushers) cross-diffuse in the direction of susceptibles (-6sr(82S/a~Z)), and
cross-diffuse away from police (6Lr(a2L/@z2)).I further m u m e that fiLJ > bsl:
a pusher would rathm amid arrest than convert a susceptible to a new drug user.
With all constants
the assignment; of initial spatial distributions for the
subpopulations is all that remains to specilFy the model.
Imagine, then, that everything transpires on a. &rwt 12 blocks long. At time
zero, the susceptibles occupy the middle four blocb, and we 1000 strong at every
point, Up at blacks 8-12 are the infectives, initidly numbering but 100 ,zt; each
point. And wagt down at block 1-3 are the cops, initially a-t t o k n levels of 25
per poht, We trwk the spatial. evolurtion of each a o u p over 50 time intervals in
fiwre 5.4.
FIGURE 5.4 Drug War Reaction-Diffusion Model
Tbe susceptible, infective, and police evolut;io~sare shovvn in the left, middle,
and right graphs, respectively.[l071 At to the levels and positions are as noted above.
Haw do things evolve?
A SPATIO-TEMPORAL STORY
Seeing that there is a large concentralion of susceptibles and few cops d a m the
s t r e t frarn them, the infectives cross-di&se to the center, Many susceptible^ we
conve&ed into ixzfectives, so the smcept'xble populat;ion falk and the infective one
rises, now swelliq nrith "converts" "into the micldle blocks. This buleng problem,
["l~)~ere,theva-tucr?s@re:p
= 0.005,p=0.5,7=0.03,t = 0.0001, b = l.O,bss =0.03,61f =0.01,
= 0.02, liLz= 0.006, Ss1 = 0.001, lilt = 0.006.
ifa7)~hese
were genaated in lGiathematim (Wolfram, 1991) using the Numerical Method of Lincs.
X thank &be&
Axtell for his misLance.
Lecture S
FIGURE 5.5 Drug War Reaction-DMusian Model: Overhead View
however, inspires a reaction, in the h r m sf dramatic inerernes in, police, who crossdiffuse from their initial barrwk at the end of the strmt into the hem%of the
problem in the center, This surge in poEic
vident; in the peak of the rightmast
graph-literally scoops away the ideetive mamd. By t = 40, there is hardly a
problem. Hence, m before, the polke "ither awaySb&er%h& p a i ~ t lewing
,
the
suseeptiblets to coxlt;inue in their untroubled dausion, as shown.
An owerhead view of the same pr~cessis oEered in figure 5.5. Here, the higher
the numbers at a point, Ghe lighter the s h d e . We c m clearZy see the surge of
pushers, followed by the police response, the hollowing-out af the pusher mound in
the center, a ~ the
d withering lzsva~yof the palice.
A nonlinear reaedion-digusion model allows us to generate a plausible spadiotemporal story of basic interest;.i1081
i l o s l ~ y point here is tZI& there exist parameter wXuw and initial conditions under which the
spadio-temporal stow emerges, A aperate study would marnine the robustnw of this result
under a wide range of pasameter and initid valua.
3 oo
Nanlinaar Dynamics, MathematicalBiology, and Social Science
PART If!. EMSTtCXTIES AND THE
DEMAND
COMPONENTS OF
Qonomkts will have recognizd one glaring oversight in the models thus far
developd-the price of the dug is not repraentd* Clearly, P-the rate at which
a contmt b e w e n a pusher and ai. not-yet-addieted susceptible raults in a use-z~
likelihood
s , that som~fonewho is not yet a&
w e n & on price. GetePis w ~ ~the
d i c k d wiU ""just say no'bhould rie;e (P should fall) with price. So, for that group
we -urn@
@ ' ( P )< 0 .
(5.22)
I f we hfdbw w u m e price to be determined by supply and. demand, then, in prineipie, the effect of drug interdiction operations on h s t uses of the drug can be gauged,
as shown in figure 5.6. Interdiction shifts supply in (further discussion below), inc r e a a the equilibrium prim, reducm @, and in principb changes the q u i l i b r h
level of drug consumption (from Q0 to Q').
By contrwt, wiLhin the addicted population, the quantity deatnnded k not
sensitive to price; the demmd came i~ a verticd fine, as shown in figure 5.7.
FIGURE 5.6 Interdiction Increasing Price
Lecture 5
FIGURE 5,"irddkted Demand
price
effect on the
Zrt sueh a market, supply interdiction merely incm
equilibrium drug quantity. The result is simply more street crime, as addicts do
whatever is necessary (e.g., rob and loot) to come up with the price.
Now, figure 5.6 assumes the market to be eomprM only of not-yebaddicted
susceptibles, while figure 5.7 arsumes it to be comprised only of addicts. In fact,
both groups are present. These two components of total demand are represented in
figure 5.8,
FIGURE 5.8 A Drw Wlai&et
*I Q2
Nonlinear Dynamiw, Mathematicat Biology, and Swial Science
The diagram encompnsses an addicted pool, with its dislinmislhing feature,
a vertical demand curve (%)oddiets), represeneing complete inelasticity of demand
with respect to price. The addicted group h= ""f~tto have" QQ,and that" that;.
The Da,, cuwe represeds the nonaddieted population, with its downward-sloping
demand curve.. Here, price does aEect the demand by potential first users of our
drug, Total demand (&,tal) is the horizontal sum of these two components. Finally,
we imagine SO to represent the current supply curve in this economy. Then the
equilibrium price and quantity in the market is (p,
QO).
Let; us use this fiamwork to campare the short-term eEwt of two supplyoriented policies, returning later to the longer-term hsue of demand adjustment,
The twa policies are, of course, interdictionlprohibition and legaXization,t"~l Assuming (perhaps generously) that interdiction is techniedfy femible, it s h a s s u p
ply inward to S" Legalization---whose aim is to drive, underbid, the cartels out of
business-would shift supply outward to S&.
First, under interdiction the equilibrium price rises to P< Since addicts have "got to
have it," they will do wh&ever is xlecasary to raise the money required to support,
their habits, and we can expect an inereme in strwt crime. Xn the normaddicted
group, the quantity demanded falls as price increases. So, interdiction produces
fewer first users but mare c.rime.
Legalization shifts supply out to SL. The equilibrium price falls to P&,so addicts
need less money than they did before (at So) to satisfy their habits; not all of the
erirnc; in which they had been engaged would now be necessary, and we would expect
to SW street crinze deerese, New users, however, may be expected to increm (and
to increase their consumplion) as f?xperirnentationbecomes cheaper (and, ohiously
less risky legally) under legalization. So, this policy produces less cT.i;nze, but more
new users. All of this is enicapsulated in the purely heuristic diapam in figure 5.9.
The debate is, predictably; wide open becaua no oxrc3 real& h a w s much about
this cuwe, about the factors underlying it-price elwticities, propensities to engage
in crime, eEectiveness of interdiction, and so o
r about the social welfare do be
associatd with diRerent points on the curve. As a result, the most fundamental
issue is open: Do we want d m g prices to .rise dramaticall@or fall dramaticallg?
But, at least the analysis reveals those fators about which advocates of the
diEerent policies are, in fact, disapeeing.i1lo"l modest claim, to be sure.
f"@l~ora thorough historicat dimumian and policy analysb, see Star= (1996).
i1"1 ""Just swing no" rduem crime indirwtly bsause DLOLal
shi&li left, rdueing the ~?quiIibrium
price, and hence the incentive to engage in ~trmustcrime,
lecture 5
FIGURE 5.9 The tsgatization-Interdiction Trade-off
EDUCATION AND P FOR THE NONADDICTED POPUUTIQN
further point concerns demand among the nonddicted susceptibles, where the
likelihood that; somwne will "ust say no" probably depends on prices. While the
incorporation of supply; demand, and price into the model dlows us to connect,
through P, the "epidemic dynmics" "veloped earlier to the alternative policies,
maMng P a function salelg of price is not completely satis$ing, for the following
reascln: Imagine an a11 out, %wefront"war on drugs" that simultaneously cut supply
through interdiction and cut demand (here in the new-use senm) through education.
The equilibrium price could be left unaffected. If P were a ftmetion solely of price,
then it, too, would be unaffected, and the drug epidemic dpamics would be exactly
M they were before, which swrns implausible. One would therefore want ,& to depend
on price and education ( E ) in some (perhaps pasametric) way such that
A,
Many "by" "candidates for @(P,E), could be constructed h r sinnulation purposes.
But, basic empirical, work is essential here, W elsewhere. Indmd, it would =em
that educatLian Is the prime way to avoid a very disturbing long-run scenario under
legalization.
IQ4
f\loniinear Dynamics, Ma%i~?matical
~ioiogy,ancl Sociat Science
LEGALIUTfOPI, EDUCATION, AND THE LONG RUM
Speescally; tz potentially mrious problem is that with repeated use over t h e , the
price elasticity of demand for individuals initially in the nonaddicted pool will go
toward zero: they may bwome ddicds.
FIGURE 5.10 Legalization and Lang-Run Price
In turn, we c m expat inerewing levels of ddictim (a) to inerewe the total
qumtiey demanded at any price and (b) to reduce the price elastic@ of aggregate
demand. The h$ddemand crxwe, thus, would be expected to shift righward and
bemme snom vedical, W shown in figure 5.10. Counterintuitivelty, in, this case the
equitibrium price could rise despite increased supply. That is, we could have (PL>
PO),as shown. Street crime, driven by rising drug prices, could in principle actually
inereme under legalizadian! The magnitude and direetion of price changw, it mmt
be ernphwhed, would b e p a d on. the s u p p l p r ~ p a mdo iegalization, the rate at
which new users become addicts, and sa far(;h,
But, the obviow way do undercut %histwt: of watution-whose liklihaod X
do no$ cldm to &iurr&t is dueation- The aim of education, in a dwhnieal eeonorxzic seme, muXd be to "flatten" the individuaik iindiEerence cuwes ( b e ~ w n
drug wn~umptionand engagement in: other forms of rtrj~reation)such that there
is no pokt of tangency betwen the indigerenee curve and the budget comtraint,
'Leaving only the corner solution, "just say no," as fewiibe.tlflf The production of
horizontal indifference curve8 for fur coats is the d m of animal rights wtivists, for
exmgle; there is then na pair of positive prices-for real arrd imitation hr-at
which the indigerence curve is tangent to the individud" budget constraint, and
he or she "use SW 110' t;6 fur.lllzJ
In, any event, it swms char that, if Iegalizatio~is to avoid the long-run problem
s u e d above, it may be necessav tO increase educalclon, vefy substa~tiafly
It should be emplrwized that;, in addition to all sorts of implicit assumptians concerning the? factors dhcumd above, positions an legalization reiimt hndmental
a-titieudes on the appropriate role of government in. remtating individual: choicm
generally, a crucial question not addressed here.
Obviousl;y, this thwretical exercise does not purport to resolve any policy issue. R,stther, iil;
to focus the debate by helping to identify the empirical imuw
that d w w e highest priority, by encouraging aplicitness in the statement; of assumptions, and by oEering very prelhinary, but testable, made18 of the dynamic
iXlf;erwtionof care variabl?~.
Finally, fi-om a purdy irntellectuaj standpaint, the discussion sugesds the relevance to social science of scrtemingly distant areas like matlkematicaj, epidemiiolom
and eeasystem modeling. And it tries to illustrate how simple wnjlinear models are
built and axzalyzed.
E1l1I'rhf: id= that the individual will comume where the indifference curve and blidget con~traint;
axe tangent is develop4 as follows. W inragine that an iirtdividud berivm utility from the consumption of quantity qd of drugs airtd quantity q3 of some alternative commodity, and that total
utility is a function (with all nice behaviors) of thaw, %(gd, qS). On an indiEerenee, or isoutility,
= 0,memkg that the slope of the indBerence
came, dzl r 0,m that ( i ) (au/aqd)dqd3-(du/aq3)dq,
cume in (Q, qS) spwe is given by (ii) (&d/&3)
= -(@zl/bq3)[(bu/@qd).
But i f the individual mmi m i m utility subject to a budget constraint B = Pdqd P,qS, then a t the constrained
( i i ) must equal: the price ratio, making the indiBerence curve tangent to the budget constraint. To
see this, form the Lapangian for the comtrdnsd problem: L =; %(qr,qz)- X j B - &(ld - F)3q3).
The first-order conditions far a maximum being (aL;/Bgd) == ( a f ; / d q 3 =
) 0, we immdiately obtain that f&/aqd) = X& and (au/i)gj)= XP,, making the right-hand side of (ii) equal the price
ratio on division. Qbviousliy, for the tangency point to be unique, the indiEerenee c u r v a must be
strictly c onva, which is among the tzlthaviora of a nice u-function.
il1211ndeed,one might go farther and argue that the goal is to ensurethrough education--that
indifference c u r v a for harmful commodities are positively sIoped. In other words, people wouId
have to be compensated with higher q u a n t i t i ~of some other good t o induce them t o consume
the harmful commodity I thank Steven McCmo11 for this idea.
+
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LECTURE 6
ntroductionto Non inear Dynamical
Systems
In this lecture, I want to collect some bmic, and very powerful, results from
the quditatiw thmry of nonlinear autonomous diEertmlia1 dynamical systems, primarily in the plane. In a field as vast as nanliaest-r dynamics, any essay of the
prest?nt len@h must be selective. In this ease, the story begins with liaearized &ability andysis for hyperbolic equilibria and proceds t o develop s o w diagnostic
tools for nonhyperbolic ernes (including the use of polar coordinattes, Lyapunov
Eunctions, and Hmiltonian formulations). The distinctly nonlinear fienomenon
of the limit cyde is then discussed and Hilbertk-&ill unsolved-16th Problem
is stated. The Poincar&Bendkson and Hopf Bifurcation Theorems are presented,
as well as an introduction to Paincar6 maps, which beautifully connect the world
of continuous system (Bows) to that of discrete systems (maps), Tools for precluding periodic orbits-the Bend
n and Bendhan-Dulw negative tests-are
then pre8ent;ed and applied to a Kolmogorov system which, naturally, proedes a
forum for Kolmogorov's Thearem on cycks-. Powerhi as they are, none of these
methods give much insip;ht concerning k m the local equilibria and limit eyelw fit
together glabdly-in the phase plane as a whole. Irrdrirx theory penetrates dmply
into this quest;ion, to reveal topological "con~ervationlaws" of great interest. 1
present some of the hadamenLa1 rwults in this area, and give an index theoretic
proof s f Brouwerk famous fixed poixtt dhmrern on the disk. Extending these ideas
108
Nonlinear mnamics, Mathematical Birolwy, and Social Science
ta closed surfaem (tw-manifolds) like the sphere ibnd torus, the lecture conclude8
with an informal pre~mkationof the m1ebrat;ed Pairreclr&Napf Index Theorem from
diRerentiaf.t a p o l o ~ ~
NONLINEAR AUTONOMOUS PUNAR SYSTEMS
We cornider the nonlinear autonomous systernf11131
where fi and fi are C' on 'R2. The system (6.1) defines a vector field F =
( f 1 ( 2 1 , 2 2 ) ~ f 2 ( 2 1 , ~ 2 ) ) fmm R2 to R2.Recall that the Jacobian of F at a point
In the linear homogeneous ease where F(%)=: Az,DF(z) iS just A. One major
difXIerence bewwn linear and nonlinear systenns in the plaae is that the latter may
i1'31~uch of the orbit theoq dmelopd below r w b on the wumption that the nonlinw firstorder initial value problemAg/dz = f ( z , g), f continuous on SZ = fa,bj x [c,dj, y(zo) = go-hw
a unique mlution y, defined and continuous on a cl
subinternal of [a,bj containing a;.a, A basic
thwrem is that if f is Lipf~chitzon R, such is the case. A full dkcussion wouXd rquire development
of metric s p s e raults quite foreign to the rest of this may, notably (a) that the space cfa,61 of
functions continuous on an intern1 fa,b]--the space that would contain m y mlutlon-is complete
with metric
p(s, v> = m m
LEIa,Y
- y(gZf;
and (b) Banachk thwrem that a eontrmtion mapping on a complete m&rie space
a unique
f i x 4 point;. T h m theorem in hand, however, aistenceuniquenm is direst, Fkst, one obmmm
that y is the unique solution to the initial value probliclsm above if m d only if it is the unique fixed
point of the i n t e p d operator, P : e[a,bf
efa,b] defined by
The proof then consista in eablishing simply that if / is Lipchitz; on Q, then P (for Picturd)
is a cantrm%ionon e[a,bj. As Kreyazig puts it, '"the idea of the approach is quite simple: [the
iaitiai wjue probbmf will be eonve&d to an integral equation which definm a mpping T , irnd
the eonditione of the thmrem will imply that 2" is a contraction such that its fixed point becomm
the solution af our problem." Our P is just Kreyssig's %. For detailed proofs of dl the many
s u b s i d i q claim involved, me, for aaxnpb, Krqszig (1978), Naylor and Sell (1982), Waltman
(1986),Groetseh (19801, Gamelin and Grmne (1983), or Marden (39741,
have multiple equilibria whereas the linear homogenmus system & == A=c hats the
origin as its unique equilibrium.
Suppom no-rv that; = (21, z2)is an ewilibrium of the nonlinear systtem (6.1).
Assuming fr and fi have partial derivativm of atf arders on an open set containing
2, W can. expand ewh in a Taylar serim about 3. Retaining only the linear termg,
we obtain the so-called linear variational equations:
Since (Zl,Zz) is an equilibrium of (6.1), we have fi(51,Z2) = f2(z1,z2) = 0.
Defining the deviations yl = (sl- f 1) and yz =t (z2- z ~ ) (6.3)
,
becomes
the familiar linear hornogenmus problern. If, far apositowy ease, we assume D F ( z )
to have disCinet real namero eigenvalucts, Xi and Xz, with (perhrce linearly independent) eigenvectors vl and e,the general solution of Ctj.4) is
Given the vector initial condition y(0) = 90, we can determine the c-va1ur;s; indwd,
with the matrisr exponential, we abfain the succinct form:
Uou will doubtless recall from previous tnrork that if both eigenvalues are negative reds, or have negative real pmt in the cornflex (conjugate) c=, then the
origin is a globally ~ y a n p t o t i c d ystable equilibrium sf (6.4).ffi4If both eigenvalues are, in fact, negative (positive) reals, then the origin is a stable (unstable) node.
fn the complex ewe, if real parts are negative fpositivct),then the origin is a stable
(unstable) spiraL It is a center if the eigenvalum are purely imaginary, and a saddle
if they ase real with XI < 0 < Xz. All of this is summarized in figure 6.1, whose
axes are y = Tr[DF(%)jand P = Det[DF(%)].
You daubtlws dsct reed1 that there are the fu&her, repeat& eigenvalue, cases
wbere issues of multiglicity mise,~ll~~
As an exercise, you might e@oy &monstrating
f11410nstability and wymptotk &ability, see Him& and Smale (1974; Swtion 9.2).
[f"bm for example, Hirwh ;and S m b (X9'14), Waltman (1986), Braun (19831, or Borelfi and
Golemibn (198'1).
f"M3
Nonlinear Dynamics, MathematicalBiology, and Social aienw
fornndly a bmdy fact that emerges from this figure, nmely that the origin is stable
if the B w e is negative and the Determinant is po~litive.
FIGURE @
Global
,I Stability of the Origin In the Linear Casa (Distinct Nonirero
Eigenvafues)
Source: Bassd on Edetstein-ashat (1988, p. 190).
Of course, all of this hasa to do with (5,4), when we are re;tiIy interated in
(6.1). If, somehow, we knew that the behevior of the linearhation (6.4) faithfully
repremnled the behavbr of (6.2) in the vicinity of Z,then we could conclude from
the gEaBal stabil* of (6.4) at the ori@n,the bomb stability of (6. l) E&I.Xndwd, at;
e x h equilibrium Z of (6.1), we could siimply inspect; the eigendum of the Jaeobian
D F ( z ) and classify exactly as in the linear case. Amazingly, for s certain class of
equilibria we can do just th&! With th& thowgh in mind, we m a b the follotving
definition.
Definition. An equilibrium 2 of the, (linear or nonlinear) system 2 =. F(%)is hyperbatie if and only if nll eigenfticlua of the Jwobian evaluated at, 2, DF(je), have
nonzero real parts.
Now id is a very useful fw&th& for hmerbofic equilibria, finea;riza,t;ionis refhble.
Zf f is a hyperbolic equilibrium of (tj.l),then its type (mde, focus, saddle) ltnd its
stability cormspond exactly to the eype and stability of the zero equilibrium of (6.41,
its local linearkation. This is a consequence of the Hartman-Grobman Theorem.
Theorem 1 (Hartman-Grobman). If 3 is a hyperbolic equilibrium of j. = F(%),
ithen there is a neighborhaod o f 2 ia zuhieh F is tol~ologicalkyequivalent to the linear
vector field f = DF(%)z,
Here, two dynamical sy&ems $ = f (X) and i -. $(S), degned on open sets U
and V of R~ are topologically equivalent if there exists a homeomorphism (a oneto-one conlinuous mapping with continuous inverse) h : U -4 V mapping the orbits
of f onto those of g md preserving direction in time.
This gives us a "recipe," if you will, for local stability analysis of hyperbolic
equilibria of nonlinesr systems: For eaeh such equilibrium 2, compute the eigenvalues of the Jacobim, DFCZ), and classi& as you would classify the origin in the
linear system i TL. Dlj-((g)z.Neighborhod stability analysis, as the technique is
a h know, is among the most ubiquitous tools in all of applied mathematics, It
appears frequently in, physics and enginering and is a cornerstone of mathematical
e c o l o ~ epidemiology,
,
and population pnetics. Linearized stability analysis also
pemades important subfields af miJ1Chematical economics, ev~lutianarygame the-.
ory, and other meas of social science, including the thwry of arms racw, wars, and
revolutions, which topics I discuss in other lectures. Precisely because examples are
so abundant in the literature, X thought a "nonstock" application would be nice.
A particulasly ingenious one arises in connection with traveling wave solutions to
reaction-diEusion equations, Since these figure in two other lectures of mine (on
revolutions and drugs), X thought ""kill two birds with one stone.""
TRAVELING WAVES, HETEROCLINIC ORBITS, AND
LINEARIZED STABILIW ANALYSIS,
Among the m&famous reaction-diffusion equations is Fisher's (1937) equation
governing the spread of an advantageous gene in a populdion. "With D a difision
constant, a a parameter memuring the intensity of mlcctiazr, and p(%,t) the g;ene9s
frequency at paint IC at time t , Fisher's equation is
W wish t o e&&lish whether f 6.7) admits traveling wave solutions consistent with
bioXogically realistic assumptions. We posit travelirrg wave solutions of the forxm[1161
P(x) = p($, 1) with z
==
z -d .
f l l ( E I ~ h diSCuw60~
is
paralXels Melstein-Kfsshet (1988).See &so Mumay (1989).
'l32
Plfanlinaat Dynamics, Mathematical Biolof,ly, and Social Science
By the chain rule,
BP -a ~ - aap ~so (6.7) becomes
t t ~a~ a~
a nonlinear m a d - o r d e r o r d i n ~ ydigefential equation. Defining -S
r a c e (6.8) sls the sysLern
==
U/&,
we
Notice that s uaw plszys the role normaJiy play& by tt so that, in phatse slpwe, z
is changing dong-is parmetrizing-traj~toriw m d eau w ~ u m e(ZXl real valuw.
By coatrmt, we place some definite conditions on P . First, since P is the relative
kequency of our advantageous gene, we require that it be bounded:
(i) O < P ( z ) < 1f o r a f l z , ( - o o < z < + o o ) .
Second, when we say the gene is advmtagwus, we mem that it will ultimately
dominate the pool. In &her wads,
(ii) P ( z ) -, 1 as z
-+ -oo(since
t
-+
m).
Symmetrically, and hdEy, we m u m e that
=
How, the dynamical system (6,9) has two equilibria in Lhtt PS plane:
z2
=
(h,%)
=
0,o).
T~US,
P
=
:
o
%e
2
,
and
P
=
I
a-t
z2.
(p1,s1)
= (o,a)
Hence, if we rzre to satis@ conditions (ii) and (iii), we umd an orbit connecting these
equillbria. Such orbits are termed hetemlinie. Technically, a heterocbnic orbit is
one whose a and u limit sets (defined below) are distinct equilibrium points,
Two centers clearly cannot be heteroefinie. NeiLkrer can a pair of s i n k (aterae
tors) or a pair of sources (repetlors). h t h e r , we nwd one unstable and a ~ stable
equilibrium, so %fiat%het r a j e c b ~ yemanating born one may be attrwted to the
ogher.
We will h a w that Fkherk equation (6.7) d m i t s a %ravelingwave solution p r e
cisely if we can show that; (6.9) admiLs a cert;ain hetemclinic orbit. Enter linearizd
stablity a~alysis!
The Jwabian of (6.9) is
and at
32,
Now, the eigenvalues in each case are hncdions of the parameters a, c, a d D. One
can. e ~ i l yshow, by computing the eigenvalues, that if
(6.m)
the origin, gl,is a &&h node (XI, Xz < 0) and 22 is a saddle goink ((X1 < O < Xz),
which is just the sort af configuration we need. Thme equilibria are indeed connected
by the desired hetemclinic orbit, shoarn in 6 p e 6.2.1113
FIGURE 6.2 Weteroclinic Orbit
one u ~ t & l eequif1171~obe precise, t h e f& that the system (6,9)hw exatly one stabb
librium does not alone atabIish that the two are connect& by a heteraclinic orbit, The unstable
equilibrium could be sumoundd by a, mmi-stable limit cycle (see below). Xn fmt, for the system
(16.9),ttrk c m Be ruld out by BendiarsonJsnegative test (sec? below).
"l4
Nontinear Dynamics, Marthamaticticaf Bi~togy,and Social Sciiencs
Findly, interpreing (6.10), we have dsa learned that; the travelillg wave" spwd
a very nice byproduct of the analysh, This value
is boundcld below by 2
is a ""bifurcation" paint (we discussion belw) alw, in that the phase portraits
are not topologically orbitally equivalent; for
orbit.
There are a a m b e r of things to relish in this example, quik wide from the
mathematical propagating wave iLself. The reiilsoning, originally due to F ~ h e rKol,
mogorov, and others, is a mawel of indireclion. We sta& with a nonlinear partial
diEerentiaI equation which we never solve; we posit a traveling wave solution whose
substitution into the original equation converts it into a nonlinear system of ordinary differential quations, which we also never solve. Rather, we make a small
number of remned wsuntptiom about the bounds and wppkotic behavior of P,
deduce $)h& there mwt be a heteroclinic orbit, and use fhearized stability analysis to establish the requisite parametrsr rdationship, which happens to provide the
minimum wave s p d as a byproduct!
~
the more general qu&ion
For a fulfctr account aE pomibili t i under
SW Fife
(1979), Britton (1986), and Smaller (1983).
Finally, befare l e a ~ a this
g exmple, notice how the nonlinectrity sf f is emential
to the $raveling wave sdution. (For in~tance,if yau aga-in b@n by positing such
a wave solution aad try to carry through the same derivation as above, but BOW
with f linear, you will obtain a Inear analowe of the system (6.9), with a, single
equilibrium and, hence, no prosptxt of a heteroclinie orbit.)
Returning to the main plot, however, the central point is that linewization is
an extremely powerhl toof whom applications are really quite far Bung, as this
example suggats. PowerEul ens it is, linearized &ability andysis is not omnipotent.
Xf it were, nonlinear dynamics would be a pretty small field.
WHEN LtMEARIZATIQN FA1LS: SOME NQNHVPERISQLIC
EQUlLIBRIA
As m exmple of a failure of lineasizatian, cornider the syskrnil1B~
Lecture 6
What is the aart;ure of the zero equilibrium? Applying the recipe, we compute the
Jwabim matrix and evduate at; 2 = 0 to yield:
The linearized systtem at 2 is thus tht: classk harmonic oscillator fmiliar from
elementary p h ~ i c s : =: 22;k2 =. - X I . The charwteristie polynomid is
whose solutions w e the purely ima@nary eigenvaluw Xr,z == f i.
Since Lbe eigeavdues have zero real parts, they are ozo&yperbolic. Now,in tbe
linear case, imaginary eigenvdues would indicate a center, neutral stability (like
the harmonic oscillator). Is that true h e ?
Xf we convert to polar coordinates, the system (6.12) becomes
with equifibriurn p = 0. Clearly f is gloM1y ~ p p t o t i c a l l i ystable if a < 0, neutrally
stable (a center) if a = 0, and unstable if a > 0,as indicated in figure 6.3.
Clearly, then, as a generd propositiort, linearkation is not rdiable for noahmerbolic fixed points. However, for critical points with purely innaginay (as agaimt,
foir exampb, zero) eigeavalues, we do have the follawirtg result of PoincetrB's.
l16
Nonlinear Dynamics, Malhstnatiiticat Biofagy, and Social Science
Theorem 2 (PoincmB). A enter equilibriwm of the Iinw&xed system f6.4) .is eiMer
a center or a foeas of the of;,ginal nonknew system (6.2).[1191
This is understandable ixr 1ig.b of our prweding results. Since the eigenvalua
are b a g i n a y , nod= md saddles are precludeb. We have ju& displayed a fscus, So
a corntruetitre proof will be done owe we find a center quilibrium of a nanlinear
system that is also a cenwr of its linearhation (try the Loth-Volterra predator-prey
model, equations (4.6) above).
AB a nrethodelogicd point, %hisexample illustrates the usehlnms of polar COordinates in some cases (the presence of the term st s; is always suggestive).
Andher way to analyzc! the nonhyperbolic equilibrium 3 = O of (6.12) is to reason
as follows, We ;itre worrying about whether the representative paint on the solution
(zl
($1,z2(t))moves tovanl the equilibrium, the origin, over time. So let us look at
the Euclidean dbtanee
+
/4I=
or, quivalently, the square of the difjtilllce,sin= if the s q w e approac_kt;s zero, so
mmt the distance itself, Accordingly, define
How do= V change with t i m e m y the chain rule:
But, with iland k2 fmm (6.12), we have
wbieh a g r m with our prwious re~uft:the equilibrium (F = 0)is R global m p p t o t i c
attrator far a < 0, a center for a = 0, and a repellor for a > 0,
The fitaction V , above, is an example of a Lyapunov function. And, in fact,
m have just used Lyapunav's Direct Method, so-called b e c a m m avert3 able to
determine the stabiliw of equilibrium directly, th& is, without having to salve
(6.12). The mom general result is given in the fallawing:
lal@l~untley
and Johnson (1983, p, 117).
Theorem 3 (Lyapunov). Let Z be a frzed point of k = f ( X ) , z E RZ and let
V : W C R2-+ R be difl'erentiable on some neighbarhood W o f 2 and satisfy:
( i ) V @ )= 0;
) 0 if a; # Z;
(ii) V ( % >
(iii) p(,) < O on W - ( g ) .
Then 5 is stable. If
< O in (iii), then 2 is asymptotiw~llystable.[1201
+
In the previous example, V = zf +S:. Quadratic forms (V = azf bslsz+W;)
are often good canditdatm, Wiile the method is very pawerEuZ, there is no hrrnula
for constructing Lyapulnov functions; this is a bid of an 81%.
The typical Lyapunov function is a bowl-lib surface w h w level sets lie in
a subset of the plane, Gmme-trically, the theorem says simply that m p p t o t i e
sdabifity requirm trajectorim to cross these level sets in the inward direction, EM
shown in fiwre 6.4.
FIGURE 6.4 kyrnptatic Stability
118
Nontinear @lynamics,Matt-rernaticalBiolagy, and Social Seisncs
FIGURE 6.5 Close Up
This swms eminexltly reasonable. Can we get from there to a more Eorrnal argument"?I2lfLet us "zoom in" on a paint where this inwad crossing takm place, zls
shown in figure 6.5.
Let Q, be the mgle betwwn V V ( n o m d to the level set of V) and the trajwb e ' s tangent, ( k l ,J2f. Rwa11 that for two vectors a and b, a . b = llalf - llbtl cos@,
where 6 is t;he angle bet wee^ a and b. As long as the flow points inward, we must
have r/2 < QI < 3 ~ 1 2which
,
implies cos sP < 0. But since I 'TV11 m d Il(kl,kz>lt
are positive, V11 * ( k l ,k2) < 0,but V I I (kl, k2)= and we are through.
As a second exmpfe of Lyaprtnovk direet method, let us establish a nice general
property of gradient systems, which, are impaftant in many ~plications.By way
of defi~tian,let U(%)be a real-valued digeremttiable Eunetion on Rn, Then
v,
t -dU/8zi; the veloci-ty of G equals the.
is a gmdielat system. For eveq i, d ~ i / d =
negative partial af U with respect to Z i . Far physicists, U iS most; naturally interpreted as a potential of some sort. But., one can think of (6.17) as stipulating a
rule of acljustment for each. xi. For example, in the backpropaga-tian neural nteb
work, connection weights are adjust& in proportion La ;an error grizdienteix22JIn
evolutianary game theory, phenotmie frequencies are asfijusted (by selection) in the
direction of fitnms grdiemts.
Gradient systems have the following:
Propem A gradient
minimum them.
s.jlstem is asymptotieaklg stable at
if U has an isolated local
Lecture 6
Proof. We demonstrate that V = U ( I ) - U ( Z ) is a Lyapunov function. We need
only establish the properties given in Theorem 3. First, V ( $ ) = O by construction.
Now, to call 3 an isolated minimum is to assert the existmm of some neigbborhood
W of in which V(%)> O for z f 2, which is the second Lyapunov property. Third,
we show that p(z) < O on W - {z). Computing,
(by the Chain Rub)
(since gradient)
Since they we d a t e d to Lyapunov functiam aad arise in some of thrr other
lectures, f will brieAy &cuss Harnaknian flows. A planar d p m i e a t systenn 2 I=.
f (z) is said to sdmit an autonomous Hamiltonian formulation if there exists a C'
h n c t b n H : R2-+ R such thaL
In that case, H is said to be a Hamzlloaian of the systern.fa24fAs a pneral proposition, we can consider H ( z ( t ) )along trajectories of (6.18). We differentiate H with
mpeet t o time exwtly EIS m did the Lyzrpunov funetion &m:
Hence H is constant, or consermd, d o n g trajectories of (6.18). Total mechanical
energy is the archetypal Hamiltonian from classical physics. Notice that, with F =
(ft, fi), V H * F = 0, wheress in the gradient case V U F < 0. For this reason,
wt.1 speak, af gradienk fields as dksipatiare in mntrast to consernative Hamiltonia-n.
fields.
112.a]~or
the definition of n dimmsional Hamilton
be even, we J d w n (1989, val. 1, p. 20).
3 20
Nonfinear Dynamics, Mathematical Biology, and Social Science
For another perspective, recall that the divergence of a vector field can be
interpreted as the rate of expansion per unit volume of a fluid whose Row is modeled
by F. In the event F is Hamiltonian, we find:
div E'=Vel"'
since H is Cl and so the mked partials are equal. Wmi1toni;zn flows are volume
gre~rving,a result sometimes known as LiouGlte's Theorem,[12@
Now, it is evident Erom "Ce above considerations that f-ramiltonim Aows cannot
have sinks or sources as equililaria since constancy along trajectories m u l d then
clearly be violated. Centers, by corttrmt, are admissible, and, less obviously, so are
(certain) saddla.[126j
&turning, then, to the issue of local stability analpis, suppow we encounter
a system we know to be Hamiltonian +hether or not we can display H)and we
have an equilibrium where the eigenvalues are purely irna@nary. The equilibrium
is nonhyperbolic, so lineariz&ion fails, but we can irtstantly conclude that it is a
center. m y 3 From Poincarh's Thmrem above, we know it is a center or a focus.
And, since the system is HarniXtonian, are h o w it is a center or a saddle. Hence, it is
a center. Nonlinear systems e h i b i t behavisrcs quite unlih those we haw discussed
to this point.
LIMIT CYCLES
To introduce the central, and (for autonomous planar systems) uniquely nonlinear,
phenome~oaof the limit cycle md some of the waciated thary, csnsider the
fallowing system-a vmiatisn on f 6.12).
In polar coordinates, this %&W the form:
gX251~uckenheimer
and IEEolmm (1983, p. 47).
i 1 2 S I ~~~ W ~ Q(1989,
X L vol. 1, p. 237).
For X 5 0,i. < O and. solutiom spiral to the origin as t
are t h r e cm- to camider:
+ m.
But, if X
> 0,there
This tells us that trajectorim beginning outside the circle, rZ = X, wind inward
while traj&ofia (the origin aside) be@nning inside that circle wind ou-rd,
and
that as t -,W, all these trajectories spiral toward the circle r2 = X, itself a periodic
orbit of (6.21). Bwause it is, in this =me, an attrwtor as t ---,oo and a cycle, the
orbit r = 6is called a stable limit cycle. For the time reversed system, the same
object is aa unstable limit cycle, for obvious rewons.
While the more modem dmhnical definition (in %er- of w-lidt sets) involvw
hrther appar&us, Mianorskrlyk s a k w immediately clear the di&inction betwwn
limit cycles and the orbits surrounding neutrally stable centers. (I italicize the
mlevaat phr~tse.)
""A limit eycb is a closed trajectary (hence the trajectory of a perladic
solution) such that no trajec;tory suficientty near it i s also cdosd. In other
words, a limig cycb is an isolatd closed trzljectory*Every trajectov beginning sufficiently near a lim& cycle approaches it either far E -+ oo or
h r t --+ --m, that, is, it either winds itself upon, the limit cycle, or unwinds
from it. If all near& trajcxtories approwh a l h i t cycle C as t -+cm,we say
that C is stable; if they approach C as L + --so we say that C is u~stable,
If the trajectories on one side of C approwh it while t h a s on the other
depart from it, we sometimes say that C is semi-stable although fmm a
practical pain$ of view C must be considered unstalble" "inorsky, 1962,
pp, 71-72).
The ftabie Xidt cycle iis the bwic model h r all self-sustained oscilators-those
which return, or recover, $0 some Eu~dannentalperiadic orbit when perturbed from
it. As Hirsch md Smde put it, "hr a periodic soluti~nto be viable in. applied
mathematics, this or wme relatd stability property. must be satisfied." "271 The
stable oscillations, ""b&i@ af %hehuman heart (which returns to some normal
rrate &er we raise it by spriding), eyclm of predator-prey systems, and various
electrical circuits are thrert among myriad examples. Business eyclw and certain
periodic ciutbreah of social unrest (SW lecture 4) are, quite possibly, athers.
122
Nonlinear Dynamiss, Mathematttimt Biotagy, and Social Science
HILIEIERTS "1H PROBLEM
Quite aside from their prwtical importance, limit cycles also occupy a venerable
, the Second
position in the history of We~tieth-centuqmathematics. Xn XNQat
International Congress of Mathmaticiw in Paris, David Hilbert presented his
famous 23 prsblew. The second part of Hilbert" 16th problem may be &ated
as follows: Deternine the mainrum number and position of Eirnit cyclm for the
syst;em
where f and g are nth-degre polyxlomids
Defining the Hilbert numbers
H(n) =
{number of limit cycles of (6.22))
,
tke problem is to deterrnhe H(%)for arbitrary n. f t is not hard to estabXish that;
H(O) = H(1) = 0.But, for n 2 2, we know precious little. Il'yashenko (1991) has
shown that H(%)is finite. We also know that H ( 2 ) > 4 and that H ( 3 ) >_ 6. And,
a&er nearly a century; that's about iLr1I2q
Small wander that in 1947, Minorsky could write, "Perhizps it, is not too great
an exaggeration to say that, t b principal line d endeavor of nonlinear mechanics at
present is a search for limit
In today's world of chaotic dynamics and
strange attrwtors, this would be m exaggceratian, But limie w l e s remdn eminentity
tvczrthy quarry and-by way of Poin~aremags-they lead to the! study of discr&e
dynamical system^, where low-dimensional &m c m be found. (In mtoaomaus
digerentiabte systems, chaos only arises i s dimensions three or peater.)
The main $hwret,icd tool in the search for limit cycles is the Paincar&Bendhrtn.
~
statement requires us to define w (omega) limit
Thmrem, w h 6 contemporary
points.
The bwic idea of w and rw limit points is simple, Axly point to which a trajec-y
converges in forwrd time is an w limit point and any point to which the timereversed trajectory converges is an a Xirnit point. The twhnical definition is just a
bit m m discriminaling. Let r(t)= (zlt),y ( t ) ) be a trajedory of
Definition. A point P E RZ is an w limit point of 'l if there exists a sequence (t,)
such that t ,
4
oo as .n
m and
The set of aH such P s is called an w limit set; cr limit paints and limit s&s are
andogously deGned, with t, appromlking --m rtzLher than +ca.
A limit cycle, tlren, is rigarausly defined as a peodie olrbit that is the w or
a limit set of other orbits.(l301 Denoting the w limit set of a trajectory 'I by w ( r ) ,
we state the celebrded Poincm&Bend
n Thmrem, perhaps the centerpiece of
nonlinear dynamics in the pla~1e.tl3~j
Theorem 4 (Poincar6Benbkon). Let I"@) Z= ( z ( t ) ,y(t)) be a trajectory of (6.29)
such that, far t 2 to, f ( t ) remains in a closed and bounded =@ionof the plane
containing. no eqztilibrpltzcm pints. Then either I' or ~ ( f " )is a p e ~ o d i corbit.
The proof rests on the fundamental Jordm Cuwe Tkorem: a simple closd
cume divides the plane into two wnnected rqions and is their common boundary;
one region (catled "the inside" ') is bounded and the other (citlled "the outside") is
unbounded. On the face of it, nothing could appear more obvious. Yet, the proof
is
In other words, it is hard to really put the basic concepts of "inside'"
and "outside" on a firm footing. How many other ""obvious" things must we not
understand?
The main difieul-l;y in. applying the Poinear&Bendima Theorem lies in. - t a b
lhhing an equilibrium-free closed a ~ bda u n d d "trapping" region. Sometimes it is
emy, as in the system (6.20) above. fn this c m , we first imagine a circle eenterd
at the origin with radius ro < 6.Since i > 0, all trajectories cross this circle
outward. For a circle of radius rl > 6,i < 0 and all trajectories cross inward.
Hence, every trajecbry that, a t t = to, is inside the closed and bounded region
betwen the= circles, the annulus ro 5 r r l , remains there for dI t > to, The annulus ~ontaiasno cquilihria of (6.20). Hence, by the PoinearBBendixson Theorem,
it must contain a periodic orbit.
In ewes of this sort, the thmrem makes good intuitive sense. A trajectory
starting inside the circle r =
is spiraling out. It cannot intersect itself (by
uniqueness) and it call% '"Lscap$" so it winds out to a periodic orbit.
As 1 said, establishing a compactf13331trapping region can be hard. The other
limitation is that the Poincar6-Bendixson Theorem is false for autonomous systems
<
rigorous, we should carefully define the "distance'"orn
a tra ectory ta a
set, as in WaItman (1986, pp. 137-38 and 142).
i1311~hisstatement parallels \n/aftman (1986, pp. 143-44).
1232jSmGamefin and Grmne (1980).
11;331~n
wbitrasy metric space A is ampact if and only if every open cover has rs finite subcover.
The quivalenee to c l a d and bound& in Rm is a thmrem (Heine-Boref). See M a d e n (1974).
[1301~0
be completely
124
Noniinaar Dynamics, Mathamaticaf Biology, and Social Science
of dimension greater than 2 (basically because we lose the Jordan Curve Theorem).
There is another powerful theorem concerning limit cycles that applies in higher
dimensions. The theorem also concerns bifircation, a concept central to the study
of dpannicizl, systems m d chaos.
As the simple& introduction to this deep mea, begin with a vector field of the sort
we have encountered: i: = f ( X , p ) , z E 7Z2 and p is a single real parameter. When we
vary p continuously, how does the overall flow in phase space, z,(t), behave? And,
specifidly, are there poinls-p value~-at which the bait, topological, structure of
the phase portrait changes? At: such points-the so-called bifurcation points-the
field f (z, p ) is "structurally unstable" in the sense that "nearby fields" f (X,p + E)
have a diAFerent topological structure. The phase portrait at p, in other words,
is not topoXogicdly quivalexr&to the phwe portrdt at8 p -t- E , In turn, and now
more prwisely, a field f is said to be ""sructurally stable" if and only if them is a
neighborhood (in the space offiedds) off seteh that all fie& in the neighborhood an:
top~k);gicatkyequiztatennt to f. Recdl that two fields are topologically equivalent if
and oniy if there is ~l homeomorphism mapping; the orbits of one onto those of the
other and preserving direct;ion in time, (Under this interpretadion?""phslse portrails"
really identify topological equivalence classes of fields). So, in a family of diferential
quatiom
(6.24)
2 = f ( s , p ) , z E R",p E R ~ ,
we take?the followingfl34J
Dsfinition. A vdue p0 for wEch the flow of (6.24) is not strueeurdly stable is a
bifurcation vdue of p.
Now, to redly discuss structurd stability; we must nail down the term "n&ghborhood of a field," and this requires that we impose a metric structure on the
relevaml set of vector fields. This; c m rigorously be done in a general metric space
context, but will no%be done here. Smalek essizy; "What is Global Analysis?"fl3sJ
remains a wonderful stwting; point, while Hale and Kog;ak get to a "big" thmrem
(on genericity) about as quickly af passible.113@l
Keeping things intuitive, it is hardly deba;t;abke that if, for some p, the phme
flow has a single point attractor bud, for even slightly larger p, it h~ a stable limit
1'341~uckenheime-rand Halm= (1983, p. 119).
male (1980, pp. 84-89).
t'3"~ale and El%&
(1991, p. 393).
cycle, then a bihrcation occurs at p. .And this is. the behizvior considered in the
Hopf Bihrcation Theorem,la"7f
Thsorem 5 (Hopf Bifurcation Thmrern in R ~ )Suppose
.
the pummet&zed system
2 = f (z,p), I E
p E R has a fized point at the origin for all val%es of
the mal prameter p. Fudher, suppose the e.igenve;lues Al(p) and X2(p;L)of the (pdependent) Jaeobian o f f , at zero, are purelg imaginaq for p --- p*. If the reat part
of the eigenvalzles, ReA1( p )(= ReA2( p ) since X 1 = Xz), satisfies
and the orZ9.L-stzs asynzpt~ticallystable when p = p*, then:
(a) p
= p* / E IS a bifircatiort pint of the sgstern;
(b) f o r p E ( p ~ , p *S) m e p1 < p * , the migin is a stable foms;
(c) f m- p E (p*,p2) s m e p2 > p* the w i g i n is an unstabke focus
surrwnded by a stable limit cycle whose size inmeases with p .
Despite the thmrem's forbidding appearance, it is aplicable wiLhout too much
a p n y First, the thcoren posits complex conjugate eigenvaluw that are purely
imaginary at the bifurcation paint p = p*, which is named in part (a). Of course
p* is named that for a reascln; the dynamics undergo a notable change & that point.
We have that
d
ReXI(p") = O but ---( ReAl(p*)) > 0 .
dp
That is to say, at
the real part of the eigenvalue is zero, but its rate of change,
the slope, is pmitive; SO, the value must be negativt3 a lialt? to the left, and positive
a, little to the. right, of p*. On either side, then, the equilibrium is hyperbolic and sa,
by linearbed stability analysis, we have the stable and unstable foci predicted in (h)
and (c). Overall, then, we would expect a change of stability as ReX1(p) "crosses
the imaginary axis." It is, however, the birth of a stable limit cycle in particular
that is surprising, and harder to prove. Indeed a big-league proof requires material
(center manifold and normal form theory) beyond the scope of this lecture. But,
there is a "physical" way to think about it. Compare the earlier dynamical system
(6.12) with the Hopf case. In the first, the origin is not asymptotically stable and
the unstable focus (to the right of the critical point in this case) simply spirals
out; no limit cycle takes shape. But, in the swand, Wopf, c-, where a limit cycle
does form, the origin is mymploeiedly stable. &s 'Yorce of attraction,'Yif you will,
while too weak to bbck the passage of &(X) across the imaginary axis (it has
f"']~his statement parallels hrrorvsmith and Place (W90, p. 205). See also Marden and MCCrxken (19761, and Guckenheirnm and Holm= (1983, pp. 15142).
126
Nonlinear Dynamics, Mathematical Biofogy, and Smiaf Science
'"m~mentum"ReAf(p*) r 0 reedf), is sufficient to prevent an unbounded escape
of the orbit and so (by uniqueness %&in, as in PoinearbBend
emerges. The thwrem is e a y to use.
HOPF EMMPLE #l,Csrrsider the dynamical system
The Jacobian at p is
Evduated at the equiEibrium Z .= 0,the p-dependent J~ztebianis
The cbaraeteristie equation is
Let us uow cheek if the Hopf bifurcation conditions are met,
First, the eigenvalues of the Jacobian at zero are purely imwinary if p -- Q, So,
p* = 0. Second*s i n e ReX(p) = p, its derivtktiw with mpact to p is 1, so
Finally, we need to d e c k thtzt, for p .= p* = 0, the origin, 2 == Q, is nsymptoticdly stable, Since the eige~valuesat p r= p* are purely imagfnary, the equilibrium
is nonh~erboXicand Iinearized stability analysis fails. tyapunw k digme m & b d
Then, with r2 = sz + zg, we obtain
And, ad p = p* = O we have
Hence, by the Hopf Bifurcation. Theorem, p = p" i s bifurcation point and for
wme p > p", the origin is an unstable focus surrounded by a limit cycle who= size
grows wikh p, m shown in figure 6.6,
FIGURE 6.6 Bifurcation to a Limit Cycle.
Source: Bwed on Wiggins (1990, p. 2741,
HOPF EXAMPLE ## 2 (The Van der Pol asclllat;~).Another, very famous, example
is the Van der PoI osciXla.tor
or, equiv&lently,
k l = 22,
At 2 = 0, we have
128
Nantincaar Dynamiw, Mathematical Biotczgy, and Swial Science
with characteristic equation
P ( X ) = X ~ - ~ ~ X + ~ = ~ .
=p f i
. Regarding the H ~ p bifurca;tioa
f
require
The eigenvalues are
meats, the eigenvalues of the Jacobian at 3 = 0 are again purely imaginary if p = 0.
So define p' = 0.ReX(p) = p once more, so we again have
Finally, at p = p*, the bifurcation point, the origin is nonhyperbolic, so linesrization
fails. But, once more, it's Lyapunav to the r ~ s c w !T&&ng
And, a&p = p* = Q t
P = -z:z$ < O on
- {Q).
Hence, 3 is asymptotically stable. Just so you won't think all limit cycles are circular, the Van der Pol oscillator is shown in figure 6.7.
FIGURE 6.7 Van der Pal Qscittator
Lecture 6
129
Por the Van der Pal equation and its vmim$s (in the family of so-called Lienard
equations), the st~tualcon,structio~
of a eomp& invariarrt (trapping) R$ eoxltaining
no equilibria, as called for by PaincarkBendhon, is fdrly arduous. The Hopf bifurcation thwrem demom%ratmthe wktmet; of a stable limit; cycle quite patinlegsly
(though, unlike aa explicit construction, it says little about its shape).
Now, the examples above are all cases of stable limit cycles in R2;
orbits wind
bward them-they are a t l r ~ d o r sThere
.
is a very ingenious way to rc3prc;sent these
orbits he-d points of discrete maps in a lower dimensional space. The method,
like so much else, is due to Paincard, and bears his name.
POINCARE MAPS
The bwic idea is easily caaveyed in. the plane. Far a soplristica$ed trc?;atment see
Wiggins (1990). Fbr illustr&ive purposes, imagine a circular stable limit cycle centered at the origjn surrounding an unstable focus, as shown in figure 6.8.
FIGURE 6.8 Crossings
Given some initial point zo on, say?the positive z-axis, we can follow the trajecbry
s
say, at zt. We call zx the point af fir&
around until it crass- the z - ~ again,
return, of zo. Then, zz is the- first return of zl , and so forth. For a given cross
seekion 7.2 ( t h w arc: edled hinear4 sec~ioas),here the gositive z-
f 30
Nonlinear Dynainics, MaUlsrnaticai Biology, and Ssciat Science
map, which. we denate TT, is called the Poinear6 map.f1381Tbe idea is to reduce the
study of continuous time flows in two dimensions to the study of wsociated discrete
time syskems (maps) in one dirnmsian. Very elegand simpfifieations retiult.
Far example, the trajectory through a point z* is a periodic orbit if and only
if s* returns to itself under the Poincard map;that is to say, nfz*) =: s*. Demonstrating the existenm of a limit cyele or ather priodic orbit (a continuous euwe)
thus rducm ta exhibiting a bed pint of the discrete Pairzcar8 map, Xn turn, a
limit cycle if stable if the fixed point of the Paincar4 map is stable. &re formally,
a periodic o r b i f l with z* E l? is iaspptoticdly stable if X* is an aymptoticaily
2 I , l? is unstable.[sQJ
stable k e d point of n-that is, if I7"(2*)< 1. If flE(s*)
I. To see this, imagine a &able Emit
Conversely? if 'I is stabb, then
cycle whose Poincard map has the axed point z*. majectories beginning outside
the limit eyele w(l") are winding dawn onto it. By the uniqueness of orbits, the
crossings z, are apgromhing z* manotonicdly kern above. Thus,
n'(z*)<
Since s,+l
=.
E(%) and s* = fllz*),we have
as was to be shown.[X4QI
The Poincar! map connects the world of coatinuous Bows in some dimension to
the world of discrete maps in a s p a e one dimension lower. The lower dimensional
entity '%"talksabout" f i e higher dimension& one. A sort of "inverse probiem" is
worth ment;ioning. Given any discrete m*, of which continuous Aow (or sows) is
it the Psinear4 m a p n h e theory of suspensions treats thi~.[l4~1
Weft, what dow one of these Paincare maps wtualty look f i b ? We take an
example from Guckenheimer and Holmm.[ld2l Let the dynamic& system in RZbe
t 1 3 8 1 ~ subtlety we will ignore isr that the map may depend on the section. Set? Wiains. (1990).
f13@1
~ a md
b KO$& (3991, p. 376).
Jean-Pieme Langlois for helping me tidy this argumcimt,
[ldalf~rn
Amowsmi"c &and Place (1990, k t i o n 1.7) and J w b n (1989, vol. l, Stx-Lion 4.10).
i142f ~u&enheirnerand Holm- (1983, p. 23).
(240]~
thank
Lecture 6
As the Poinar4 secticln, take $he positive z-axis:
and the section, becannm
A product syst;ern, (6.26) is easily solved by elemedstry methods for the A m
Since 4 = 1, it is clear (separating variables) that the period equals 2n so that the
Paincar6 map is
- 112
We have n(l) = 1, a fixed point, so there is a limit cycle of radius 1. And, since
n l ( l )= e-4v C 1, it is stalble.
Despite t b tools we have developed-the usc: of polar coordinad~,the PoincarB
Bendixson and Hopf Bifurcation Theorems, and Poincarb Maps-finding limit cycles can be hard (consider Hilbert's 116th problem). Precluding their efistence can
be easier, even when the dynamical sptem looks, at first glance, very forbidding.
NEGATIVE TESTS
For example, consider:
j.
= 2eysinyy - y 1 / 2 + x= f(5,y),
tj
= sez cos z2 y = g(z, g ) .
+
Does this dynamical system have any cycles in z27Well, the presence of some
trigonometric Eunctions mighL embolden one to venture a tentative yes, But, at least
to me, this ~pecimen1o0h pretty inscrutable at first glance. Amazingly, there is a
theorem that will let you decide the issue virtualfy at sight! It is called Bendhson's
negative test. To state the theorem, let f and g define a vector field on a (simply
connected) region D C RZ.
Call that field G! = ( f , g). Then, we state:
132
Nonlinear Dynamics, Mathematical Biology, and Sacial Science
Theorern 6 (Bendboa's Negative Test). If div $2 bras &ed sign in a =$ion D, then
$2 has as cyctes in I),
We will prove a more powerEuE rmuIt short;ly but, first, lock bow simple thh
make8 the probbm (6.27) above. At sight, we obtain
and, voild, there are! no eycles. System (6.27) k of the form:
/as = @@z/8y= 0, the %S! contribute nothing to the divergence of the
Since
field and, t;hus, m rn&ter honr wild they may look, are irrelevant to the question
of q c b s . As f men,t;ianed, we wtudly have a stronger retsult .
Theorsm 'S (Bendban-Dulsbc), Give%the $@stem
where f and g are smooth (C') in a simply connected region D, let B(z,y) be a
smooth funetion zn D s ~ c hthat
has @etd
sign in D. Then f6.28) has no cdosed t r a j e c h ~ e sin D.
Proof. Assume there is a cXod trajectory, dD, which isr the boundary of a simply
conrrect;ed re@on D. By Grwntk Themem,
The right-hand side is zero since, faeto~ngaud B and expanding the differentialsf
So, if there is a closed trajectory @D,the righbhmd side aE (6.30) is zero. Suppose
now that expression (6.29), the inLegrand of the double integml above, does not
e h a ~ g esign. WiGhout Iom of generality, suppose the iegrancf is always positive an
133
Lecture 6
D, Then, clearly, the doubb integral will be positive, Loo, and we will be forced to
deduce that;
As a simple applcation of Bendbon-Dulac, let us prove bgozink negative
eriteriani""3 for inter~ctingspwies, bearirrg in mind, ias always, that "qecies" could
be interpreted in mpiad other ways (e.g., a
s, chemical coaeentratians), This
theorem applies t a Kalmsgorov-type dynamical system. T h a e have the form:
In effect, lnz plays the rob in (6.31) that z pla,ys in (6.11, since (6.31) implies
S/z = f (z,
g), but k l z is dldt(lns). It is the per capita growth rate S/s that is
given by f . Now, species are sdd to be sev-regvlating if the per capita gromh rate
decrewa as the species population i n c r e ~ sIf.
then, af course, each species is self-regulating. With this background we state:
Proposition (hgozin). A Kolrmog~trovsystem in did each s p i e s k self-regalat-in,g
has no cycles in Wle population quadmlat (z, y > 0).
Now,W seek some real Eunction B(%,y) such that. Bendkon-.Dulac applies. Caming up, with such Eunetioxls is a bit of an art, like ixrventing tyapunov functions.
Following Het;h~ste,I'~~1
we try a function of the farm:
Specificalfy, let; j
=.
-1. Then, &&era little algebra, we find that
134
Nonlinear Dynamics, Mathematical Biology, and Social Science
Since X and y are assumed positive, so are 11%
and lly. But, Bf l a x and 8glBy are
negative by hypothesis. Hence
and we are through. U
Clearly, the same reasoning applies to Kolmogorov systems in which each
species is everywhere self-amplifying, or autocatalytic:
Then, the system (6.31) has no cycles in the population quadrant.
In summary, as a corollary to Bendixson-Dulac: for Kolmogorov systems, if
either (6.32) o r (6.33) obtain, there can be no cycles.
Notice that the generalized Lotka-Volterra equations of lecture 1 are of Ko1mogorov type.
X1 = X l ( ~ l +a11x1 012x2) ,
~2 = x2(~2 a2lxl+ a2zx2).
If a11 and a22 are negative, each species is self-regulating; if all and a22 are positive,
each species is self-amplifying. In either event, there are no cycles. (In the predatorprey variant where cycles do occur, a l l = a n = 0.)
There is a powerful theorem of Kolmogorov governing the stability of those
systems that bear his name. These, recall, are
+
+
= x f (X,Y)?
Y = Y ~ ( xY),
Following May (1974), we state the result as follows: Kolmogorov systems have
either a stable equilibrium point or a stable limit cycle provided f and g are C'
functions of X and y on X, y 2 0 and the following relations h0ld[145]:
(i) -Sf
<O,
ay
(ii)
Of +y- @f C O ,
X-
ax
ay
69 < 0 ?
(iii) ay
[146]~ee
May (1974, pp. 87-88). May adds that "The theorem also usually holds when certain of
the above conditions are qualities (=) rather than inequalities (C or >). Such cases need to be
dealt with on their merits, but can often be seen to be eensible limiting cases of more general
predator-prey equations which do obey the above criteria" (1974, p. 88).
Also, there must exist quantitim A, B, and C such that
(vi) f ( O , A) = 0 with A
0,
(vii) f (B, 0) = O with B > 0,
(viii) g(C, 0) O with C > 0 ,
(ix) B > G .
=l.
Plwing the theorem in context, May writes,
""Inmare biological terms, Kolmogarov's conditions are rough& that (i) for
any given populatim size (as rnewured by numbers, biomass, etc.), the per
capita rate of increse of the prey species is a decreming function of the
number of predators, and similarly (iii) the rate of inerease of predat;ors
decremes with their population size. For any given ratio between the two
species, (ii) the rate of i n c r e ~ eof the prey is a decreasing funetion of
population size, while converse& (iv) %h&Of the predat;ors is an iwreasing
Eunction. It is also reguired that (v) when b&h popufations are small the
prey have a positive rate of increase, and that (vi) there e m be a preddor
population size suEciently 1 x 8 to stop further prey increase, even when
the prey are rare, Condition (vii) requires a critical prey popul&ion size
B, beyond which they emnot; increwe even in the absence of predators (a
resource or other self limitation), and (viii) requires a critical prey size C
that stops fusther increase in predators, even if they be rase; unlas (h)
B r C, the system will coXfacpse9"CMay, 1974, pp. 88-89).
For rt @ven ECo1xxrog;orov system, then, the programme is clear. First, establish
whekfier the system sat&&= the thwrem's Irypothwm, an c3ve~tudiil;yMay accounts
as quite likely. He writes, "What h a b e n lacking in the literature is not the derivation. of the above theorem, but rather the realization that it applia to essentiauy all
the conventional models people urn." If the system meets all the conditions, then
it posmses either a stable equilibrium point or a stable limit; cycle, Our sdandiard
linearized stabililty analysis at the equilibrium will then reveal whether that point is
stable, "whereupon we have the complete global &abiii$y character of this system
faid hare."^^^^^
INDEX THEORY
We have seen that nonlinear vector fields xnw have multiple k e d poi~w.And we
have developed a theofy of stability allowing us to elmsie hmerbotic and (to some
f 36
Nontinear Dynamics, Mathematical Biolcrgy, and Social Science
extent) nonhyperbolic fixed points as stable or unstable nodes, foci, saddles, and
centers, and to rule out and (with some cre&ivity) to d e h d limit cycles. But, the
theory is completely local; we have dewloped no theary of how these entities "can
combine, or "fit together." Far instance, could one evter encounter a limit cycle with
a sixlgle focus and tovo saddlm inside?
There is a beautiful and penetrating theory that allows us to answer many such
questions. It is called index theory and originates, once more, with Poincar4.[1471
Among other things, index Lhwry reveals what amount to toyalogkd cansew*
tion laws-people even speak of the ""consemation of topological charge'? !It also
reveals that, from a particular topological perspective, centerg, nodes, and foci are
equivalent!
Ta begin, we recall the planar autoxlomous system
The w&or field (f,g) is tangent to the solution Bow in phase spwe at each point
(s,g). Now, let C be a simple closed curve (not necessarily a trajectory) that does
not intersect any fixed points of (6.34). This last proviso ensures th&, on C, J and
g are never simultaneously zero, so that f gZ # 0,a condition whose importance
will soon be evident. Since the field (f,g) is tangent to the flow for all (z, y), it is
tarmgent to the flow for all ( X , g) on C. So imagine basing a small arrow at some
point of C, letting the arrow point in the direction (f,g). Now9 slide the arrow's
base exactly once around C coun%e?relochise,allowing the arrow to w u m e the
(continuously changing) direction ( f , g ) at each point;, Since the terminal point
is the initial point, the terminal dirmtion of the iarrow eoineidiw with the, initial
direction and, thus, having returned to its initial position, the arrow must have
rotated through 27r radians m irrtegral number of times: that imteger is the indez
of C. Qhiously it depends on, the field.
For example, if the field is evervhere flowing right to left, the arrow will not
rotate at all and, for m y cloed curve C, the index wilE be zero. By contrast, if the
curve C is itself a simple closed trajectory of (6.34), then, at any point an C, the
tang@&to the flow i s the tangent to C, so that the index is 1, as illwtrated in
Ggure 6.9, which shows iz variety sf singularities and their indices (in pwentheses).
+
[147j~incar~
index
k
is defind in b i n c a r d (1881, 1882, 1885, and X886),. For an x c a u n t , see
Lefsclnetz (1977, pp. 195-96). For developments of index theory, see d%oJzteksan (1989, vol. 1, pp,
243-501, Jordan and Smith (1981, pp. 65-14), Cuckenheimer and Holm= (1983, pp. 5&53), and
Amof" (1991, pp, 30S18).
FIGURE 8.9 Fbld Dirstions and Indices.
"--%
/
--..
*
--m
'-%
/
%--"
/ \
Circulation ( + l )
Siah (+l t
Source (4-1)
Saddle (-1)
Spiral (+l)
(~2)
Source: Based on Guiltemin and hilack (1974, g. 133).
To put this on a more formal fasting, we will m u m e that f and g and their
first p;artids are continuous on the relevant sts-bmicalliy ss there is no problem
invoking Grwn's Theorem, which looms large here.
At any point ( X , g), the slope! of the field =tor is &/dz =. g / f . Hence, relative
to the X-=is, the angle of inclination, @, of the field vector must satkEy
Defining zc = g f f : we find
+
whence our concern that f 2 92 # 0. Now, it is clear that the total change in if2
over one circuit around a simple ccIosd cuwe G is given by the line integral.
138
Nonlinear Dynamics, Mathsmiatical Biology, and Social Science
But, as argued above, thh Is an integer multiple of 2n,the integer being the index
of C. Hence, denoting the index I(G), we arrive at the relation
Rom here, it is a short step to the next basic result:
Theorem 8. het C be a simple closed eume that neither contains nor interswb ang
eqailib+.tsnz point8 of (6.34). men I(C) = 0.
Proof. Let R be the (simply connected) interior of C. Then (assuming f and g to
be G') Green's Theorem yields
But, biEerentiatioa will quicHy show the integrmd on the right to be mro. Cl
So, the index of a closed curve containing (and inter~cting)no equilibria is
zero. Now, wh& happens to the index if we deform C? If the index Es invariaxld
under (certain) d&rma;tions of C, then W really sbuld no&think of the index W
characteristic of G at all; but, then, what is the ixldex really about? We begin to
answw the qumtion .with a corollav to Theorem 8.
Corollary 1, Let G be a simple dosed eecsue and let C& be a sim*
ctased came
sumozmding
T f them i s 720 qailibrizlm of (6.34) on either carme or i~7~
the region
between them, then I(Cl) = I(Cz). Deformation has no eflect.
a.
Proof. The proof is reminiscent of the standard complex variables proof of Cauchy'~
Tlsmrem for multiply connected domains; we "cut" a multiply connected dorndn,
leil-ving a simply connected one3 to which the more bltsk theorem applies. So, as
at point A with a
s h w n in figure 6.10,let us cut the annulus between Cl and
segment; K, Sta&ing A, we intepale counlerclochise around G ,then down K - ,
around C2 (now clockwise, reversing sign), and out K+ to the start. The clo~ed['~8]
curve just trwed neither intersects nor eontdne any equilibria, so its index is zero
by Theorem 8. That is,
[2481~hroughout,
f will ask your indulgence for a slight mathematical, indiscretion. Strictly s p e d ing, if vcre cut the annulus with a lino mgment, then the path traced in the proof is not a simple
cXos& curve since K+ intemmts K - m e w h e r e . To mske things right, ane couM take ewntialty
the above approach, but s t a by
~ snipping ouf a swath of width e 3 0, with sidm K- ~ n K"'",
d
and then m&@ a riprouts limiting cwe Z ~ SE 4 0.
Lecture 6
The integrals along K- and K+ cancel, and we have our result. O
FtGURE 6.10 Deforrnatian
We now h a w that the index of a ebsed curve containing rro equilibris is invariant under deformations, sc, long as equilibria are not intersected. Since the index,
%herefore,h a little to do with the cume, what; is it truly refiecdirzgUt is really
the fixed point structure inside the eume, not the eume itself, that; is reAwted. We
make the following definition;
Definition, The index, I b ) , of a singulx point, p, is the index af m y simple closed
curve surrounding p that neither intersects nor enclsm equilibria other than p.
Using an, argument similar to that emplayed above, we shall prove:
Theorem 9, The index of a closed curue equals the sum of the indices of the equzE i b ~ u mpoints it encloses,
Proof. We prove the result for $WO equilibria, p1 and B. With C the outer closed
curve, surround p1 and p2 with eurms Cl 8nd C2,m d make cuts K and L, as in
figure 6.1 l.
l40
Nonlinear Dynamics, Mathematical Biology, and Social Science
FIGURE 6.11 Sum of indices
If we integrate Rom A to B, down L-, around C2,and out L+; then from B to A,
down K - , around Cl, and out K" to the start, the cfasd curve described neither
ialersctcls nor enciosw any equilibria? so its index is zero. Writing this out in fulX,
but noting that .LCcaneels L- and that K+ caneels K-, we are bft with
which quite evidently generafizes to the rault we seek;
where the fi's are singularities e n e l o d by. C. B
With one more theorem uve will be able to extrwt s o m very unexpmted rexufts.
Lecture fi
"Cl
Theorem IQ. The indea of a ctosed periodic orbit is 1,
Proof. We nzerely formalize the plausibility a r g u e a t made earlier. Were, C is a
closed prsriadic orbit (mshown in figure 6.9 above). At every point of 6,
therefore,
the field veetor is precisely $=gent to C. In one circuit around C,the tdal change
in the tangexl(;vector%-and hence the field vector"-angle of elevation, @, to the
z-axis is 271"radiam, or
But this line inlegraf is, by definition, 2x1(6). CJ
Thmrerns 8 and 10 imply:
Corollary 2. A closed p ~ o d i ctmjeetory mwt szlwovnd an quilib~um,
Proof, Posit, to the contrary, a closed periodic orbit C enclosing no equilibria. Since
closed, its index must be zero, by Theorem 8. But, since C is a periodic trajectory,
its index i s is, by Thwrem 10, yielding a contradiction. O
More general is the following '%conserv&i.~nlaw,"
Gorollary 3. Suppose a closed qoerz'odie Cmjeetory slsmuncds C centem, N nodes, F
foci, and S saddles. Then,
Proof. 1E3y Thwrern 9, the index of the surrounding orbi-t;equals the sum of the
indices of interior h e d points; m d as obsewed above, every center, node, and focus
explaining the l@&-hand
contributlcss 3-1 to the sum; every s d d l e contributa ---l,
side. But, by Thmrem f 0, tfre surrouding orbit" index i s 1. O
So, for example, one will not encounter a limit cycle with exactly one saddle and
one focus inside, or a center and a node, as thwe arrrangements viafate (g.37). Interpmed slightly dierent15 (6.37) o@ema sewnd negative eriterion far limit cycles.
If a region is papulated by mnters, nodes, foci, and s d d k and (6.31) is violated,
then there cannot p ~ g ~ i b lbe
y a surrounding periodic trajectory; Or, assuming C
to be a periodic orbit, is the ghme poI-trait shown in, figure 6.12 impossible?
142
Nonlinear Dynamics, Matfiernatia1Biatogy, and Social Science
FIGURE 6.12 lmpossibls Phase Portrait?
Another very unexpected result, with applications in mathematical economics,g'mj
is:
Corollary 4. A closed periodic trajectory must enclose an odd total number of centew, nodes, foeri, and saddles.
f roof. Suppose to the contrary, an even number, 272, n 2 1. Then, in the notation
of Corollary 3,
C+N+r;"+S=2rz.
But, by (6.37),ure have
Cit-N+F-S=1.
Adding, we obtain
2 ( 6 $- N
+ F) = 272 + l , an obvious comxtradic'eion.
CT
As a final result of this sart, aassume, as usud, that l" is a, closed periodic orbi*
interseetiag no critical points. Then, from the conmrvation law (6.311, the falbwing
corollary rdatiag index thmry to bifurcations bums trivially.
Corollary 5. If, @at a bibreation point, a saddle is created (destroyed) inside I?, then
a node, fclcus, or eenter must be created (destroyed),
To conserve ""dopologicaE charge" (i.e., the sum of the hdicw) new sinplarities
mu& arise or disappear in pairs having opposite indices. (The parallel to particles
and anf;iparticlesin high energy physics has been suggated). In faet, the coroHary
goes through if r( is a simple clmed curve, not nwessztrily an orbit* Far this, much
stronger, result, see Birhoff and ftslta (1989).
While t k s coroflitry flows eEordbssly from the basic conservation thmrem, stop
and consider how utterly unapproachable it would be without index theory,
BROUWERS FIXED POINT THEOREM
As demaastrated in Theorem 9, the index of a curve I(C) is a topological invariant
in that continuous deformations of C do not aEect the index so long as no equilibria
are intersected. Here, we imagine G as being deformed over same spwific underlying
vector field V, An equivdent, but digerently powerful, perspwtive is to fix C and
continuously deform the vector field V , ensuring as usual that no singularities are
brought i&o contact with C. Qf course, we have not defined "con_tinuousdeformation of a vector field." This is a hndamental notion in topolom and has a special.
name.
Definition, Let f ,g : X -,Y be,
Then f is ham6 topic to g if t h e ~ eexists
a continuous map H ( s , t ) , O 5 1 5 1, such that H ( z , 0 ) = f ( 2 ) and H(%,
1) = y(z).
With this $efinition,[l5" it can be shown--and
it s e m s entirely plausib
Fact: The index of a curve is invariaat under continuous deformations-horn*
E the vector field so tong as no equilibria are brought in$o contact with the
curve in. the course af the deforrxz~tion.
Accepting this faet, index theory provitclw an efegant way to prove Brauwer's
celebratd fixed paint theorem on the closd disk in the plane. Although he does
not use quite this terminoltog, the proof faflows Arnoldd.[1521
Theorem 11. Every smooth mapping f : t)4
of the ctosed disk
into itself has a
&ed point,
Proof. We imagine the disk-whwe closed boundary we denote BD-to
at the ori@n. Define now the vector field
V(.) = f ( a )- 2 .
be centered
(6.38)
Clearly, the k e d points of f are, identically; the equikibria of V. The Tltmrenr
will therefore be proved if we c m establish thaL the index of aD in Ir is I. Why?
[lso]~erctX and Y are topatogiocal spwm. See hydctn (1988).
ilGL1SO=@ tratment~rquim sm06thnm. See Guilledn and P a l l d (1974).
ffml~rnol"(1973, p. 257).
144
Nonlinear Bynamics, Mathematical Biologysand Social Science
Because then, by Tbwrem (10), L1 =e int(8D) must contain an equilibrium of V
(else I(i3D) would be zero). But th& equilibrium, as just noted, is a fixed paint; of
f . SO,we slate;
Claim, The index of d D in V is 1.
Now, by the Fact above, the claim will be proved if we can provide a vector
field homotopic to V in which I(aD) = l. lCiQ that endl define
H defines ;;l, hamotopy b e w e n the fields H(%,Q) = -s and W ( z ,X) = V(LE),and
for no t E [Q,11 does H(%,t ) have singularities on OD.But I(@D)in the field -s is
obviously 1 since -s is just the field with all vectors pointing to the center of D. D
To summarize the proof, f : b -+ b ha4 a fixed point if and only if V has an
equilibrium (by construction of V). In turn, V has an equilibrium in D if I(aD) in
V is 1 (by Thwrem 10). And I ( a D ) in V is 1 if I ( 3 D ) is 1 in any field homotopic
to V. With H(%,t)defined as above, H ( z 2Q) = ---X is homotopic to V and I(aD)
in -z is obviously 1.
Penetrating rzs the kfieory is when applied to vector fields in the plane, even more
startling results emerge when. we consider vector fields on more general objects,
notably closed surfaces like the sphere m d torus.
A GLIMPSE BEYOND THE PLANE
In pa&icular, X will conclude with a very infarmd presrsnt;ztian of the beautiful
hincar6Hopf index theorem, which nicely connects our work on index theory to
digereatial t o p o l o ~ .
to begin is with Eulerb sell-hewn formula
Perhaps the most e l ~ s i c a place
l
(which was apparently known to Descartes)il54 that for closed convex polyhedral
~urfaees"likC?'' the pyramM and cube,f1Sd1if V, E, and F are the numbers of vertices,
edges, and faces, then
V-E+E"=2.
Now, think of a sphere surrounded by a closed ""approSmatixre;"' polyhdral surface,
all of whose hces are triangles that fit together '"nicely"-so
that trimgles intersect
only h a, common vatex or an entire common edge. By (6.39), if you count vertices,
subtract edges, and adcl fma, you will again find that the sum is 2. This number, of
11531~eeFrkbet and Fan (1961,g, 2 l).
ilriclI~n
toplolawt we do not urn the word "like" fifigh%ly.Far a punetiXious chwwt;eriz~ttionof ex&Iy
thme polybdm ta which the rmult applia, see Armstrong (2983).
course, does not change if we diow ""rubber trianglesD-which we imagine pressing
.
is the
down to exwtly cover the spher and count curved fwes and d g e ~This
idea of a t~angulationof a surfze. You could &so i m q i m etching curved triangles
all over the spherical surface in the ""nice" w w 1 menitioned above, as shown in
figure 6-23.
FIGURE 6.33 The Triangulations of the Sphere.
T'o be precise, hawever, a triangulation af ia (compact) surfme S is a covering of
S by a finik family of dosed sets (G),
e x h of whi& is the homwmorphic image
of a triawle in the plane,i3=l And, again, we require that ttvo distinct "triangles"
(irnaga) be disjoint or have in common either a single "vertex" aor one entire "edge?"
where thme are aXsa understood to be image objects. Lewing out the quotes, on
this understanding, we can count V - E+F just as before. It is a deep fact that for a
given surfwe, S, the number thus obtained is independent of the triangul~tioxl;im
it is a ehnracteristic of the surfwe itselg-the so-called Euier cXzmwteri&ie % ( S ) .
The n-dimensional definition is
, and a2
where a, is the nurnber of "fwes af dimension i."[157j For n = 2, a ~al,
are, rapeetivelly; the numbers of vefiiees (dimension 0 ), edges (dimension l), and
~ z s 5h/lmwY
]
(1967).
['"l~t is1in fact, a thmrem (WO,
1925) that a triangulation iu pmible,
i-simplexm.
146
Nonlinmr Dynamics, Mathematical Biology, and Social Science
faces (dimension 2) in the trian(;ul&ion. So in two dimensions, we reeover Euler's
alternating sum:
X(S)==V-E-~-E'.
And, the generstfized Euler theorem is that; for all surfaces hornearnorphic to the
spheresuefit as the cube and pyramid ab
he Euler characteristic is 2.
Naturally, we will be interested in the Euler characteristies of other smooth
closed surfaces like the torus. And we expect the Euler characteristic, being topolo@cal, to depend on the conneetedness of the surfwe, a prope&y captured in the
secalled genus.
Mie met the notion of eorzncsctedness in proving that the index of a closed euwe
equals the sum of the indices of enclosed singularities-we used a line segment to
"dice'"
multiply connected domain, ereating a simply connecbd one to which
a more basic theorem applied. But, the number of slices nwded w a clearly a
topologicai invariant having to do with the coanwtedness of the domain.
Similarlx for closd two-manifolds (which we will define shortly) like the sphere,
torus, and so on, one e m ask: What is the m~xirnumnumber of, now, simple clost?d
curves o ~ l ecan draw an the surface vrithozlt dissecting id. into disconnected parts?
This number is the genus of the manifold. Far the sphere, it is clearly zero, since
even one simple closed eume will dissect it. Two simple c l a d curves are neded
to dissect the brus, so its genus is l, and sis on, ;ss shown in figure 6-14.
FIGURE 6.14 Classification of Oriented Closd Two-Manifolds
Gem= O (hommmorpbicLo a sphere with no haadles)
Gmus l (homeomofpKcLo a sphere witH one hmdk)
&nut?,2 fhommmofp&cto ai sphere with two harxdfes)
a n u s 3 ( h a m o m a ~ to
k a sphere with tkm haadlee)
Source: Based on Guillemin and Pollack (1974, p. 124).
i r s a l ~ h Eufer
e
~Gfxerizcrteristicillso depends an the an'entability of t h e surfwe, a topic into which
we will not enter here. All surfwes are ws,sun?& to be orient&, even though X will o&en say so
explicitly just to drive borne t h e point that i t matters. The cXwic nonofient8ble surfme is, of
course, the Moebius strip,
The figure rc?flec%~
a hndamental ctassification theorem in topalogy: every cornpact o ~ e n t e dboundqkss two-manifobd is honteonaorphic to one of Ihese surfscms-a sphe.au?*.t;h n 3 O hanrlkes,lf5@1We know from above that the Euler charwteristic
of the sphere is 2. Its genus is zero. The general relationship is:
Theomm t 2 . Oeentd closed surJitceslX64 of gelazds g have Eulier cltaracterislie
2 - "2,fI"ZS
We S-m to have wandered -Ear from index theory, which concerned singularities
of vectar fields, To procmd h r t her and connect all of this to index thmry, we nwd
to be able to define vector fields on sudwes of the sort we have discussd. And we
are going to want to do calculus on them, which puts us in the world of diEerential
tspalogy. Obviously, we all learned calculus on the Euclidean plane. And we will be
able to do calculus on objrtets that are "looelly Euclidean" which is certainly how
practitioners think of manifolds: surfwes that we, locdly, smooth dtlformatiom of
the plane. In fact, all the closed surfwes we have been discussing are twenranifolds.
~ ~ each
I
poirrL of
Technically, a two-manifold is a connected Hausdorff s p ~ e , i M,
which has a neighborhood homeomorphic to an open set in R2.This is what one
means by "looclly Eucfidean." And, in turn, a vector field on M is simply a smooth
(C') assignment of a tangent vector to each point z. E M, just as in vector calculus.
But, it will pay to be a bit Dore painstaking. Following Smde (19691, we iassociate
the so-called tangent
to each point a: E M a twedimensiond wetor space, %(M),
spwe of M at z, For a twemanifold, like the sphere, T z ( M )is the plane t a n e n t to
M at s. A vector field V ( z )on M is a C' assignment to each z E M of a tangent
vector-that is, a vector lying in % ( M ) .
At singularities I where V ( f )= 0, the field may exhibit sinks, saurces, centers,
saddlm, and more cmplex behaviors-ph~ portraits-on M , Technically, we do
not know how to compute indices on M. But, M is m object th& is 1ocaIly horn*
morphic to the Euclidean space, R', where we do know how to compute indices. So,
we imagine doing the same thing on a small surficGe element of the tangent, space
T z ( M )E&
As Guilfemin and Polfaek put it, '"~ooking at; the manifaid localXy,
~259~Xnded,
for two-mnifalds, diffmnnorphic.
i1""11'21hough, again, we will nat delve into it, the general relationship between genus g, EuIer
chig~~ttcteri~tic
X , and oritsntabilib is
fzsxfl;w
ie~aurantand b b b l n s (1963)for a very nice intuitiw development.
[ l c j z ftapolag~cal
~
space is Hausdorff if it siltisfim the f o l b w ~ n gcondition: Given two distinct
points z and g , there exist apen sets Q1 and O2such that zr E Q1 and y E Q z and Ox n Qz = Q1
See &yden (1988, p. 178).
~ ' " ~ ~ b i again,
s,
1s standard procedure in surface integration, far example.
148
Noniinear Dynamics, Mathematical Biology, and Sr>cialScience
we m emefiidly a piece of Euclidean spMe, so we simply r e d off the index as if
the vector field were Euclidean."
With all this in pfme, we can state the Paincw6HopE index thmrem. AS you
d g h t imagine, there are mmy formulations; we fallow Ateksmdrav".
lt?eorem13. (PaincarBHapf). 1Tf, on a given o ~ e n t e dtwo-manifold, a eontin~ous
vector field is defined hauzng only a f i ~ i t enumber of siwular points, then the sum
of their indices equab &fie Euler charneterzstic of ithe manifold.
This is remslrkabte because the Euler charaederist;ic ia a bpologicaf prapefiy of
the manibfd only, and would appear ts have absolutely nothing ta do with flows,
or vector gelds, on the manifold. How does the manifold, M, already "how" 9-0
much about the singularity structure ctf vwtar fields definable on M? C l e ~ f ydwp
~
things are afoot,
One immediate consequctnice of the theorem is %hatvector fiel& having no
singularities are passible only an manifolds with Euler characteristic: zer
1; namely, the toms (and, in fact, the Klein batdfe, a nonorientable "'one-sidedn
torus). Notably, it is hpossible to construct a nonvanishing fidd on the sphere. TMs
result is sometimes aflFeetionately called "the hairy ball theorem," the interpretation
being that any attempt to wmb smooth a "hdry balf" must leave at lertst one babfd
spot, Bdbnms can be avoided on, the brus, as shown in figure 6.15.
FIGURE 6,15 Hairy Batt Theorem,
Twa bald spots
Qna bald @pot
Hetry torus
Source: Bamd on Armstrong f 1gm, p. 198).
[lci416uililerninand PoIlwk (2974, p. 133).
i1551~ee
Abhandrov, et al. (1963, p. 216)
Another interprf:tat;ion on the sphere is that "somewhere on the surfwe of the earth
the wind isn't btoaring." Ebr m interwting application of the thmrern to chemical
and ecological aetworks, see Glass (1975).
Mlk certalniy cannot prove the hincar&Hopf index theorem here. But, if you
will grant that there is mmethiw generic &out a certain map, the su-called Lefschets, map, then at least; a vefy suggestive emmple will be in hand. Following
Guilternin and Pollack (f974), consider the genus 4 twumnifold in figure 6-16,
FIGURE 6.1fi A Two-Manifold of Genus 4.
Source: Based on Guillemin and Polltack ($9'74, p, 125).
Their dmcription, of the map indicated by arrows in the figure is clear, Bnd appetizing!
' W e keonstructl a Lehchetz map on the surfwe of genus k: m follows, Stmd
the surface vertically on one end, and coat it evenly with hot fudge topping.
Let ft(z) denote the oozing; trajechry of the point r?: of hdge as time t
pwsm. At time 0, fo is just the identity. At time t 3 Q, f t is a Lefs~kedz
150
Manlinear Dynamics, Mathematical Biology, and Social Science
map with one source at the top, one sink at the bottom, and satddlepoints
~t the top rand bottom of each hold"(Guillernin and P o l l ~ k1974,
,
p. 125).
As 1said, this oriented maaifoEd is obviously of genus 4. Notice that in this case,
the genus (the number of handles) is also the number of holes. Now, let us sum the
indices.
By our previous w r k (and the crucial fwt that a manifloid is locally Euclidean),
we can directly count plus two for the sink and the source, and minus axle for each
sddle; that is, minus tTwo for each ""hof$\of which there are, in general, h. So, the
global sum of the indices is 2 - 2h. But, the number af holes i~ the genus of the
manifold. So, by Theorem 12, 2 - 212 is also the Euler chaaeteristic, as predicted
by Poincar&Wopf!
Many of the topics treated in t h m lecture are far more dvanced than the mathematics applied .in other ledurm, While certain of the G O ~ ~ C ma;v
B
seem highly abstract;,
the history of science shows that applications of pure mathematics are hard ta
anticipate. Who imagined that complex numbers, non-Euclidean geometries, a ~ d
infinitedimemiond spaces wauld find powerful applications in physies?E1"F.l Against
that background, it; muid be naive to preclutctc: the applicability of even the most
abstrwt; mathematics t;o social scieace. And meanwhile, of course, the mathematics
is its awn reward.
VVigner9s%say, "The Unremnable ERwtivenw of Mathern~%it;ies
in the
Natural Scit-mcw." "igner, 1963).
I I s s j ~ nthis, see Eugene
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Bethesda, MD.
Wilson, Edward 0. 1975. Soeio biolwy, Cambridge, M A : E41award University Press.
Wilson, Edward 0. 1978. Qn Haman Natane, c b p , 7, Cmbridge, MA: Wamwd
Universiey Press.
Wolfram, Stephen. 199%.MathemalSz"ea:A System for Doling hthernatics by Cowputes; 2nd ed. &adin%, MA: Addison-Wesley.
Wrmgham, Riehard W, 1988, "W= in Evolutiorrwy Berspwtive." h Xnme~ing
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Vol. I. Reading, MA; Addison-Wesley.
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ndex
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A
Adaptive Glynmic Model, 2, 19, 28-30
adaptation in, 34
adtzptive withdrawl in, 32
attrition in, 33
c s e s of, 35-36
overview of, 30-31
prowcution in, 32-33
simul;ationsof, 37-40
Azmrafl Z T ~ Warfare, 20
altruism, 3, 42-43
animal behavior, 2'
arms racm, 2, T-8, IQ, 69, 1x1
a;nalio&;yto drug epidemic, 94
m d c o f l ~ t i v e~wuri@,41
bisfo@caf, 52
nanfhear models of, fib66
see dso Richardson model
itttrilian, 20, 29
stalemate, 56
Axtell, b b e r t , 4
Axelrod, Robert, 41, $4
B
Bendixson-Dulac negative t s t s , 107, 131134
bifurcation, 84, 142
to a limit cycle, 324-128
bre&point-s, 25-26
Brouwer's fixed point thmram, 4, 143-144
C
casualty-exchange ratio, 27, 56
catastrophe thmry, 2
e h a s , 57, 80, 87, 122
Clark, Calin, 25
clmification theorem, 147
coevolulion, X2
rnatftematical theory of, 14
moth-bat vs. air war tmtics, 12-14
collective seurily, 2, 41-44, 47-48, 50-51,
64
eonnwtionism of, 53-55
effect on competition, 66
rigorous degaitirsn of, 41
community xnalrix, 95
competition, 14, 69
Complex System Stmmmer School, l
D
damping, 46, 97
deformations, 138, 143
demity; 22-23, 27
drug &diction, 3
drug epidemic
am race component, 94
model, 89-99
drug wm, 97-98, f 11, 103
and ducation, 103-2 0 4
model W epidemic, 89, 91. Y3, 95
SW also interdiction
see also Eegdizatian
dynmical malogiw, 69-71
epidemics, 711
revolutions, 71
dynamic& system, 1, 3, 41, 107
and collwdifr~lwuriky, 42
E
wonomics, 70, 108-105, 111, 242
epidemics, 3, 9, 15, 69
mdogy for revolutions, 15-11, 71,
74-75
analogy to explssive socid ~ h m g e71
,
drug model, 89-93
d y n m i c d aandogies, 6911.
herd immunity, 116-17, 82-83
infeelivers in, 1516, 12, 90
Kermxk-McKendrick; threshold
model, 15, 72-73, 91-92
model of, 72-73, 90-91
Ptaqztes and Peoples, 69
social change, 71
technology, 80
traveling wavw, 75
vitd dynamics, 76
muation of logistic gror;vlh, f 5, 131
Euler, 144-147
external thre;zts, 46
G
Gause, G. F., 14
Gawek flour beetle experiment, 32
Gell-Mann, Mufray, 71
Glwnost, 17
glob& stability, 109110
gfoboeop, 47, 65
vs, Rieharctson's sodeE, 48, 52, 65
grdient systems, 118-119
Growing Artifictal Soezeties, 4
1EI.
Hamiltonian, 78, l f 9
Aows, 119-120
Hartman-Grobman Theorem, f 10-411
herd immunity, 82-83
Eilbertb 116th Problem, 4, 101, 222
Erfopf Bihrcsation Theorem, 3, 107, 125-129
1
Index Theory, 3-4, 107, 135-143, 147-150
caroXlmy on bifurcation, 142-143
Poinear6Hopf Index Thmrenr, 3,
108, 144-146, 148-1.19
indirection, l l 4
interdiction, IQ0^1Q3
J
Jordan Curve Theorem, 123-124
K
Kermmk, 1S, 72-73, 92-92
Kolmogorov's Theorem, 107, 134-135
L
Lanchmter, R d e r i c k William, 20
1,anchwter Square Law, 21
La~~ehwter
Thtliary, 2
tanchester quation, 2, 8, 20
ambush varicsnt, 23-24
and Gause, 14, 2425
attrition in, 20, 513
demity in, 22-23
linear digerential, 22
reinforcement, 24
square model, 20-21
Langlois, Jean-Pierre, 84
tefxhetz map, 149-150
Xegdttzation (drug), 402, 104
limit cycle, 3, 107, 12&121, 1123, 129-131,
241
Hilbctrtk 16th problem, 122
Minorsky's definition, 121-122
see also negative tmts
linear homogenctous problem, 109
linearization, 118114
failure of, I f 4- 115
Iineilrizd stability andysis, 3, 95, 101
LiouvilleTsTheorem, 120
Lipschitz, f 08
Loth-Volterra
ecosystem equations, 2, 9, 17
predator-prey model, 76, 93, 96
vs, Richardson's mod&, 2, 47
Lyapunov function, 3, 1Lei-119
M
nzallizemilticd biology, 1, 7-9, 15, 76
Loth-Volterra model, 47, 76, 93, 96
see Loth-Volterra c?qu&ions
mathematical theories of arms r w a
see Richardsoxlk model
mathematical tools, 1
May, Robert M., 11, 1%-135
McKendrick, 15, 72-73, 91-92
model
Adaptive Uynmic, 2, 19, 28, 36-35
agent-bed, 4
as illuminating abstraction, 9
class struale, 77
collective security, 48, 58
drug epidemic, 90, 97
globocop, 48, 58
Kermmk-McKendrick threshold, 15,
72-13, 92-92
model (contud)
Lmchester nonginear attrition, 23
linear with variants, 48
Loth-Valterra, 47, 76, 93, 96
movement in war, 28-29
noniinem with variants, 58
of complex graces=, 1
reaction-diffusion , 75, 19, 97-99,
111-114
Richardsonian comaetit ion, 48
SIRS. 80
SIR epidemic, 73, 92, 95
two-counlry, 43
war, 19
see Rchardson model
mutualism, 10, 69
mutualistic gopulstions, I l
N
negative tests, 131-434, 141
nonltinear arms r x e models
see arms races
nonlinear autonamous system, 108
nonlinear dynamical systems, I , 3, 107
EIopf Bifurcation Theorem, 3, 107
125-129
Index Theory, 3-4, 107, 235-143,
147-150
limit cycles, 3, 120
Lyapunov hnctions, 3, 116-119
Paincar8 maps, 3, 307, 129-131
TFZaineizfBBendixson Theorern, 3, 107
122-123
PoincarBHopf Index Theorem, 3, 108,
144-146, 148-149
stability malysis of, 3, 109-120
theory, 1
nonlinear reitction-diRusion model, 75, 79,
97-99, 111-l f 4
P
Poinearci? map, 3, 107, 129-131.
PoincarBBendixson Theorem, 3, 107, f 22124
PoincarBPXopf Index Theorem, 3, 2 08,144
146, 148-149
Poincardk Theorem, 116
political gievanca, 45
principal of competitive exefusion, 14, 25
R
b g o z i n t s negative test, 133
reation-diRusion equdions, 75, "1, 92-99,
111-114
see Fis;herk equation
reciprocal, mtivation
see mutualism
reciprocal d t r u b m , 41
removal capacity, 79-80
repression, 80, 84-86
revdution, 2, 7, 15, 69, I11
as epidemic, f 5 1 7 , 71-72, 74
deeentrdzed totalitarianism, 82-83
models of, 1516, 12-86
removal capacity; 79-80
reprmion in, 80, 84-86
SEW model, 80
traveling wavw, 75
Richardsonk model, 2, 8, 20,4548, 58, $4
bmic, 45
wonomie fatigue in, 11, 46
externd threat t e r m , 46
gtobacap and, 48, 52, 58, 65
gievancw in, 45
simple analytics of, 46-41
twecountry, 43
vs. collective swurity, 48, 58
vs. Lot h-Voiterra wossysdem m d e l ,
10, 47
Robimon , Miehaet , 2 2
S
Samuelson, P a d A., 70-71, 78
SIR epidemic model, 73, 92, 95
socid change
explosive, 3, 86
social dynmics, 1, 86
mciai, systems, 1
and nonlinear dynmics, 71
stability andysis, 3, 95, 107, 109-114,
117-119, 124-125, 134-135
stability criterion, 1X.
St~km'Thearern, ?Q
str~~ctural
stability, 124
T
totdi tarimism, 82-83
traveling wave solution, 2 11-112, l14
V
Van der Pol oscillator, 127- 128
Verdun, 36, 40
vital d,mmics, 76
W
WilXtmsn, Paul, 72-73
war, 2, 7-8, 19, 28, 69, 111
attrition and withdrawl in, 20
attrition rates, 20, 30
cwudty-exchange ratio, 27
drug, 91-99, 103
fixing operations in, 36
grremina, 36
mathematicd modeh of, 19
pain thresholds in, 30
stmdoff in, 36
Weis, Herbert, 22-23
withdrawl rakes, 32 -35