Delay-Line Models of Population Growth
Author(s): E. R. Lewis
Source: Ecology, Vol. 53, No. 5 (Sep., 1972), pp. 797-807
Published by: Ecological Society of America
Stable URL: http://www.jstor.org/stable/1934295 .
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MODELS
DELAY-LINE
OF POPULATION
GROWTH'
E. R. LEWIS
Departmentof ElectricalEngineeringand ComputerSciencesand theElectronics
of California
ResearchLaboratory,University
Berkeley,California94720
A bstract. Replacingthe usual principleof conservationof numberswith the equivalent
of flow(or rates) leads to a veryheuristicapproachto modelingtime
principleof continuity
lags in populations.The approach allows the directuse of unequal time lags (or age classes
Furthermore,
of unequal lengths)withoutthe implicitassumptionof a stable age distribution.
for simple,linearnatalityprocesses,it allows directestimationof the biotic potentialand of
thefrequenciesof inherentoscillations.
the age dependence of natality and N is a vector
Pure-birth processes have been described by a comprisingage cohorts. Since eq. (2) representsthe
fairlylarge number of diverse mathematical models aging process in implicit form, it occasionally is
(see Keyfitz 1967 and Goodman 1967). In light of coupled with an explicit representationof aging, the
the complex and stochasticnature of various natality basic von Foerster equation2 (von Foerster 1959,
processes, one should not be particularly surprised Sinko and Streifer1967, Fredrickson 1971):
at the ever-increasingdiversityand complexityin the
DN(t,7)
DN(t,z)
(4)
published models. In fact, one finds that the most
Dy~~ ~
abl
at Dt
sophisticatedand complex models are those describing the natalityprocess that is probably the simplest N(t,y) is population density at age 7. Death proand most primitive,binaryfission(see Bronk, Dienes, cesses, migration terms, and density dependence of
and Paskin 1968). When more-complex birth pro- course can be added to any of these models (somecesses are treatedwith similar attentionto detail, the times at the expense of analytical manageability);
models promise to be extremely complicated and and the effectsof time delays, many of which are
carried implicitlyin eq. (2), (3), and (4), can be
difficultto manage analytically.
Most deterministicmodels of pure-birthprocesses incorporatedin eq. (1) by convertingit to a simple
differenceequation such as that proposed
are elaborations of one of three basic models (for differentialan excellent review of stochastic models, see Bha- by Tognetti and Mazanov (1970) for certain insect
rucha-Reid 1960): the differentialform of the Mal- populations:
thusean equation (see Pielou 1969):
dNMt (5)
o'cN(t -T)
dt
dN
dt
where N(t) is the numberof adults at time t, oxis the
INTRODUCTION
where o is the intrinsicbirthrate and N is the total
population; a form of the renewal equation (Lotka's
equation) (Feller 1941):
dN
d=
G(t)
+
())
dN
d
(2)
where 7 is age, oc(7) is the age-dependent natality,
and G(t) is the continuingcontributionto the birth
rate by the initial members of the population (those
alive at time 0); and a discrete approximationto the
renewal equation, the Leslie-Lewis vector difference
equation (Parlett 1971):
Nt + T = ANt
(3)
where A is an extremelysparse matrixthat describes
'Research sponsored by the Joint Services Electronics
Program, contract F44620-71-C-0087 and the National
Science Foundation, grant GK-3845. (Manuscript received December 28, 1971; accepted March 24, 1972.)
rate of egg production per adult, and T is the duration of the egg stage.
Equation (5) points up a difficultyinherent in
eq. (2), (3), and (4). Certain specific, reasonably
well-fixed,discretedelays often are importantaspects
of pure-birthprocesses. Equations (2) and (4) are
designed to deal with a continuumof age dependence
and thus a continuumof time delays, and eq. (3) is
designed to provide a discrete approximationto that
continuum. When discrete delays such as gestation
periods, nonreproductivelactation periods, and ovulation intervals are to be incorporated in a model,
equations designed for a continuum of delays are
neither economical nor especially revealing. On the
other hand, a delay such as any of those mentioned
above generally exhibits a stochastic nature, and
representingit as a single number may be a gross
2Sinko and Streifer(1969, 1971) have modifiedthis
variable,
equationto includemass as a thirdindependent
size as well as age.
thusincorporating
Ecology, Vol. 53, No. 5
E. R. LEWIS
798
simplification.However, in most models of population dynamics,random variables are reduced to their
mean values; for example, in eq. (2), o(*') usually
is definedto be the mean value of natalityfor a given
age at.One is no less justifiedin similarlyconsidering
the means of discrete delays.
In thisnote I shall discuss two methodsof modeling
natalityso that discretedelays are retainedexplicitly.
One distinctadvantage of these formulationsis that
one often can deduce from them directlythe r-lationshipsbetween the delays (or their mean values)
and the biotic potential (innate reproductive capability) of the species involved. Furthermore,these
formulationslend themselvesrather readily to block
diagrams and are thereforeeasily visualized and quite
heuristic.Another advantage, in contradistinctionto
the Leslie-Lewis formulation,is that this method allows direct use of unequal time delays (equivalent
to age classes of unequal lengths) without the implicit assumptionof a stable age distribution.A major
disadvantage (with respect to eq. ( 1 ) and (3) ) is
that they include continuous delay lines and therefore inherentlydo not have a finitenumber of state
variables. They are not as difficultin this respect as
eq. (2) and (4), however.
MODELING
DISCRETE
input functions but do not otherwise alter them.
Equation (4) describes a delay line whose input is
the densityof newborns (number per unit time) and
along which moves a waveform (population density
given in individuals per unit age) that does not
change shape as it advances (Fig. 1). In its more
complete form,with death and migrationterms,the
equation becomes
DN
DN
+
(4a)
N + rN = ?
DX
where 8 is the reath rate and r is the migrationrate.
This equation does not describe a pure delay line;
migration and death will alter the waveform as it
advances. The equation was derived from a basic
conservation principle: at any age other than zero,
the number of individuals is changed only by three
processes: aging, death, and migration(von Foerster
t
+
N0
N,
N2
N3 N4 Ni+1(t+l)
N=i)
SalN ja2N2la3N3ta4N4
DELAYS
The von Foerster equation provides a dynamic
picture of the process of aging in a population. In
the form of eq. (4), where death and migration
terms have been eliminated,it describes a very simple flow process such as one mightfind along a conveyer belt moving at constant speed or through a
telephonewire carryingmessages at a constantspeed.
Material or messages are loaded at the input end and
flowsmoothlyalong a route toward some destination.
A stringof material or messages thus is distributed
over the route, and at any point along the route one
finds,unaltered, the material or messages that were
loaded at some time in the past. We can lump all
such flowprocesses into the general categoryof pure
"delay lines," which introducetime delays into their
N0 N (t+)=
FIG. 2. A discrete(segmented) versionof the delay
line of Fig. 1, withthe input(births) determined
by the
weightedsums of the delay-linecontents.The open sethe delay line; the circles
quence of rectanglesrepresents
representscalors that apply weightingfactors (&x's) to
the contentsof the delay-linesegments;and the long,
open rectangleis an adder that sums the weightedcontents.
1959). In the renewal equation (2) that generates
the input function,N(t,O), to the delay line of Fig.
1, the contentsof the line are sampled continuously
along its length.This is most easily diagramed (Fig.
At t=t,
2) from the discrete approximation representedby
N*1O
_
_
_
_
_
_
________
eq. (3), where the delay line is segmented and the
N(tl,0)__
densityfunctionin each segment integratedto yield
a single number.
0
x
In the case of a single, discrete delay, one may
Ht2-H
Ata latertime
reduce the number of terminalson the delay line to
t=t2_
two, one input and one output. Furthermore,with
N(t 0)
!
such delays it is most convenientto convertthe usual
FIG. 1. Von Foerster'sdynamicconceptof a deathless principles of conservation of numbers to an equivpopulation.The open-endedrectanglesrepresenta delay alent principle of "continuity"of the flow from secline, along which waveformstravel withoutchanging tion to section of the delay line, completelyanalogous,
shape or amplitude.In this case, the waveformsN(t,x)
representpopulationdensity,and distancealong the de- for example, to the continuityprinciple applied to
lay line represents
age (x). The velocityof travelis one. flow processes in fluid mechanics. Rather than con-
LateSummer1972
799
DELAY-LINE MODELS OF POPULATION GROWTH
sidering the number of individuals at a particular
age, we shall consider the rate at which individuals
pass througha particular age. Thus, for example, if
we happen to be ignoringdeath, we shall apply our
conservationprinciple by demanding that the rate at
which individuals emerge from a 2-year delay is
precisely equal to the rate at which they entered 2
years ago. In other words, the rate of emergence of
2-year-olds today equals the birth rate precisely 2
years ago. By applying the conservationprinciple in
this manner we shall be able to constructwith ease
very heuristic systems diagrams of our population
models.
Consider, for example, a population of idealized
bacteria, each of which undergoes fission every 10
min. Using classical approaches, we could describe
this population rather simply with either eq. (1) or
eq. (3):
Integrator
aNCI)
NMt):faNt) dt
-,,,
FIG. 4. The classic model for the idealizedfissionprocess of Fig. 3.
cion is confirmedwhen populations with several discrete delays, such as those of mammals, are considered. The relationshipbetween a and the various
delays becomes an end resultof modeling ratherthan
a startingpoint.
While it may be a more fundamentalrepresentation than eq. (6), eq. (8) also has somewhat more
dN
(6)
-d = N;x = [(log 2)/10]
complicated solutions, which are difficultto express
precisely
in closed form. One can discuss this differor
in termsof Fig. 3 and 4. The sysence
conveniently
(7)
Nt2Nt-10
tem of Fig. 4 has a single state variable, N(t). If
the unit of time in each case being 1 min. On the N(t) is known for any single value of time, the beotherhand, if the process of fissionis sufficiently
con- havior of the system can be predicted precisely for
tinuous or if the population is sufficiently
large that all time. It is simply
we can definea continuousrate of fissionat any time,
N(t) = Noeat
then we can say that the rate of fissionnow is precisely twice the rate of fission 10 min ago. We may In order to predict completely the behavior of the
representthis view diagramaticallyas shown in Fig. systemof Fig. 3, one must know the entire contents
of the delay line at some moment in time. In other
one must know the formof the functionN(t)
words,
10-minute
delayline
over a complete 10-minspan. Justas the initialvalue
of N is constrained to be positive for eq. (6), the
of N(t) over the specifiedsegmentfor eq. (8)
slope
A
dNI 2|
dN
is constrained to be greater than or equal to zero
dt t
dt -Omin
dt -Omin
(assuming no death or migration). In serving as a
boundary condition, this segment of N(t) carries
informationabout the proportion of the initial population participating in fission. Since we initially
FIG. 3. A delay-linemodel of an idealizedfissionpro- assumed that every bacterium participated,we must
cess. In thiscase the fissionintervalis 10 min.
conclude that in this case our boundary conditions
are compensatingfor a deficitin our model. Such a
3. The current products of fission enter a 10-min deficitis not presentin the classic differenceequation
delay. On emerging from this delay, they immedi- (eq. (7)). However, this formis no more fundamenately undergo fissionagain. Therefore,we may write tal nor useful than that of eq. (6) when several
a thirdequation to describe the population:
delays are involved.
Even when the contentsof the delay line are specdN
dN
(8)
ified
at some point in time, it is difficultto express
dt
_
t-iO
the future(or past) behavior of the systemof Fig. 3
For contrast,a diagramatic representationof eq. (6)
in analytical form. On the other hand, in this paris presentedin Fig. 4. The major distinctionbetween ticular example it is not difficultto construct the
the two representationsis the replacement of the behavior graphically. One simply draws the known
integratorin the classic model by a delay line. When segmentof N(t), thenrepeats it forsuccessive 10-min
one computes the appropriate value of a (= log 2/ intervals,amplifyingit successively by factors of 2
10), he begins to suspect that the model of Fig. 3 (see Fig. 5).
is the more fundamentalrepresentation.This suspiFrom the known segment of N(t), one also can
800
E. R. LEWIS
N
f//
Ecology,Vol. 53,No. 5
average measure of the innate reproductivecapability
of the species representedby eq. (6) or the systemof
Fig. 3. Therefore,as is often done, we shall definea
to be the biotic potential of that species.
DELAY LINE MODELS THAT LEAD TO DIFFERENTIALDIFFERENCE EQUATIONS
-~~~~~~~~~~~~~~~~~~~
v
Z
I
In probably theirsimplestapplication to vertebrate
populations, delay lines can be used to representthe
mean time (T1) between birth and the initiationof
offspringproduction and the mean duration (T2) of
the productive period. If one ignores age-dependent
fecundityand assumes that the rate of production
of new females, N], is directly proportional to the
total number of productive adult females, NPt, he
can write
/
~~~~Bl
explog2t]
Pf
-_
kNpf
(9 )
Assuming no deaths,
0
20
10
30
40
50
t- T1
Np=
t (min)
FIG. 5. One of many growthcurvesfor the model of
Fig. 3. Ordinate,population;abscissa,time.
find two exponential functions (as shown in Fig. 5)
that precisely bound the behavior of the population.
These functionshave the form
to
+ Bi exp [a
log 2
A + B2 exp [a
(10)
Substitutingeq. (9) into eq. (10) and decomposing
the integral,one obtains
t- T1
t-(T1+T2)
= k rNvfdt
-k
Npvf
-
t]
f Nfdt .
t- (T1+T2)
cc
NPfdt.
-
1
cc
Differentiating
both sides with respect to t leads to a
simple differential
differenceformof eq. (9).
t]
dtvf = k[Nf (t - T1) - NPf(t - T,T2)1]
where the unit of time again is I min, and A = 0 if
(hla)
a =
all of the initial population participates in fission.
One can approximate the behavior of the system of Substitutingeat for Nvfin eq. ( 11) or ( 11a), one can
find the Malthusean approximation and a transcenFig. 4 ratherwell with any functionof the form
dental equation relating the biotic potential, a, implicitlyto T1 and T2:
A + B exp [a t], a -lo 10
where B lies -betweenB, and B2. Such a Malthusean
approximationwill intersectthe actual behavior every
10 min and will never be an error by more than
100%. Malthusean approximationswith values of a
that differfrom (log 2) /10, on the other hand, will
diverge progressivelyfrom the actual behavior and
the maximum error will increase without limit. The
exponential coefficientof the nondivergingMalthusean approximationcan be found by substitutingeat
into eq. (8):
fromwhich
This value of
a,
a
= k e-aT(l
-
e-aT2)
.
(12)
One also can construct a systems diagram directly
from eq. (11) with its decomposed integral. However, a simpler, but completely equivalent diagram
(Fig. 6) can be constructed directly from simple
considerationsof conservationand continuity.In the
absence of death, the rate at which the population
of productive females changes is completely determined by two terms-the rate of addition of newly
maturing females (i.e., the rate at which females
emerge from delay T1) and the rate of removal of
meat
2mea(t+10)
sexually senescent females (i.e., the rate at which
females emerge from delay T2). The delay line T2
contains all the productivefemales and the integrator
log 2
simply provides a running total of that delay line's
contents. The scalor k relates the total number of
then, can be taken as a long-term productive females to the rate of production of new
Late Summer1972
DELAY-LINE MODELS OF POPULATION GROWTH
Delayline
T,
Delayline
T2
-
~~~T
Nf
I
Pf
=~~~~~~
Nf
801
L_~
k
Integrator
~
T2
eT
e-Ta
-
~~~~~~~~~~~~~dN
d
FIG. 6. A delay-linemodel of natalityin an idealized
Npf
deathlessvertebratepopulation,Nf is the presentbirthIntegrator
rateof females;NPfis the totalnumberof sexuallyproFIG. 7. The model of Fig. 6 witha nonselectivedeath
ductive females; T1 is time from birthto sexual maprocess added. The poisson death rate is 8 (individuals
turity;T2 is the durationof sexual productivity.
per individualunittime).
females,which then enter delay T1. The characteristic differential-difference
equation (11 a) can be
found by direct analysis of the systemdiagram, beginningat the rightof the summingpoint.
The effectsof an ideal, nonselective(i.e., Poisson)
death process can be added rather simply to the
model. If the fractional death rate is 8 deaths per
individual per unit time, then one can account for
the deaths in each time delay, T, by introducinga
survivorshipscalor, e T, in series with it. Thus the
rate at which surviving individuals emerge from a
delay T is e-5T times the rate at which they entered
it T time units ago. The rate of change of productive
females now has three components: the rate of addition of young females survivingto maturity:
(e-5Tl)
Nf(t -T),
DELAY-LINE
the rate of subtractionof females survivingto sexual
senescence:
)Nf(t -
(e-(Ti+T2)
T-
T2)
and the rate of attritiondue to death:
Npf(t)
These processes are representedin the systems diagram of Fig. 7. Beginning at the rightof the summing point in this figure,one easily can write the
characteristicdifferential-difference
equation for the
system:
dNpf =
dt
-e-5(Ti+T2)Npf
(t
-
T
e-1TNPf(t
- T-
T2)
-
-
T1)
Npf (t)
(13)
The transcendentalequation for the biotic potential
becomes
a
(e-5T,)e-aT,
_ (e-- (l+T2))e
type (Bellman and Cooke 1963, Hale 1971). This
will be a common result in population models whose
dynamic propertiesare determinedboth by discrete
delays and by mass-action phenomena. (The models
of Fig. 6 and 7 contain two first-order
mass-action
phenomena: a first-order
natalityprocess and a firstorder death process.) Many population systems are
representedmost naturally by models that incorporate mass action; and when delays also are important
in such systems,then models often lead to retarded
differential-difference
equations (see Wangersky and
Cunningham 1956, Tognetti and Mazanov 1970).
Such equations generally are extremely difficultto
manage analytically.
a(TI+T2)
(13a)
Systems models, such as those of Fig. 6 and 7,
that contain both delay lines and integratorsin any
loop inherentlylead to differential-difference
equations. The two population models of Fig. 6 and 7 led
to differential-difference
equations of the retarded
MODELS
THAT LEAD
LINEAR DIFFERENCE
To
SIMPLE
EQUATIONS
If one looks into the fine structureof the natality
process, one generally finds discrete processes with
fixeddelays. Thus, for example, it often is more natural to discuss a reproductiveintervalthan a reproductive rate. The latter is only approximatedby the
reciprocal of the former;and in some cases, such as
that of binary fission, the approximation is rather
gross. One can model such reproductive intervals
quite naturallyand easily with delay lines. As an example, consider a population of idealized fin whales.
On reaching sexual maturity(at age T1), a female
whale becomes pregnant,producing one calf at the
end of her firstgestationperiod (TG) and becoming
pregnant again at the end of her nonreproductive
lactation period (TL)- She continues this process as
long as she survives,producingone calf every TL plus
TG years. Half the calves born at any given time are
female. The proportion of female calves surviving
to become sexually mature adults is Yi (This factor
may include the effectsof infant mortalitydue to
deaths of lactating females.) The survivorshipsfor
pregnantand lactatingfemales are YG and YL respectively.With simple bookkeeping (conservation) one
easily can constructan appropriate systemsdiagram
(Fig. 8) from the verbal model. Beginning at the
E. R. LEWIS
802
Ecology, Vol. 53, No. 5
P
b
8. A verysimplemodel of fin-whale
natality(see
textfor details).
FIG.
left in Fig. 8, one requires a delay of T1 and a survivorshipscalor Yi between the rate of productionof
newborn females and the rate at which maturing
females are added to the productivepopulation. The
rate of addition of maturingfemales is summed with
the rate at which already-productive females are
completinglactation,the sum representingthe present
rate of impregnation. Between impregnation and
birth, a gestation delay of TG and a survivorship
scalor YG are required. The total birthrate is divided
by 2 to yield the rate of productionof females,which
is the input to the firstdelay line, T1. After giving
birth,the females are delayed by TL and scaled by
YL before reenteringthe reproductivepool and becoming pregnant again. From the diagram one can
write a characteristicdifferenceequation, the exponentialsolutionof which will yield the biotic potential
of the idealized whale population. For convenience,
we can write the equation in termsof the output of
the summer (i.e., the rate of impregnationP):
P = (l/2)yIyGP(t
-
TG - T1)
+ YGYLP(t
-TG
-
TL).
(14a)
Substitutingeat for P, one finds
1 = (1/2)yiyGe
a(TG+T')
+ Y;yLe
a(G+T
L)
(14b)
which is a transcendentalexpressiongiving the biotic
potential implicitlyin terms of the delays and survivorships.
C
TotaI
birth
rate
Birth
rate
offemales
One mightwish to include even more finestructure
of natalityin the model. For example, adult females
that are neitherpregnant nor lactating do not automatically and instantaneously become pregnant.
Therefore the ovulation interval becomes a potentially importantfactor. Again, one can model this
with delay lines. Consider the same baleen whale
population, but with an ovulation interval To and a
probability5 of impregnationat any given ovulation.
This leads to the systemsdiagram of Fig. 9. Baleen
whales (and most other groups of mammals) show
seasonal variation in fecundity,rate of impregnation,
or both. This can be incorporatedin the scalor 5 by
making it time dependent. Finally, there is some evidence (Laws 1961) that nulliparous baleen whales
have ovulation intervalsof as much as a year, while
FIG. 9. A slightlymore completemodel of fin-whale
natality.P representsthe rate of impregnation
(see text
forremaining
details).
(b-
~a
FIG.
10. An even more completemodel of fin-whale
natality. T'o represents the ovulation interval for nulliparious whales, and P'is the probability of impregnation
of an ovulating nulliparious whale (see text for remaining details).
the ovulation intervalof primiparousand multiparous
females generally is I month. This dichotomy can
be included in the model quite easily, as shown in
Fig. 10. Characteristic differenceequations can be
obtained easily for the systemsof Fig. 9 and 10. In
the case of Fig. 9, we must sum the effectsof three
loops:
(t T,,T
(11PS)P(t) = PYGO/2)7170
+ (1-
P)YO
+ PYGYIP(t
?t-T)
Y
- T(;- Tj)
(15)
where the firstterm on the right representsloop a;
the second termrepresentsloop b; and the thirdterm
representsloop c. The correspondingtranscendental
equation for the biotic potential is
1 = (1m2)YonthG.eTh
+ yo(l1 -
P~tT0)
-
P)e-aTo
+
d+iTo)
P(t-
YGYL~e--a(TG+TL)
TLO)
The system of Fig. 10 is more complicated, having
four loops. Analysis proceeds most easily if the rate
of impregnationis decomposed into P' and PI' and
the effectsof loops a and b are summed separately
froc9 the effectsof c and d. For a and b, we have
(1/P')P (t)
I (12)YGY1YoP (t -
+ PI (t - TG- Tj)
+ ( 1-1 P)Yyo
T1)
TG-
I,
(16)
DELAY-LINE MODELS OF POPULATION GROWTH
Late Summer1972
and for c and d we have
( 1/ )P (t) = y(,,yl[P (t-
+ P"(t - Tc- Tl)]
+ (l-)y P7
TL)
TG-
(1/2)
-
-
(1/2)
'1("oyo
+
1
P(tf
1
-
-
To
0
PYY
P"(t)-
P"(t + TL
T7)
_(1-
')
(1
-
7)
,"
P"(t-
-
P (t
+ (1/2)YlY'(YG
TG
TL)
-
+ T+
+ T,
P"f(t)-
),
L
-
To -To
T)
-
(18)
Tj)
TG -
pf(t
P"(t + T(;, + T.-T'o)-Y
?5,
?
To + TG + TL,)
(t-
P"(t
()I/2)Yyl'oyG
TI)
T1 +
TG
and the solution can be substitutedfor each P' term
in eq. (16):
Next we can solve eq. (17) for P'(t):
P"(t + T(; + T) )-
P"f(t?
PYGYL
Y
P"(t-To
YGYL
P"(t-T.) 1-T
(17)
p
1
~
P'(t)
803
- T
+ TG + TL)
Th YLr,
"(
from which one obtains a 10-term transcendental
equation for the biotic potential.
Althougheq. (14) and (18) may seem rathercomplicated, they are in fact simple linear difference
equations, for which analytical techniques are well
established (see Goldberg 1958). In addition to biotic potentials,the various oscillatorymodes inherent
in the models also can be evaluated by application
of these techniques. Returningbrieflyto simple binary fission,the characteristicequation was
The solution by this method for simple binaryfission
becomes
dN
t
to + 7=
2T
dN
dt
to
(21)
where the choice of to is completelyarbitrary.Geometrically,this solution representsa sequence of discrete points at integral values of T. In order to
describe the behavior of the equation completely,
one would need an infinitesum of such solutions,
one for each point on a continuous curve representdN
dN
ing dN/dt over a span of T time units.
dt ]t
dt
t-T
On the other hand, one can connect with a con(where T is the fissioninterval). This can be rewrit- tinuous curve all the points of a single solution of
ten as follows:
the form of eq. (21). Clearly, one such curve is
t
given by
let 7 __
dN
and
=x
dt
then xT = 2x].
y(to + t) = y(to)e(
/log 2
t
)t
T
(22)
Furthermore,by appropriate choices of to, one can
obtain two solutions of this type that form, respecfor all nonnegativeintegralvalues of -. The solutions tively,the upper and lower bounds of all such soluto linear differenceequations such as this always take tions intersectingthe actual behavior of the system.
This is precisely what was done graphically in Fig.
the form
5. However, since these solutions representsdN/dt,
xT = nIt
(20)
they must be integratedto provide the bounds of N.
where in is a root of the characteristicpolynomial of Thus, if
the differenceequation. The polynomial is found
Iog 2
t
quite easily by collecting all terms of the difference
Y(tni,+ t) = y(tM) e t Z )
equation on one side and substitutingeq. (20). Thus,
describesthe upper (or lower) bound of all exponenfor eq. (19)
tial solutionsintersectingdN/dt, then
x-
n1_-
2x
0
m-2
m
0_
l
2mT-1
2.
= 0
(19)
z (tnt + ?0 )
T
lo( 29)
log 2 y(tJn)e 10g
=Z (t ) et
log 2
a.7
t
Ecology, Vol. 53, No. 5
E. R. LEWIS
804
describesthe upper (or lower) bound of all exponential solutions intersectingN. Clearly, in the case of
simple exponentials, integrationdoes not alter the
functionalform of the bound. The upper and lower
bounds, taken together,form an envelope confining
the behavior of the system.
In the case of a linear natalityprocess with more
than one delay, the difference-equation
solution generally will have oscillatory terms, and these can be
incorporatedas temporal fine structurein the envelope of the systems behavior. Presumably, in any
practical model, each time delay can be expressed
as an integertimes a common divisor of all the delays (i.e., the common divisor becomes the unit element of an integral domain to which all the delays
belong). The order of the resultingdifferenceequation (and its characteristicpolynomial) will be given
by the longesttime delay divided by the greatestunit
element T (the greatestcommon divisor of all delays
in the model).
Consider a simple population of idealized herring
gulls. Beginningat age 3, each adult female annually
produces a brood from which n females survive to
full fledge (age 2 months) and n y survive to age 3.
From the time of full fledge until the end of her
life, each female faces a constant probability of
death, which can be expressed as a fractional death
rate, 8 (deaths per bird per year). Employing delay
or complex conjugates. The positive real root will
representa growing term if the root is greater than
unity,a constant term if the root is equal to unity,
or a declining term if the root is less than unity. A
negative real root represents oscillation at an apparent frequencyof 1/2T (i.e., the term nT is positive for even x, negative for odd :). If the magnitude
of the root is greaterthan unity,the oscillations will
grow; if it is equal to unity,the oscillations will remain constant; if the magnitude of the root is less
than unity,the oscillationswill be damped. The same
statements hold for a conjugate pair of complex
roots; but the frequencyof oscillation will not equal
1/2T.
Thus Jr generally will contain one nonoscillatory
term and one or two oscillatoryterms:
either
J,= A ?ml7 + BIm2fT cos 7C + CInm3f cos As
(25a)
or
Jr= Aim,1f+ Bfm2jTcos(0 ? c)
and, the envelope of J will be given by two equations
of the form
Jenv= aeat + beftcosjtt+ ceytcosjtt (26a'
or
Jenv= aeat + beftcos(Ot+ c) .
+
3 yrs.
+yr.
I
(25b)
where C E Z+;
(26b)
From the envelope of J alone, we cannot determine
the envelope of N (in the sense of a curve that periodicallyis tangentto the actual solution). However,
since J cannot be negative, we can determine the
upper and lower bounds of all possible envelopes:
t
11. A delay-linemodelof an idealizedherring-gull
population.J is the rate of nesting(see textfor remaining details).
N -
lines, one easily arrives at the systemsmodel shown
in Fig. 11. The greatestcommon divisorof the delays
is 1 year, and the correspondingdifferenceequation is
( 2 3)
+
e-aJTIl
J,-nyJT,_3
N - Nmin
FIG.
where J is the rate of nesting,r is an integerin the
domain for which 1 year is unity,and
y =exp[-( (17/6)8
The characteristicequation is
(nyJ'c1-,
Nmax
Ninax)dt
(27a)
t
=f (nyJ"CI-V
NW11l11)dt (27b)
_00
where Tenvdescribes the upper bound of J and J",,v
describes the lower bound. Evaluation of NlllaXand
Nmin can be achieved quite easily with the aid of
Laplace transforms.From (27a), for example,
= ny Y UJl""-)/G + 8)
(28)
Y(Nniax)
where s is the complex variable of the Laplace transform.The functionNmax can be found directlyfrom
e-5m2
(m3
(24)
0,
ny)
eq. (28). Qualitatively, the effects of the factor
and the solution to eq. (23) takes the form
1/(s + 8) on Tenvare threefold: (1) The relative
= alml? + a2m2r + a3m3T
Jamplitudes of the various components (oscillatory
and nonoscillatory) are changed, the slower or lower
where ml, in2 and m3 are the roots of eq. (24).
By Descartes's rule we know that eq. (24) has at frequencycomponentsbeing favored. (2) The oscilmost one (the number of reversals of sign) positive latory components will be shiftedin phase (but not
real root. The remainingroots must be negative real in frequency). (3) An additional nonoscillatoryterm,
-
--
DELAY-LINE
Late Summer 1972
MODELS
OF POPULATION
will appear. For a quantitative example of
these effects,consider the second term in eq. (26b),
De-5t,
beftcos(Ot + c)
The factor 1/(s + ') convertsthis term to
bVeftcos(Ot
+ c + 11') + d'e-5t
LJi-2
where
805
can be replaced by a vector state equation consisting
of a k X k state projection matrixand a k-orderstate
vector. In the case of the herringgulls we can write
-Ji
Ji-1|
(29a)
GROWTH
1
e-5
1
?
0?
i =,r+l
~J
ny
0
?
1
0
(30)
IJi-1,
Ji-2 ji=,
-
Because mass action could be ignored and continuity
could replace conservation,the elements of the state
V(E + 3)2 + 02
vector are rates ratherthan age cohorts.
0
If, on the other hand, we had taken the mass ac.
(29c)
tan--'
IIJtion approach, the elementsof the state vector would
Such shiftingof amplitudeand phase, of course, does be age cohorts and the elements of the state projecnot alter the rate constants and frequencies of the tion matrixwould be different.
Since natalityin individual gulls is assumed in the
various terms; therefore,we can obtain the biotic
potential along with the frequenciesof various oscil- model of Fig. 11 to be preciselyannual, it is simplest
lations directly from the solution of eq. (23), or to formulate the corresponding Leslie matrix by
analogous equations fromotherlinear natalitymodels. assuming that the entire population is synchronous.
Define t = 0 to be immediatelyafteregg laying (each
nest has a full clutch), and r is an integerdefinedby
RELATIONSHIPTO THE LESLIE MODEL
b
bf=
(29b)
If we are contentto ignore the temporalfinestructure embedded in the initialfunctions(i.e., the initial
contents of the delay lines) of the models of the
previous section, and instead concentrateon the envelopes and the natural frequencies contained in
those envelopes, then we effectivelyreduce the number of state variables in the models to a finiteset.
In fact, the dimension k of the state space becomes
simply the order of the differenceequation, which
N(,
[N|
0
noe-0
7o] E
I.
0
1
e-a
0
0
L 0
LNkrlT
where y' =y-e
n(e-0
0
s
.....
. . .
which can be simplifiedas follows:
-)
i = 0
+ 1 is given by
(N3)
,
+
...
+ (N1C)r1
No
noe-5
N|
I
I
I
~~~~~ (31)
. ..e-6 0 ]LNk
0
r
k
.-y'noe-U
+
X
where (NX)r is the numberof x-year-oldfemale gulls
alive at r. The parameter n has been factored into
no (the number of female eggs per clutch) and yo
(the survivorshipof those eggs to full fledge). The
resultingstate equation is
e-g
X-y'no E
j=O
=
(e-
k
lim E: (e-8>-1)
Kx
jY'no
1 e-eX-
e-6
y'noe-26Xkl2-2y'noe-36Xk-3
k
- y'noV-,
E (e-6X
j=2
(NO) +1 = noe 6[(N2),
0i
- (5/6) 6.
-.
-k+1
The number of eggs, No, at
e~~~~~~ 0
Owing to the transcendentalnature of the model
(which was employed as a reasonable convenience
ratherthan a logical necessity), k in the Leslie formulation is not finite.Nonetheless, the characteristic
polynomial and eigen values (A's) of the matrix are
easily found. First, fromthe method outlined in Pielou (1969: 30) we have
Xk+1 -
=
t/T; T = 1 year.
-
0,
,3/Xe-?-2
')i
j=
+yn1eyO
+ y'no+ y'noe-
1
0
_-6X1
e
0
y'no + y'no -y'noe-6-1
-1 - y'noe-26X-2
+ y'noe-
= 0
ny = 0 (ny = y'noe26)
Thus the Leslie formulationleads to the same characteristic polynomial as the delay-line formulation.
The Leslie matrix is a bit more cumbersome than
that of eq. (30); but this is due simply to the transcendental nature of the model. Since the dimension
of the state space is three,we could have simplified
the Leslie approach by considering only three cohortsN1, N2, and all adults NA:
NA =E
(
Ecology, Vol. 53, No. 5
E. R. LEWIS
806
/ 1
'
0
)2 =(
NA
T+1
n~e-6 '
o
e
e-6
0
NE
3
1'
28)(Ni)
(32)
NA
e-a
The characteristicpolynomial can be found quite
easily by noting that for any eigen value A, and its
correspondingeigen vector,
(NA),+,
-=(NA)T=
e 5(Ne)T+
e 6(NA AT,
from which
e-6
(NA)T
6
(N
and
In the case of the fin-whalemodel of Fig. 9, however, the greatest common divisor that can serve as
a unit elementof the time delays is, at most, 1 month
(the ovulation interval). Therefore,an algebraic state
equation, whethergenerated by the delay-line or the
Leslie approach, will be of awkwardly high degree
for analysis. Furthermore,if any of the timz delays
is carried as a parameter with unspecifiedvalue, as
it might be in sensitivitystudies, then variation of
that parameter will lead to variation in the degree
structureof the polynomial ratherthan to variations
of its coefficients.In other words, the time-delay
parameter is not carried explicitlyin the characteristic polynomial of the algebraic state equations. It
is carried explicitly,however, in the transcendental
characteristic equations (such as eq. (13b) and
(14b) ) derived directlyfromthe characteristicdifferential-difference
or differenceequations of the delayline models. In this case, therefore,the delay-line
model has very distinctadvantages, the most important of which is that it allows us to deal simply,
directly,and realisticallywithnonuniformage classes.
THE
(NA)
()+ 7 1 =
Be-6
-\
(;
e-a 6
N,)
This allows the matrixto be displayed in the standard
Leslie form
0e n0
Xe-O 8
e6
1
and the characteristicpolynomial can be obtained
directly(Pielou 1969):
X3
)X4
-
e-63
_ e-6X2-
-
-6
~~~00
- nYX = ?
ny = 0.
Thus, if we assume that natality in the gulls is
synchronized,we can develop a simple Leslie formulation that leads to precisely the same characteristic
polynomial as the differenceequation derived from
the delay-line model. If time is considered discrete,
both methodslead to a three-dimensionalstate vector
and to projection matrices that are similar and contain simple biological parameters.Thus, if one wishes
to consider time as a discrete variable, the two approaches are equally easy and, accepting the assumption of synchronizednatality,equally intuitive.This
is true of all linear natalitymodels in which the time
delays have a common divisor that can serve as the
unit element of an integraldomain to which all time
delays belong.
INTEGRAL
APPROACH
DIFFERENTIAL
VERSUS THE
APPROACH
In most population models, natalityis viewed as a
mass-action phenomenon, with the present natality
rate taken to be determinedeither by the total number of sexually productivefemales or by the weighted
sums of the numbers occupying various age classes.
This approach, which has been quite successful,is an
application of the classical state-variablemethod; in
which the rates of change of state variables are related (by a state-transition
matrix) to those variables
themselves. Most population models incorporating
time delays also have been based on the state-variable
approach. In these cases, however, the present rates
of change of the state variables are related not to the
present values of those variables but to their values
at some fixed interval of time prior to the present.
Once this has been done, of course, the notion of
state variable is no longer particularlyuseful; and
one must thinkin terms of state functions.Nonetheless, the models generated by this approach can be
useful (see Wangerskyand Cunningham 1956, Frank
1960, Lefkovitch 1966, Kiefer 1968, Pennycuick,
Compton, and Beckingham 1968, Tognetti and Mazanov 1970). In general, these models are characterized by their propertyof relating the present natalityrate eitherto the total population at some time
in the past or to the weighted sum of age cohorts at
some time past. Conceptually, using this approach,
one convertsrates to populations by integration,then
delays the products of integrationto generate future
rates. Since this approach, strictlyspeaking, is not a
state-variableapproach, and since continuoussystems
models generated by this approach can be identified
Late Summer 1972
DELAY-LINE
MODELS
OF POPULATION
by the presence of integratorsin the loops representing natality, it might well be called an "integral"
approach.
In this paper a second approach has been introduced. Here, ratherthan introducingdelays into the
population or the population vector, we have introduced them into the rates themselves. Present rates
are related to rates at key times in the past rather
than to total numbers.In its purestform,this method
leads to systemsmodels (Fig. 3, 8, 9, 10) that are
completelydevoid of integratorsin the loops representing natality. Formally, these pure delay models
are specificforms of the renewal equation, in which
the weightingfactor [I(7.) in eq. (2)] is a sequence
of Dirac delta functions.Because of the lack of integratorsin the systemsmodels, this approach might
well be called a "differential"approach.
The models of Fig. 6 and 7 representa hybridapproach. Natality is taken to be a mass-action phenomenon,but the time delays are applied to the rates
ratherthan the populations. Fundamentally,this approach is quite similar to those of Wangersky and
Cunningham (1956) and Tognetti and Mazanov
(1970). Conceptually,however, it differsin that the
time delays are placed ahead of the integratorsinstead of after them. Formally, the hybrid integraldifferentialapproach representedby Fig. 6. and 7 is
a specificformof the renewal equation in which the
weightingfactor is a sequence of steps.
It is not my intentionto propose that either the
differentialapproach or the integral approach is inherentlybetterthan the other and thereforeleads to
more realistic models. I do assert, however, that the
two methods differheuristically,that depending on
the circumstances one or the other will be more
natural, and that the proper choice therefore will
facilitate the modeling of the particular system at
hand. Clearly, where dynamics are most easily described in terms of mass action, as of course they
often are, the integralapproach is indicated. On the
other hand, time delays generally (but not always)
can be modeled most easily by the differentialapproach. Where time delays are coupled to mass
action, the most natural overall approach may be
differenceequahybrid,in which case a differentialtion will result. Certain linear natality processes, as
we have shown, can be modeled with a purely differentialapproach, in which case simple difference
equations result.
GROWTH
LITERATURE
807
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