Variational Principles and
Conservation Conditions in
Volterra's Ecology and in Urban
Relative Dynamics
Dendrinos, D.S. and Sonis, M.
IIASA Collaborative Paper
November 1984
Dendrinos, D.S. and Sonis, M. (1984) Variational Principles and Conservation Conditions in Volterra's Ecology
and in Urban Relative Dynamics. IIASA Collaborative Paper. IIASA, Laxenburg, Austria, CP-84-049 Copyright ©
November 1984 by the author(s). http://pure.iiasa.ac.at/2528/ All rights reserved. Permission to make digital or
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OF THE AUTHOR
VARIATIONAL PRINCIPLES AND CONSERVATION
CONDITIONS I N VOLTERRA'S ECOLOGY AND
I N URBAN RELATIVE DYNAMICS*
D i m i t r i o s S. Dendrinos**
I d i c h a e l S o n i s ** *
November 1984
CP-84-49
Contribution t o the MetropoZitan Study:
14
* P a p e r p r e s e n t e d a t t h e Second World C o n g r e s s
of t h e Regional Science Association,
J u n e 1984.
Rotterdam, Netherlands
-
* * D i m i t r i o s S. D e n d r i n o s
***Michael S o n i s
P r o f e s s o r o f Urban P l a n n i n g
Associate Professor of
The U n i v e r s i t y of Kansas
Geography
Lawrence, Kansas 66045
Bar-Illan University
USA
52-100 Ramat-Gan - ISRAEL
Funding from t h e N a t i o n a l
Science Foundation under
C o n t r a c t #SES-821-6620 i s
g r a t e f u l l y acknowledged.
P a r t o f t h i s work was done
while a t University of
Kansas a s V i s i t i n g F o r e i g n
Distinguished Scholar,
S e p t . 1 5 - 0 c t . 1 5 , 1983 u n d e r
t h e S p o n s o r s h i p o f t h e MidAmerica S t a t e U n i v e r s i t i e s
Association.
C o Z Z a b o r a t i v e P a p e r s r e p o r t w o r k T w h i c h h a s n o t been
performed s o l e l y a t t h e I n t e r n a t i o n a l I n s t i t u t e f o r
A p p l i e d Systems A n a l y s i s a n d w h i c h h a s r e c e i v e d o n l y
l i m i t e d review. V i e w s or o p i n i o n s expressed h e r e i n
do n o t necessarily r e p r e s e n t t h o s e of t h e I n s t i t u t e ,
i t s N a t i o n a l Member O r g a n i z a t i o n s , or o t h e r o r g a n i z a t i o n s s u p p o r t i n g t h e work.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
A-2361 Laxenburg, A u s t r i a
CONTRIBUTIONS TO THE METROPOLITAN STUDY:
Anas, A. a n d L.S. Duann ( 1 9 8 3 )
Dynamic F o r e c a s t i n g o f
T r a v e l Demand.
C o l l a b o r a t i v e P a p e r , CP-83-45.
I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s
( IIASA) , A-2361 Laxenburg, A u s t r i a .
Castis J.
(1983)
Emergent N o v e l t y and t h e M o d e l i n g o f
S p a t i a l Processes.
R e s e a r c h R e p o r t , RR-83-27.
IIASA,
Laxenburg, A u s t r i a .
L e s s e , P.F.
( 1 9 8 3 ) The S t a t i s t i c a l D y n a m i c s o f
Socio-Economic Systems.
C o l l a b o r a t i v e P a p e r , CP-83-51.
IIASA, Laxenburg, A u s t r i a .
Haag, G . a n d W. W e i d l i c h ( 1 9 8 3 )
An E v a l u a b l e T h e o r y
C o l l a b o r a t i v e Paper,
of a C l a s s o f M i g r a t i g n Problems.
IIASA, Laxenburg, A u s t r i a .
CP-83-58.
N i j k a m p , P . a n d U. S c h u b e r t ( 1 9 8 3 ) S t r u c t u r a l Change
i n Urban Systems.
C o l l a b o r a t i v e P a p e r , CP-83-57.
IIASA, Laxenburg, A u s t r i a .
Leonardi , G.
(1983)
T r a n s i e n t and A s y m p t o t i c B e h a v i o r
o f a R a n d o m - U t i l i t y Based S t o c h a s t i c Search Process i n
C o n t i n o u s Space a n d T i m e .
W o r k i n g P a p e r , WP-83-108.
IIASA, Laxenburg, A u s t r i a .
F u j i t a , .M.
(1984)
The S p a t i a l G r o w t h o f T o k y o
M e t r o p o l i t a n Area.
C o l l a b o r a t i v e P a p e r , CP-84-03.
IIASA, Laxenburg, A u s t r i a .
A n d e r s s o n , A.E. a n d B. J o h a n s s o n ( 1 9 8 4 )
Knowledge
I n t e n s i t y and P r o d u c t C y c l e s i n M e t r o p o l i t a n Regions.
W o r k i n g P a p e r , WP-84-13.
IIASA, Laxenburg, A u s t r i a .
J o h a n s s o n , B. a n d P. N i j k a m p ( 1 9 8 4 )
Analysis o f
Episodes i n Urban E v e n t H i s t o r i e s .
Working Paper,
WP-84-75.
IIASA, L a x e n b u r g , A u s t r i a .
W i l s o n , A.G.
(1984)
T r a n s p o r t and t h e E v o l u t i o n o f
Urban S p a t i a l S t r u c t u r e .
C o l l a b o r a t i v e Paper,
CP-84-41.
IIASA, L a x e n b u r g , A u s t r i a.
Anas, A.
(1984)
The c o m b i n e d E q u i l i b r i u m o f T r a v e l
Networks and R e s i d e n t i a l L o c a t i o n Markets.
C o l l a b o r a t i v e P a p e r , CP-84-42.
IIASA, L a x e n b u r g ,
Austria.
B a t t e n , D., P. N e w t o n a n d J. Roy ( 1 9 8 4 )
Nested
Dynamics o f M e t r o p o l i t a n P r o c e s s e s a n d P o l i c i e s
Melbourne.
C o l l a b o r a t i v e P a p e r , CP-84-47.
IIASA,
L a x e n b u r g , A u s t r i a.
-
(1984)
N e s t e d Dynamics o f M e t r o p o l i t a n
M a c k e t t , R.L.
Leeds.
C o l l a b o r a t i v e Paper,
Processes and P o l i c i e s
CP-84-48.
IIASA; L a x e n b u r g , A u s t r i a .
-
14.
D e n d r i n o s , D.S. a n d M. S o n i s ( 1 9 8 4 ) V a r i a t i o n a l
P r i n c i p l e s a n d C o n s e r v a t i o n C o n d i t i o n s in V o l t e r r a ' s
E c o l o g y a n d in U r b a n R e l a t i v e D y n a m i c s . C o l l a b o r a t i v e
Paper, CP-84-49.
IIASA, Laxenburg, Austria.
15.
(1984) The Changing Economic Structure of
B a t t e n , D.
M e t r o p o l i t a n Regions: A P r e l i m i n a r y C o m p a r a t i v e Analysis..
C o l l a b o r a t i v e P a p e r , CP-84-50.
I IASA, L a x e n b u r g , A u s t r i a .
FOREWORD
The M e t r o p o l i t a n Development P r o j e c t was i n i t i a t e d i n
1983 a s a c o l l a b o r a t i v e s t u d y .
I n 1984 e f f o r t s have been
c o n c e n t r a t e d o n c r e a t i n g a m e t h o d o l o g i c a l b a s i s f o r a more
focused r e s e a r c h phase s t a r t i n g i n 1985.
One o f t h e
p r i o r i t i e s i s t o a n a l y z e t h e s p a t i a l dynamics of i n t e r a c t i n g
populations.
This papaer c o n t a i n s an a p p l i c a t i o n of V o l t e r r a ' s
e c o l o g i c a l model t o t h e i s s u e o f i n t e r u r b a n p o p u l a t i o n
inoveraents.
I n t h e paper it is argued t h a t t h e Volterra
p a r a d i g m i s u s e f u l a l s o f o r r a o d e l l i ng o f human p o p u l a t i o n s .
The i s s u e of f a s t u r b a n g r o w t h and d e c l i n e i s a n a l y z e d
w i t h i n t h i s framework.
i k e E. Andersson
Leader
Regional I s s u e s P r o j e c t
November 1 9 8 4
ABSTRACT
From a n a n a l y t i c a l v i e w p o i n t , V o l t e r r a ' s v a r i a t i o n a l p r i n c i p l e s and t h e i r
associated
absolute
integrands
growth
in
single
conditions
in
and
the
multiple
field
of
species
interaction
mathematical
under
ecology
are
r e c o n s i d e r e d and s i m p l i f i e d .
They a r e t h e n compared w i t h t h e c o n s e r v a t i o n
conditions
to
found
appropriate
hold
r e l a t i v e growth i n u r b a n a n a l y s i s .
integrands
of
geographical
in a
class
of
dynamic problems
of
The comparison a s s i s t s i n i n t e r p r e t i n g t h e
(spatial)
associations as a
"stationary
effort
f i t n e s s f u n c t i o n " a s s o c i a t e d w i t h a c u m u l a t i v e e n t r o p y measure of t h e r e l a t i v e
u r b a n dynamic s p a t i a l d i s t r i b u t i o n s .
From a s u b s t a n t i v e v i e w p o i n t , t h e p a p e r shows t h e t h e o r e t i c a l c o n d i t i o n s ,
which would r e s u l t i n a l l s p a t i a l a c t i v i t y t o be c o n c e n t r a t e d i n t o a s i n g l e
p o i n t , s o t h a t i n f e r e n c e s c a n be made r e g a r d i n g t h e c o n d i t i o n s u n d e r which t h e
activity will
disperse.
It a l s o d e m o n s t r a t e s
t h a t assuming a p a r t i c u l a r
problem f o r m u l a t i o n , i n t h i s c a s e a r e l a t i v e dynamic framework i n a n i n a c t i v e
environment, w i l l r e s u l t i n o b t a i n i n g s p a t i a l c o m p e t i t i v e e x c u l s i o n .
d e n o n s t r a t e d i n a p a r s i m o n i o u s manner.
- vii -
This is
f ntroduct ion
Mathematical
ecology
a s s o c i a t i o n s have
formalizations
been s t e a d i l y making
of
urban
inroads
and
regional
spatial
i n t o geographical analysis
d u r i n g t h e p a s t f i v e y e a r s , Dendrinos [ 2 ] , Curry [ I ] , Dendrinos and M u l l a l l y
[ 3 ] , S o n i s [ l o ] , and o t h e r s .
T h i s r e c e n t work s u p p o r t s t h e argument t h a t w e l l
e s t a b l i s h e d e c o l o g i c a l i n t e r a c t i o n s c a n p r o v i d e new and r i c h i n s i g h t s i n t o t h e
dynamic i n t e r d e p e n d e n c i e s of a broad c l a s s of g e o g r a p h i c a l systems.
C e n t r a l t o t h i s work i s t h e r o l e , d e r i v a t i o n and i n t e r p r e t i t i o n of t h e
v a r i a t i o n a l p r i n c i p l e s and c o r r e s p o n d i n g " f i t n e s s " f u n c t i o n s which g i v e rise
(or
underlie)
systems.
the
particular
dynamics
governing
the
evolution
of
such
S i n c e t h e e a r l y developmental s t a g e s of t h e f i e l d of m a t h e m a t i c a l
e c o l o g y t h e q u e s t f o r t h e s e g o v e r n i n g f u n c t i o n s was viewed a s a n e s s e n t i a l
Although t h e i n t e r e s t i n s u c h a q u e s t i o n h a s s u b s i d e d s i n c e t h e
element.
e a r l y work by V o l t e r r a on t h i s t o p i c ,
found i n Scudo and Z i e g l e r [ 8 ] ,
its
import h a s n o t deminished.
V o l t e r r a was a b l e t o d e r i v e d i f f e r e n t i a l e q u a t i o n s of a b s o l u t e growth
p o p u l a t i o n dynamics
of
e c o l o g i c a l a s s o c i a t i o n s as s o l u t i o n of
p r i n c i p l e s s i m i l a r t o t h o s e of c l a s s i c a l mechanics.
variational
Moreover, V o l t e r r a g a v e
t h r e e d i f f e r e n t forms of t h e s e p r i n c i p l e s : t h e p r i n c i p l e o f " l e a s t a c t i o n " f o r
onespecies
population
growth;'
the
principle
of
"stationary
action"
and
p r i n c i p l e of " l e a s t v i t a l a c t i o n " f o r m u l t i s p e c i e s e c o l o g i c a l dynamics.
He
drew from M a u p e r t u i s ' n o t i o n of " q u a n t i t y of a c t i o n " and i t s u s e by D e s c a r t e s
(with
the
kinetic
principle
energy)
on
of
momentum)
and by L e i b n i t z
the
integrals
of
(with
dynamic e q u a t i o n s .
the principle
Volterra's
of
main
r e s u l t , however, emerged t h r o u g h t h e u s e of H a m i l t o n ' s p r i n c i p l e of s t a t i o n a r y
a c t i o n reducing
t h e b i o l o g i c a l a s s o c i a t i o n s ( t h e dynamic e q u a t i o n s ) t o t h e
mechanics ( k i n e t i c s ) of a branch of problems i n t h e c a l c u l u s of v a r i a t i o n s .
A n a l y t i c a l l y , V o l t e r r a showed how t h e t r a j e c t o r i e s i n t h e s p a c e of s t a t e s of
e c o l o g i c a l a s s o c i a t i o n s (which d e s c r i b e how t h e system e v o l v e s o v e r t i m e ) c a n
b e found
as
functions
the stationary value f o r a
( e x t r e m a l s ) which g i v e
c e r t a i n i n t e g r a l ; i.e.,
t h e e x t r e m a l s X a r e t h e s o l u t i o n of t h e v a r i a t i o n a l
problem:
when X i s a s t a t e v a r i a b l e ,
T i s a t i m e h o r i z o n and d o t s t a n d s f o r t i m e
A s p e c i a l c h o i c e of t h e i n t e g r a n d I allowed him t o d e r i v e t h e
derivative.
e q u a t i o n s of m o t i o n i n e c o l o g i c a l dynamics . f o r a s i n g l e s p e c i e s ,
logistic
growth, ecology.
I n t h i s c a s e t h e i n t e g r a l (0.1) o b t a i n s a minimum.
Further,
he was
d e r i v e more g e n e r a l
able to
formulations
of
governing
integrands
r e g a r d i n g t h e dynamics of a g e n e r a l c l a s s of c o n s e r v a t i v e , m u l t i p l e s p e c i e s
ecological
associations
under
absolute
growth.
Moreover,
under
c o n d i t i o n s t o be d i s c u s s e d l a t e r , V o l t e r r a d e r i v e d t h e p r i n c i p l e of
v i t a l action.*
certain
"least
A l l d e r i v a t i o n s a r e d e s c r i b e d i n P a r t I of t h i s paper.
V o l t e r r a d e f i n e d c o n s e r v a t i v e e c o l o g i c a l a s s o c i a t i o n s p r e t t y much l i k e i n
classical mechanics:
manner
t h e t o t a l i n t e r a c t i o n among s p e c i e s i s b a l a n c e d i n a
t h a t r e s u l t s i n t h e v a l u e of
t h e t o t a l i n t e r a c t i o n t o equal zero.
E c o l o g i s t s d i d n o t e x t e n d V o l t e r r a ' s work i n t h e f o l l o w i n g decades.
disatisfied
by
the peculiar
conditions
s t a b i l i t y p r o p e r t i e s of V o l t e r r a ' s
associated with
c o n s e r v a t i v e systems.
the
They w e r e
e x i s t e n c e and
This, coupled w i t h
t h e i r a l m o s t e x c l u s i v e i n t e r e s t i n a b s o l u t e growth dynamics, d i d n o t p r o v i d e
mathenatical
ecologists
conservative
systems
or
the
opportunity
relative
growth
to
search
dynamics
for
where
other
the
kinds
of
existence
p r o p e r t i e s of t h e e q u i l i b r i u m a r e n o t a s u n r e a s o n a b l e o r r e s t r i c t i v e as t h e
o r i g i n a l Volterra formulations.
Although a b s o l u t e growth c o n s e r v a t i v e dynamic systems may be of l i t t l e
importance f o r dynamic s p a t i a l a n a l y s i s i n r e g i o n a l s c i e n c e , r e l a t i v e growth
c o n s e r v a t i v e dynamic s y s t e m s a r e , Dendrinos ( w i t h M u l l a l l y )
[4].
At l e a s t ,
t h e y might be more i m p o r t a n t t h a n a b s o l u t e growth i n u r b a d r e g i o n a l a n a l y s i s ,
t h a n they a r e i n t h e f i e l d of animal and p l a n t ecology.
F o r example, i n t h e
a r e a of a g g r e g a t e urban dynamics, s i n c e t o t a l n a t i o n a l growth may have v e r y
l i t t l e t o do w i t h any p a r t i c u l a r u r b a n a r e a o r r e g i o n i n a n a t i o n a l economy
( p a r t i c u l a r l y when a v e r y l a r g e number of m e t r o p o l i t a n areas o r r e g i o n s a r e
i n v o l v e d i n a b s e n c e of
primacy),
i t i s e l a s t i c i t i e s of growth t h a t matter.
C i t i e s and r e g i o n s , competing w i t h one a n o t h e r f o r economic a c t i v i t y , a t t r a c t
o r r e p u l s e growth depending on r e l a t i v e a d v a n t a g e s they e n j o y i n t h e n a t i o n a l
space.
R e l a t i v e growth h a s been argued t o be of i m p o r t a n c e i n a v a r i e t y of
o t h e r g e o g r a p h i c a l c o n t e x t s , i n c l u d i n g i n t r a - u r b a n dynamics and t h e p r o c e s s e s
underlying
innovation
diffusion.
In
these
contexts,
the
role
of
the
environment a s i t may a f f e c t a g g r e g a t e u r b a n dynamics c a n be a n a l y t i c a l l y
studied
and
the
purpose
of
this
paper
is
precisely
s p e c i f i c a l l y , i s s u e s a r e a d d r e s s e d r e g a r d i n g multi-urban
to
do
so.
More
interact ion s t a b i l i t y
and c o n d i t i o n s under which t h e e x i s t e n c e of p o t e n t i a l s c a n be shown.
Under
absolute
growth
and
for
certain
conservative
ecological
a s s o c i a t i o n s V o l t e r r a was a b l e t o d e r i v e a g o v e r n i n g i n t e g r a n d .
This paper
shows t h a t such a n i n t e g r a n d c a n be d e r i v e d f o r p a r t i c u l a r classes of s p a t i a l
(urban conservative)
w i t h r e l a t i v e growth.
systems,
d i f f e r e n t than t h e V o l t e r r a ones,
T h i s i s done i n P a r t I1 of t h i s paper.
associated
R e l a t i v e growth
dynamics i s shown t o be t h e s o l u t i o n of a g o v e r n i n g i n t e g r a n d which measures
the
e n t r o p y of
a g g r e g a t e growth
dynamic d i s t r i b u t i o n and
relative
the interaction within
s p a t i a l distribution.
the zero
The s t a t i o n a r i t y of
the
i n t e g r a l of such a n i n t e g r a n d r e s u l t s i n s t a t i o n a r i t y of a c u m u l a t i v e e n t r o p y
T h i s i s of p a r t i c u l a r i n t e r e s t t o
measure of r e l a t i v e s p a t i a l d i s t r i b u t i o n s .
geographical a n a l y s i s s f nce i t provides i n s i g h t s i n t o t h e f i t n e s s f u n c t i o n s
present
in
spatially
interacting
f u n c t i o n s t o t h e n o t i o n of
systems.
urban
entropy,
link
We
these
fitness
which we s t u d y ( f o r t h e f i r s t t i m e i n
dynamic s p a t i a l a n a l y s i s ) n o t o n l y o v e r s p a c e ,
b u t a l s o o v e r time.
It is
shoun t h a t o v e r t i m e e n t r o p y i s maximized, but o v e r s p a c e i t i s a t a minimum
in
the
(asymptotically
competitive
locale)
for
implications
stable)
exclusion
( e . ,
dynamic
spatial
a r e discussed
total
at
steady
which
agglomeration
conservative
the
state,
end
of
u n i q u e n e s s of such i n t e g r a n d i s a d d r e s s e d .
of
systems.
Part
represents
population
urban
into
one
Interpretations
and
11, where t h e s u b j e c t
of
Further, suggestions on s p e c i f i c
hypotheses f o r empirical t e s t i n g a r e presented t o g e t h e r w i t h conclusions.
P a r t I.
V o l t e r r a ' s Integrands.
I n t h i s P a r t t h e vork of V o l t e r r a on a b s o l u t e growth i s summarized and,
i n certain instances,
s i m p l i - f i e d and
extended
t o a c q u i r e g e o g r a p h i c a l and
I n S e c t i o n A t h e d e r i v a t i o n of t h e l o g i s t i c growth p a t h
economic meaning.
f r m a v a r i a t i o n a l p r i n c i p l e i s supplied f o r t h e s i n g l e s p e c i e s case.
Section
B d e a l s w i t h m u l t i p l e s p e c i e s i n t e r a c t i o n s and t h e d e f i n i t i o n of V o l t e r r a ' s
conservative systems (under a b s o l u t e growth),
notions.
In
conservative
derived.
Section C
"stationary
Finally,
the
variational
action"
dynamic
and
t h e i r demographic e n e r g y
principle
multiple
i n S e c t i o n D t h e p r i n c i p l e of
generating
species
Volterra's
interaction
is
" l e a s t v i t a l action"
by
Volterra i n m u l t i p l e species i n t e r a c t i o n is presented.
The r e a s o n s t o i n c l u d e
a summary of V o l t e r r a ' s work i n P a r t I i s n o t o n l y t o b r i n g t h i s s i g n i f i c a n t
work i n a c a p s u l e t o t h e a t t e n t i o n of r e g i o n a l s c i e n t i s t s and u r b a d r e g i o n a l
g e o g r a p h e r s who have
additional
light
not
been exposed t o i t u n t i l now,
o n V o l t e r r a ' s work
by
elaborating
b u t a l s o t o shed
on and/or
simplifying
c e r t a i n derivations.
A.
The p r i n c i p l e of "least a c t i o n " f o r o n e s p e c i e s l o g i s t i c growth.
T h i s s e c t i o n p r e s e n t s t h e f i r s t a t t e m p t t o l i n k e n t r o p y w i t h dynamics of
s p a t i a l s y s t e n s , d e f i n i n g t h u s cumulative s p a t i a l entropy.
Volterra i n h i s
c l a s s i c a l work on " t h e c a l c u l u s of v a r i a t i o n s and t h e l o g i s t i c c u r v e " found i n
Scudo and Z i e g l e r [8] (p.
11-17)
was t h e f i r s t t o d e r i v e t h e V e r h u l s t - P e a r l
e q u a t i o n of l o g i s t i c ( a b s o l u t e ) growth
i
-
x(a
- bx)
f r m t h e m i n i m i z a t i o n of a n i n t e g r a l E
-
PI(x,;,~)
0
dt
, where:
V o l t e r r a i n t e r p r e t s X a s t h e t o t a l ( c u m u l a t i v e ) q u a n t i t y of l i f e .
The E u l e r
c o n d i t i o n f o r o p t i m i z a t i o n of E is:
The i n t e g r a n d I , chosen by V o l t e r r a , is:
where ml,
m2,
k a r e appropriate constants.
entropy i s introduced.
a
-
m,
(In
3
+
dt
I n I t h e element of c u m u l a t i v e
Then t h e components of t h e E u l e r c o n d i t i o n a r e :
1)
-q
b (ln ( a
-
dX
b =)+ 1 )
and :
I f t h e t h r e e c o n s t a n t s s a t i s f y t h e conditions(').:
then:
which i s i d e n t i c a l t o t h e o r i g i n a l V e r h u l s t - P e a r l
equation.
( l ) ~ h e r ei s a p r i n t i n g e r r o r on page 16 of Scudo and Z i e g l e r [ 8 ] r e g a r d i n g k =
aml
V o l t e r r a computed t h e second v a r i a t i o n of t h e i n t e g r a l E w i t h r e s p e c t t o X and
i n d i c a t i n g E a t t a i n s a minimum.
found i t p o s i t i v e ,
Volterra
interprets
this
as
"reducing
the
I n h i s o r i g i n a l paper
movement
p r i n c i p l e of minimum," Scudo and Z i e g l e r [ 8 J p.
15.
of
population
to
a
I n compliance w i t h t h e
s p i r i t of l a t e r work by V o l t e r r a one must assume t h a t he meant "minimum e f f o r t
f o r - a d a p t a t i o n , " although V o l t e r r a does not e x p l i c i t l y s t a t e t h i s .
Note t h a t
t h i s expression E does not p o s s e s s a Hamiltonian, t h u s i t i s not a p o t e n t i a l .
B.
Conservation of demographic energy i n m u l t i p l e s p e c i e s i n t e r a c t i o n .
Volterra's
general
f o r m u l a t i o n of
the
(non-logistic)
growth m u l t i p l e
s p e c i e s i n t e r a c t i o n a b s o l u t e growth is:
ii
-
where ai
1
(ai+T
i
1a j i
xj)
Xi
= qxi
,
(2 1)
i , j = 1,2,...1
9
1
i s t h e " c o e f f i c i e n t of self-growth",
-1
i s an "equivalence f a c t o r "
bi
and ai i s
a
Coefficients a
sign.
"demographic
ij
depict
coefficient"
particular
Also, V o l t e r r a c a l l s a
i
(Volterra's
species
terms
associations
[8],
p.
239).
depending on t h e i r
t h e "gross" growth r a t e , bi a n "average weight"
and ai t h e "net" growth r a t e .
The key
n o t i o n of
Volterra's
e l a b o r a t i o n i s t h e "value of
t h e whole
a s s o c i a t i o n " V, o r t h e " a c t u a l demographic energy".:
The d i f f e r e n t i a l of V is e q u a l t o :
dv =
1 bi
i
ai xi d t
+
l1
i j
'ji
X iX j dt
(2.3)
on t h e b a s i s of which i t i s p o s s i b l e t o i n t r o d u c e t h e "demographic p o t e n t i a l
energy" of a n e c o l o g i c a l a e s o c i a t i o n :
I f t h e value of t h e second term i n t h e r.h.s.
zero
-'
1.e.:
then
the
individual
association.
interactions
Requirement ( 2 . 5 ) ,
will
not
of c o n d i t i o n ( 2 . 3 )
affect
the
total
i s always
ecological
according t o V o l t e r r a , i s t h e d e f i n i t i o n of a
"conservative ecological association."
The system of d i f f e r e n t i a l e q u a t i o n s (2.1)
and, t h e r e f o r e , t h e d e f i n i t i o n (2.5)
implies that:
of a c o n s e r v a t i v e e c o l o g i c a l a s s o c i a t i o n
a l a Volterra i s equivalent t o t h e following condition:
I n t e g r a t i o n of (2.6)
1 bi
x i
1
g i v e s us:
- a iX i ) = C o m t
o r , due t o (2.2),
2.4).:
T h i s i s V o l t e r r a ' s " p r i n c i p l e of c o n s e r v a t i o n of demographic' energy'
vhich i s
t h e stnu of a c t u a l demographic energy V and t h e p o t e n t i a l demographic energy P
([BIB P* 242)I n o r d e r f o r t h e a s s o c i a t i o n t o be c o n s e r v a t i v e ,
p. 165, employing u n n e c e s s a r i l y complicated p r o o f s ) ,
antisymmetry c o n d i t i o n s nus t be met:
V o l t e r r a proves ( [ 8 ]
t h a t t h e folloving tvo
A s i m p l e r proof goes a s follows: t h e expression (2.5) can be w r i t t e n a s :
which i d e n t i f i e s a p o l y n a n i a l of second degree i n t h e independent v a r i a b l e s
x1,x2,.
.., xI.
implies d i r e c t l y
d e f i n i t i o n (2.5)
The c o n d i t i o n r e q u i r e s t h a t a l l c o e f f i c i e n t s be zero.
(2.10).
Thus,
c o n d i t i o n s (2.10)
a r e equivalent
This
to
of a c o n s e r v a t i v e e c o l o g i c a l a s s o c i a t i o n and t o (2.7);
a r e necessary c o n d i t i o n s f o r a n e q u i l i b r i u m t o exist, but n o t s u f f i c i e n t .
best,
the
they
At
t h e e q u i l i b r i u m i s n e u t r a l l y s t a b l e , something which o c c u r s when a l l
e i g e n v a l u e s have z e r o r e a l p a r t s ; ' o t h e r v i s e t h e e q u i l i b r i u m i s u n s t a b l e .
i s a d i r e c t r e s u l t f r m t h e f a c t t h a t t h e sum of
t h e r e a l p a r t s of
e i g e n v a l u e s e q u a l s t h e sum of t h e d i a g o n a l elements of m a t r i x [ a
z e r o ( s i n c e aii = 0 , f o r a l l i = 1,2,...,
The non-zero
association
equilibrium
immediately
give
d i f f e r e n t i a l e q u a t i o n s (2.1).
s t a t e s of
us
the
ij
1,
This
the
vhich i s
I).
Volterra's
first
conservative ecological
integral
of
the
system
of
( I t v i l l be r e c a l l e d t h a t a f i r s t i n t e g r a l of a
system of . d i f f e r e n t i a l e q u a t i o n s i s a f u n c t i o n v h i c h h a s a c o n s t a n t v a l u e
aloog each s o l u t i o n of t h e system of d i f f e r e n t i a l equations.)
Folloving, is
t h e d e r i v a t i o n of t h e f i r s t i n t e g r a l , vhich has a n i n t e r e s t i n g form f r a n a n
econanic theory e tandpoint.
The
norrzero
equilibrium
*
*
s t a t e ( x l , x2,
..., xI)* of
Volterra's
c o n s e r v a t i v e e c o l o g i c a l a s s o c i a t i o n must e v i d e n t l y s a t i s f y t h e "fundamental
system" ( V o l t e r r a ' s term) :
The
non-zero
equilibrium
state
of
this
e x i s t s ) , due t o c o n d i t i o n s ( 2 . 5 ) , o r ( 2 . 7 ) ,
Conditions (2.1),
(2.7),
(2.10),
(2.12),
conservative
association
(if
it
requires that:
(2.13) combined imply t h a t :
Thus,
and, t h e r e f o r e ,
t h i s i m p l i e s f u r t h e r t h a t a t a l l time p e r i o d s :
x* bi
exp
v / n
(Xii)
i
-
Comt
,
where V i s t h e v a l u e of t h e whole a s s o c i a t i o n (2.2).
i s t h e f i r s t i n t e g r a l f o r t h e s y s t a n (2.1).
w r i t t e n as:
-
biXi
i
xi
exp v
Cons t
Thus, t h e f u n c t i o n
C o n d i t i o n (2.16)
can be a l s o
T h i s e x p r e s s i o n f o r t h e f i r s t i n t e g r a l c a r r i e s some economic i n t e r p r e t a t i o n
from e i t h e r a u t i l i t y o r p r o d u c t i o n f u n c t i o n s t a n d p o i n t .
Cobb-Douglas
type u t i l i t y / p r o d u c t i o n
function,
-*
v e c t o r of i n p u t f a c t o r s i n p r o d u c t i o n , x
t h e v e c t o r of t h e i r p r i c e s .
It c o r r e s p o n d s t o a
where 5 c a n be viewed as a
-
t h e i r e q u i l i b r i u m v a l u e s , and b as
, Q u a n t i t y expV from c o n d i t i o n (2.2)
i s then t h e
The r e t u r n s t o s c a l e a r e e q u a l t o V s i n c e (2.2)
t o t a l v a l u e added.
The s t a t i o n a r y p r i n c i p l e ,
t o be e l a b o r a t e d i n S e c t i o n C below,
holds.
t h u s may be
c r i t i c a l i n c o n n e c t i n g ecology (and t h e n a t u r a l s c i e n c e s ) t o economics.
One of t h e p e c u l i a r f e a t u r e s of V o l t e r r a ' s c o n s e r v a t i v e a s s o c i a t i o n s i s
t h e f a c t t h a t f o r t h e e q u i l i b r i u m t o exist t h e a s s o c i a t i o n must c o n t a i n a n
even number of d i f f e r e n t s p e c i e s ( [ 8 ] , p.
174).
T h i s fundamental d i f f e r e n c e
between t h e e c o l o g i c a l a s s o c i a t i o n s c o n t a i n i n g even and odd number of s p e c i e s
is
difficult
to
accept
from
an
ecological
viewpoint;
because
of
this,
V o l t e r r a ' s c o n s e r v a t i v e e c o l o g i c a l a s s o c i a t i o n s were c r i t i c a l l y c o n s i d e r e d by
biologists.
analyzed;
effect
of
I n P a r t 11 t h e c o n s e r v a t i v e a s s o c i a t i o n s w i t h z e r o growth w i l l be
f o r t h e s e r e l a t i v e growth e c o l o g i c a l a s s o c i a t i o n s t h e d i s t u r b i n g
even and
odd
number
of
species disappears.
Before,
however,
e n t e r i n g t h i s t o p i c , w e c l o s e V o l t e r r a ' s a n a l y s i s by summarizing t h e f i n d i n g s
regarding V o l t e r r a ' s v a r i a t i o n a l p r i n c i p l e .
C.
The p r i n c i p l e of s t a t i o n a r y a c t i o n f o r t h e m u l t i p l e s p e c i e s c o n s e r v a t i v e
a s s o c i a t i o n s by V o l t e r r a .
Analogical
t o t h e s i n g l e species case, Volterra considered a m u l t i p l e
s p e c i e s c o n s e r v a t i v e a s s o c i a t i o n embedded w i t h i n t h e i n t e g r a l (which w e s h a l l
c a l l t h e "cumulative a c t i o n " ) :
where t h e i n t e g r a n d G (which we s h a l l c a l l t h e " c u r r e n t a c t i o n " ) i s g i v e n by:
where
xi(t) d t
=
Xi
present.
The u p r e s s i o n
Again,
1 b iX i
the
element
of
cumulative
1 n Xi i s V o l t e r r a ' s ' t o t a l
entropy
is
infinitesimal v i t a l
i
action";
thus,
t h e c u r r e n t a c t i o n (3.2)
connected c o r r e s p o n d i n g l y w i t h :
i s d i v i d e d i n t o t h r e e p a r t s , each
v i t a l a c t i o n ( t h e e n t r o p y measure of
the
a s s o c i a t i o n ) , i n t e r a c t i o n , and demographic energy ( t h e t o t a l q u a n t i t y of l i f e )
a t a s i n g l e time period.
T h i s e c o l o g i c a l i n t e r p r e t a t i o n of a c t i o n c a n o b t a i n
a d e e p e r meaning f o r a n a s s o c i a t i o n u n d e r r e l a t i v e s p a t i a l growth c o n d i t i o n s
( s e e P a r t 11).
i n a more e x p o s i t o r y manner t h a n V o l t e r r a , we prove t h a t
I n Appendix A,
t h e i n t e g r a n d G f r a n (3.2)
E u l e r condition.
under i n t e g r a l E of
(3.1)
produce (2.1)
a s its
The f i r s t o r d e r ( E u l e r ) c o n d i t i o n d e f i n e s t h e p r i n c i p l e of
stationary action, since:
A
Having s o a p t a n e x p r e s s i o n (3.1,2)
constructed
the
corresponding
H a m i l t o n i a n H and
d i f f e r e n t i a l e q u a t i o n s e q u i v a l e n t t o (2.1).
v a r i a b l e s Xi and pi,
f o r t h e cumulative a c t i o n , V o l t e r r a
the
canonical
He used t h e c a n o n i c a l ( c o - s t a t e )
where:
--
aG
(3.4)
Pi
and i n t r o d u c e d t h e B a m i l t o n i a n H i n t h e form, w i t h G g i v e n by (3.2).:
The system (3.3)
system of
n w h a s t h e f o l l o w i n g c a n o n i c a l form:
Expression ( 3 . 4 ) i m p l i e s t h a t :
from which we o b t a i n :
and, using (3.5),
- 1 bi X i
I n Xi
i
Thus,
the
therefore,
since
- -21 1 1
Hamiltonian
H
H = V
+
P = Coast.
The
principle
of
- 1 a i b i xi
=
i
coincides
t h e p r i n c i p l e of
t h e c a n o n i c a l v a r i a b l e s Xi,
D.
aijXixj
1 1
with
total
t h e c o n s e r v a t i o n of
demographic
energy,
and,
demographic energy i s met
H is t h e f i r s t i n t e g r a l of t h e system (3.3).
pi,
"least
In
t h e Hamiltortian o b t a i n s t h e form, from (3.81:
vital
action"
for
the
multiple
species
c o n s e r v a t i v e a s s o c i a t i o n s by V o l t e r r a .
I n t h i s s e c t i o n , a s p e c i a l c a s e of multi-species
( t h e c a s e where t h e demographic work i s zero.)
i n t e r a c t i o n i s presented
We w i l l draw from t h i s s p e c i a l
c a s e f o r our i n t e ~ u r b a ns p a t i a l dynamics i n s e c t i o n C of P a r t 11.
Volterra's
definition
of
the
total
vital
action
for
a
conservative
multispecies
association is the integral:
The f i r s t and second v a r i a t i o n s of t h e v i t a l a c t i o n A w i l l be ( i n a manner
e q u i v a l e n t t o t h e one shown i n Appendix A):
and
where hi
X2,
Xi's
a r e the variations
..., XI be
of
Xi
t h e q u a n t i t i e s of l i f e f o r each k i n d of s p e c i e s .
f r a n (2.1)
L e t X1,
such t h a t hi(0) = hi(T) = 0.
Therefore t h e
s a t i s f y t h e s y s t a n of e q u a t i o n s :
where:
are
Volterra's
increase"
([8],
"demographic
p.
239).
coefficients"
or
The s u b s t i t u t i o n of
"effective
(4.4)
coefficients
i n t o (4.2)
of
gives the
f o l l o w i n g e x p r e s s i o n f o r t h e f i r s t v a r i a t i o n &I of t h e v i t a l a c t i o n A:
w h i l e hi are t h e v a r i a t i o n s of q u a n t i t i e s of l i f e Xi s u c h t h a t h i ( 0 )
0.
The e x p r e s s i o n
-
hi(T)
-
i s , by V o l t e r r a , t h e "vork of growth", o r t h e " v i r t u a l demographic vork" f o r
t h e v a r i a t i o n s h l , h2,..
.,hI.
L e t u s assume t h a t f o r some i n f i n i t e s i m a l v a r i a t i o n s of t h e q u a n t i t i e s of
l i f e t h e v i r t u a l demographic work i s e q u a l t o z e r o :
t h e n t h e f i r s t v a r i a t i o n of t h e t o t a l v i t a l a c t i o n w i l l be z e r o
Simultaneously,
bi
>
due t o p o s i t i v i t y of
t h e second v a r i a t i o n ( 4 . 3 )
2
(6 A
>
0. )
t o t a l v i t a l a c t i o n &, due t o t h e e x p r e s s i o n ( 3 . 5 ) ,
Thus,
XI,
any i n f i n i t e s i m a l v a r i a t i o n hi,
X2,
..., XI
.
4 k
)rr
h o r i z o n which a t t a i n s a minimum.
principle
problems
.
of t h e
The increment of t h e
w i l l be
of t h e q u a n t i t i e s of l i f e
w i t h z e r o v i r t u a l demographic work (4.8)
A t o g o v e r n t h e e v o l u t i o n of
variational
..., hI
h2,
i n c r e a s e of t h e t o t a l v i t a l a c t i o n (4.1).
= 0
t h e " a v e r a g e w e i g h t s " of s p e c i e s
0 and t h e p o s i t i v i t y of p o p u l a t i o n s ,
t o t a l v i t a l action is strongly positive
6A
w i l l determine a n
Thus, V o l t e r r a proposed a n i n t e g r a l
m u l t i p l e s p e c i e s i n t e r a c t i o n o v e r a time
T h i s s t a t a n e n t summarizes V o l t e r r a ' s main
of
least
vital
action
e c o l o g i c a l a s s o c i a t i o n w i t h e c o l o g i c a l dynamics as i n (4.4).
for
conservative
P a r t 11. C o n s e r v a t i o n C o n d i t i o n s i n Urban Dynamical Systems.
I n t h i s P a r t we d e a l w i t h r e l a t i v e growth dynamics, a problem V o l t e r r a
never a d d r e s s e d .
growth
E q u i v a l e n c e s a r e drawn b e t v e e n t h e a b s o l u t e and r e l a t i v e
is
which
defined
under
different
conditions,
i n S e c t i o n A.
The
n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e of a n e q u i l i b r i u m i n
i n Section B with
r e l a t i v e growth a r e demonstrated
V o l t e r r a ' s c o n s e r v a t i v e s y s t e n s exposed.
and
their
t h e i r advantages
over
Finally, the variational principles
together with t h e i r
e n t r o p i c n a t u r e a r e provided i n S e c t i o n C,
i n t e r p r e t a t i o n f o r s p a t i a l systems.
A.
P r o p e r t i e s of s p a t i a l dynamic i n t e r a c t i o n .
ecology
necessitates
c o n v e r t i n g e a c h s p e c i e s p o p u l a t i o n t o t h e c o n s e r v e d q u a n t i t y V.
T h i s i s done
Volterra's
absolute
t h r o u g h t h e u s e of t h e
growth
hi's
multiple
species
(what V o l t e r r a c a l l e d "weight e q u i v a l e n t s " ) .
As
we mentioned i n P a r t I t h i s set of p a r a m e t e r s c a n be i n t e r p r e t e d as f a c t o r
prices
in
econanic
production
theory
i n v o l v e d which are h e t e r o g e n e o u s ,
where
different
f o r example,
geographical
variable
(income,
factors
l a b o r and c a p i t a l .
when d e a l i n g w i t h human p o p u l a t i o n d i s t r i b u t e d
homogeneous
input
over
capital,
space,
etc.)
or
under
are
However,
any o t h e r
relative
growth c o n d i t i o n s , t h e conserved q u a n t i t y ( w h a t e v e r t h a y may be) d o e s n o t need
conversion f a c t o r s .
Whereas, V o l t e r r a a v o i d e d t h e c o n s i d e r a t i o n of p a r t i c u l a r
c o n s e r v a t i v e c o n d i t i o n s ( f o r example,
[ 8 ] , p.
b e c a u s e h e d e a l t w i t h t h e h e t e r o g e n e i t y of
170 on z e r o s e l f - g r o w t h )
partly
b i o l o g i c a l s p e c i e s , t h e r e i s no
need f o r s u c h r e s t r i c t i o n s t o be imposed on u r b a n s p a t i a l dynamics.
I n what
follows,
we s e t up t h e u r b a n r e l a t i v e s p a t i a l dynamics under
various g r w t h conditions.
W
e d e m o n s t r a t e t h a t s p a t i a l r e l a t i v e dynami c s
c o r r e s p o n d t o p a r t i c u l a r a b s o l u t e growth e c o l o g i c a l c o n s e r v a t i o n dynamics o f
Volterra.
Then, we proceed t o show t h e i r s t a b i l i t y p r o p e r t i e s and d i s c u s s t h e
competitive exclusion condition, a r e s u l t a p p l i c a b l e t o a l l r e l a t i v e s p a t i a l
dynamics.
T h i s d i s c u s s i o n h a s d i r e c t i m p l i c a t i o n s upon t h e a p p r o p r i a t e n e s s of
t h e d i f f e r e n t i a l e q u a t i o n s assumed t o h o l d f o r s p a t i a l dynamics; a l s o upon t h e
a c c e p t a n c e of a g e n e r a l " e x c l u s i o n a r y p r i n c i p l e , " implying t o t a l a g g l o m e r a t i o n
i n t o ' a s i n g l e s i t e , i n l o c a t i o n theory.
F i n a l l y , we p r o p o s e a c o r r e s p o n d i n g
i n t e g r a n d ( e q u i v a l e n t t o V o l t e r r a ' s l e a s t v i t a l a c t i o n p r i n c i p l e ) , and we show
i t t o a t t a i n a maximum a s i t g o v e r n s o u r i n t e r u r b a n e v o l u t i o n .
We show i t t o
be a maximum c u m u l a t i v e e n t r o p y measure of t h e r e l a t i v e s p a t i a l i n t e r a c t i o n .
Assume a homogeneous g e o g r a p h i c a l v a r i a b l e ( p o p u l a t i o n ) d i s t r i b u t e d o v e r
different locations i ( i
-
1,2,
...,I )
a t any time p e r i o d , t , s o t h a t t h e t o t a l
p o p u l a t i o n V i s i n d e p e n d e n t of time:
Along V o l t e r r a ' s
distribution."
lines,
V
c a n be i n t e r p r e t e d
C o n d i t i o n (5.1)
a s the total
the
i s more a p p r o p r i a t e f o r u r b a n s y s t e m s , a s i t
i s a l e s s r e s t r i c t i v e conservation condition than Volterra's
V.
" v a l u e of
d e f i n i t i o n of
One c a n extend t h e c u r r e n t a n a l y s i s by examining t h e c a s e where V i s a
f u n c t i o n of time.
The s p e c i a l c a s e of s p a t i a l r e l a t i v e dynamics where V
-
1
w i l l be r e f e r r e d t o as n o r m a l i z e d dynamics.
Volterra's
a b s o l u t e growth c o n s e r v a t i v e e c o l o g y dynamics shown i n (2.1)
a r e now t r a n s f o r m e d
i n t o t h e r e l a t i v e growth u r b a n ( s p a t i a l )
conservative
dynamics by a system of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s :
s u b j e c t t o (5.1).
The v a l i d i t y of such a n a s s o c i a t i o n f o r i n t e r u r b a n dynamic
s p a t i a l i n t e r a c t i o n r e s t s on t h e o r e t i c a l g r o u n d s ,
Dendrinos ( w i t h M u l l a l l y )
as w e l l a s on e m p i r i c a l v e r i f i c a t l o n .
[5],
identifies
i t s canmunity m a t r i x of
in
geographies
(predatory,
to
support
efficiently
the
competitive,
proposition
described
p o p u l a t i o n dynamics.
by
models
the
theoretical front i t
c o e f f i c i e n t s a s e t of
a l l possible
isolative,
cooperative,
commensal) among v a r i o u s u r b a n s e t t i n g s .
seems
On
amensal,
Recent e x t e n s i v e e m p i r i c a l evidence
that
aggregate
drawing
from
urban
dynamics
mathematical
can
ecology
be
and
Although t h e s e e n p i r i c a l f i n d i n g s a r e mostly r e p o r t e d
f o r s i n g l e c i t y - n a t i o n i n t e r a c t i o n s , Dendrinos ( w i t h M u l l a l l y ) [ 5 ] , t e s t i n g of
multi-urban i n t e r a c t i o n s c u r r e n t l y underway p r o v i d e s s u p p o r t f o r such modeling
effort.
These f i n d i n g s
a r e reported
in
forthcoming
papers,
for
example
Dendrinos [ 4 ] .
Two q u a l i t a t i v e f e a t u r e s of t h e model i n (5.2)
a r e w e l l known:
first, if
t h e system h a s a n e q u i l i b r i u m s o l u t i o n , i t i s u n s t a b l e , o r ( a t b e s t ) n e u t r a l l y
stable;
second,
for
it
connecting t h e model's
t o have a s o l u t i o n ,
c e r t a i n c o n d i t i o n s must hold
c o e f f i c i e n t s i n t h e canmunity i n t e r a c t i o n m a t r i x .
We
a d d r e s s next t h e q u a l i t a t i v e p r o p e r t i e s of t h i s model and t h e i r t h e o r e t i c a l
implications.
The a n a l y t i c a l a s p e c t s of t h e model a r e shown i n Appendix B , where, i t i s
proven
that
condition,
testing
things,
the
model's
parameters
under zero-self
of
validity
of
growth
such
must
for
it
interurban
satisfy
a
special
antisymmetry
t o have a s o l u t i o n .
association
allows,
Empirical
among
other
t h e e x a m i n a t i o n of any c o r r e l a t i o n between s p a t i a l r e l a t i v e impedance
( o r r e l a t i v e a c c e s s i b i l i t y ) and t h e magnitude a n d / o r s i g n of t h e i n t e r a c t i o n
coefficients.
On a t h e o r e t i c a l l e v e l i t e n a b l e s u s t o d e t e c t ( i n c a s e of a
solution)
the
dynamics.
T h i s a n t i s y m m e t r i c p r o p e r t y i s v e r y i n f o r m a t i v e ; i t shows t h a t ,
under
underlying
specification
(5.2),
integrands
governing
urban
and vhen s o l u t i o n e x i s t s ,
spatial
relative
s p a t i a l conservative
a l s o i m p l i e s t h a t t h e r e is no f r i c t i o n
dynamics a r e p u r e l y c a p e t i t i v e .
due t o a g g l o m e r a t i o n i n r e l a t i v e dynamics, t o damp t h e dynamic e q u i l i b r i u m ,
t h u s producing e x c l u s i o n a r y a l l o c a t i o n s .
B.
E q u i l i b r i u m s t a t e s of r e l a t i v e s p a t i a l dynamics and d i s c u s s i o n .
Although
associations,
Volterra
did
from
point
the
not
examine
of
view
of
zero
self-growth
urban
spatial
conservative
dynamics
a s s o c i a t i o n s a r e i n s i g h t f u l t o model r e l a t i v e d i s t r i b u t i o n dynamics.
these
Under a
r e l a t i v e framework i n t e r n a l growth and n e t m i g r a t i o n a r e i n t e r t w i n e d s o t h a t
t h e i r canbined e f f e c t i s modeled.
An i m p o r t a n t example of a z e r o s e l f - g r o w t h
normalized e c o l o g i c a l a s s o c i a t i o n was f i r s t s t u d i e d i n t h e t h e o r y of t e m p o r a l
d i f f u s i o n of c w p e t i t i v e i n n o v a t i o n s , S o n i s [ l o ] .
of
I n t h e c a s = of r e l a t i v e growth d e p i c t e d by (5.2)
the s t a b i l i t y properties
the
Volterra's
equilibrium
differ
e c o l o g i c a l a s s o c i a t i o n s (2.1),
significantly
(2.5):
from
conservative
t h e r e i s no need f o r a n even number of
species t o i n t e r a c t f o r a s t a b l e equilibrium t o exist.
I n agreement w i t h
V o l t e r r a , ,however, i f a s o l u t i o n e x i s t s t h e e q u i l i b r i u m i s s t a b l e o n l y under
competitive exclusion.
T h i s - i m p l i e s t h a t o n l y t h e c o n c e n t r a t i o n of t h e whole
(homogeneous) g e o g r a p h i c a l s u b s t a n c e ( e . g.
p o p u l a t i o n , c a p i t a l , income, e t c . )
i n o n e of t h e r e g i o n s can b e s t a b l e a s y m p t o t i c a l l y .
the
equilibrium
perturbation,
the
*
s t a t e (yl,
perturbed
*
y2,
..., y*I
) means
s t a t e ( y l , y2,
A s y m p t o t i c a l s t a b i l i t y of
that,
..., yI)
for
exhibits
any
the
small
dynamic
property:
lim
Proof
of
yimy;,
i=1,2
t h i s statement
Appendix B.
,...,I .
f o r s p a t i a l competitive e x c l u s i o n is s u p p l i e d i n
Under r e l a t i v e growth c o n d i t i o n s d e p i c t e d by e x p r e s s i o n ( 5 . 2 ) ,
a t best
n e u t r a l l y s t a b l e s o l u t i o n s a r e o b t a i n e d and i n a l l l i k e l i h o o d o n l y e x c l u s i v e
allocation
result.
of
the
Inter-urban
homogeneous
geographic
interactions,
variable
onto
one
locale
will
viewed w i t h i n t h e framework of a s p e c i f i c
environment ( t h e n a t i o n o r a r e g i o n ) u s e d t o n o r m a l i z e t h e urban s i z e , imply
a s y m p t o t i c a l l y s t a b l e t o t a l a g g l o m e r a t i o n of t h e g e o g r a p h i c v a r i a b l e o n t o one
site.
T h i s i s t h e r e s u l t of p u r e c o m p e t i t i o n and a b s e n c e of f r i c t i o n found i n
t h e a n t i s y m m e t r i c p r o p e r t i e s of t h e i n t e r a c t i o n m a t r i x .
I n modeling s p a t i a l
dynamics i n a r e l a t i v e framework, t h u s , fundamental i n s t a b i l i t y i s b u i l t i n t o
t h e system.
One by l o o k i n g a t t h e e r n p i t i c a l e v i d e n c e produced s o f a r , Dendrinos ( w i t h
M u l l a l l y ) [ 5 ] , f i n d s s t a b l e p a t t e r n s of s p a t i a l growth when c i t i e s a r e viewed
i n i s o l a t i o n and i n r e f e r e n c e t o t h e n a t i o n a s t h e environment o v e r which
t h e i r r e l a t i v e s i z e i s computed.
the relative
This j u x t a p o s i t i o n has c e r t a i n implications:
community i n t e r a c t i o n matrix of
may
r e g i o n s ) , t h e U.S.
c e r t a i n s e l e c t i v e environments (i.e.,
o n e of them f o r i t s u r b a n areas.
(5.2)
be a p p l i c a b l e f o r
as a whole n o t b e i n g
I n t h e o r y , t h e r e i s no a p p a r e n t r e a s o n why
one c a n n o t f i n d a n environment w i t h r e s p e c t t o which c i t i e s c o u l d e x h i b i t
dynamics of
t h e t y p e found i n (5.2).
Further,
t h e s t a b i l i t y p r o p e r t i e s of
p a r t i c u l a r s p a t i a l s y s t e m s may v a r y as o n e changes t h e b r o a d e r environment
w i t h i n which t h e s e s y s t e m s are viewed.
T h i s may l e a d t h e s p a t i a l a n a l y s t i n
c e r t a i n instances t o formulating i n t e r u r b a n interdependencies i n a d i f f e r e n t
manner t h a n (5.2).
F o r i n s t a n c e , o n e may wish t o i n t r o d u c e s t r o n g e r forms of
i n t e r u r b a n i n t e r c o n n e c t a n c e by i n c l u d i n g c u b i c terms i n t h e s t a t e v a r i a b l e s ,
S o n i s [ 101.
The i n f o r m a t i o n of
help
of
a
stochastic
a n a c t i v e environment c a n be accomplished w i t h t h e
matrix S
-
1
( ( s i j I which
describes
the
process
of
redistribution
intervention.
of
population
among
the
various
regions
due
to
this
This r e d i s t r i b u t i o n process a c t s i n a d d i t i o n t o the e c o l o g i c a l
dynamics p r e s e n t i n t h e i n t e r a c t i o n m a t r i x .
The a c t i v e environment smooths
o u t t h e extreme a c t i o n of t h e c o m p e t i t i v e e x c l u s i o n p r i n c i p l e and l e a d s t o a
more balanced
regfons.
f i n a l a s y m p t o t i c a l l y s t a b l e d i s t r i b u t i o n of
p o p u l a t i o n among
C l e a r l y , f u r t h e r e x t e n s i v e e m p i r i c a l s e a r c h i s needed t o a s c e r t a i n
t h e v a l i d i t y of any of t h e above t h e o r e t i c a l c o n j e c t u r e s .
C.
V a r i a t i o n a l p r i n c i p l e s f o r u r b a d r e g i o n a l r e l a t i v e growth dynamics.
I n t h i s s e c t i o n t h e main f i n d i n g of
t h e p a p e r i s p r e s e n t e d , namely t h e
d e r i v a t i o n of a n i n t e g r a l which g o v e r n s t h e e v o l u t i o n of
spatial relative
It i s c l o s e l y
dynamics a s assumed i n (5.2) when t h e y p o s s e s s a s o l u t i o n .
related
to
spatial
cumulative
This
theoretical
interpretation.
entropy
from
which
it
draws
its
i m p l i c a t i o n s u p p o r t s t h e e v i d e n c e of
e n t r o p y p r i n c i p l e i n s p a t i a l dynamics.
an
C o n s i d e r a normalized s p a t i a l dynamic
s y s ten g i v e n by:
w i t h t h e a n t i s y m m e t r i c m a t r i x B = (b
i j )- : b
=
-
b
; b
= 0
Denoting as:
o n e h a s t h e dynamical system:
I n t h i s z e r o a g g r e g a t e and s e l f - g r o w t h c o n s e r v a t i v e s p a t i a l dynamics t h e v a l u e
- 21 -
of
t h e whole a s s o c i a t i o n V i s e q u a l t o one and
the potential
demographic energy P i s z e r o , ( s e e ( 2 . 2 ) and ( 2 . 4 ) .
interurban
T h e r e f o r e , one can use an
a p p r o p r i a t e m o d i f i c a t i o n of t h e V o l t e r r a i n t e g r a n d ( 3 . 2 ) f o r t h e c o n s t r u c t i o n
of a v a r i a t i o n a l problem ( e q u i v a l e n t t o h i s i n t e g r a l of cummulative a c t i o n ) ,
5 ) as i t s E u l e r c o n d i t i o n .
which g e n e r a t e s t h e system (6.4,
We propose t h e
followfng i n t e g r a n d :
and t h e a s s o c i a t e d c u m u l a t i v e a c t i o n
w e i n t e r p r e t a s t h e urban f i t n e s s function.
stationary
cumulative
i n t e g r a l El v a n i s h e s ,
action
giving
means
rise
that
to
The v a r i a t i o n a l p r i n c i p l e of
the
the
first
system
of
variation
Euler
of
the
differential
equations
Direct calculation gives
T h e r e f o r e t h e i n t e g r a n d w e p r o p o s e through,
10)
and
the
time
derivative
of
(6.10)
(6.8)
the
i m p l i e s t h a t through (6.9,
original
condition
(6.4)
obtained , since:
whereas, t h e antisymmetry of t h e i n t e r a c t i o n m a t r i x B = ( b
ij
) i m p l i e s (6.5).
is
The s t a t i o n a r y v a l u e of
t h e c u m u l a t i v e a c t i o n E' t u r n s o u t t o be t h e
c u m u l a t i v e e n t r o p y f o r normalized s p a t i a l dynamic s y s t e m s :
Thus,
and,
t h e r e f o r e t h e " s t a t i o n a r y cumulative a c t i o n " f o r a normalized s p a t i a l
d i s t r i b u t i o n dynamics i s t h e c u m u l a t i v e e n t r o p y of t h e p o p u l a t i o n d i s t r i b u t i o n
/
d u r i n g t h e time h o r i z o n T.
T h i s i s o u r main f i n d i n g .
C o n t r a s t i n g V o l t e r r a ' s i n t e g r a n d G, ( 3 . 2 ) ,
i n h i s conservative ecological
dynamics, w i t h our i n t e g r a n d @ i n r e l a t i v e u r b a n dynamics, o n e sees t h a t t h e
three
terms
in
the
ecological
conservative
systems
(constituting
total
i n f i n i t e s i m a l v i t a l a c t i o n ) c o l l a p s e i n s p a t i a l dynamics i n t o a s i n g l e term;'
v i t a l a c t i o n , i n t e r a c t i o n and demographic e n e r g y merge i n t o a single e n t i t y ,
namely c u m u l a t i v e s p a t i a l e n t r o p y .
I t i s i m p o r t a n t t o p o i n t o u t t h a t t h e f i r s t term of i n t e g r a n d
Volterra's
"infinitesimal
v i t a l action",
p o p u l a t i o n d i s t r i b u t i o n (Shannon ( 9 1 , p.
represents
396).
6 which is
t h e Shannon e n t r o p y o f
Volterra naturally did not
g i v e s u c h a n i n t e r p r e t a t i o n of t h e " i n f i n i t e s i m a l v i t a l a c t i o n , "
( o r what we
d e f i n e d a s " c u r r e n t v i t a l a c t i o n " e a r l i e r ) , a s Shannon's work a p p e a r e d i n 1948
i n t h e c o n t e x t of
(6.6)
communication t h e o r y .
The second term of i n t e g r a n d @ i n
r e p r e s e n t s t h e i n t e r a c t i o n between d i f f e r e n t p a r t s of t h e homogeneous
geographical substance (population).
The a n a l o g u e o f
the Volterra
"least
v i t a l a c t i o n " p r i n c i p l e o b t a i n s f o r t h e r e l a t i v e growth dynamics t h e form o f
dynamic maximum c u m u l a t i v e e n t r o p y p r i n c i p l e .
In
maximum.
general
one
cannot
d e t e r m i n e whether
(6.12)
has
a minimum o r
a
However, f o r a s p e c i a l c a s e t o be a d d r e s s e d below one c a n show t h a t
t h e s t a t i o n a r y v a l u e o f (6.12)
a t t a i n s a maximum.
Let u s consider f i r s t the
i n t e g r a l ( 1 - e . , t h e c u m u l a t i v e e n t r o p y ) f o r t h e g e n e r a l c a s e (6.12) which c a n
a l s o be w r i t t e n a s
The f i r s t and second v a r i a t i o n s of t h e c u m u l a t i v e e n t r o p y El a r e
where hl, h2,
Yi =
J:
..., hI
yi d t
,
A
a r e t h e v a r i a t i o n s of t h e cumulative populations,
s u c h t h a t hi(0)
cumulative populations Y
i
= hi(T)
= 0
( s e e Appendix A).
s a t i s f y t h e system ( 6 . 4 ,
Since the
5), the f i r s t variation is
It i s p o s s i b l e t o i n t e r p r e t t h e e x p r e s s i o n s
as " c o e f f i c i e n t s o f p o p u l a t i o n i n c r e a s e due t o t h e i n t e r a c t i o n , " o r r e l a t i v e
migration ( i .e.,
r e l a t i v e t r a n s f e r of p o p u l a t i o n ) c o e f f i c i e n t s .
manner t h e e x p r e s s i o n
1 sihi
In a similar
c a n be i n t e r p r e t e d a s t h e v i r t u a l work of u r b a n
i
growth due t o t h e i n t e r - u r b a n i n t e r a c t i o n .
Next
we
draw
from
the
special
case
of
multispecies
interaction
by
V o l t e r r a , when t h e v i r t u a l work of urban growth i s z e r o t o show t h a t ( 6 . 1 2 )
o b t a i n s a maximum.
I f f o r some c h o i c e o f t h e v a r i a t i o n s h l , h2,
t h e cumulative populations Y
1 sidi
i
=
1 (1 bijYj)
i
,
hi = 0
1
>
0
,
,
of
i s equal t o zero,
of cumulative
and, s i m u l t a n e o u s l y , due t o t h e p o s i t i v i t y
t h e second v a r i a t i o n ( 6 . 1 6 ) o f t h e c u m u l a t i v e
e n t r o p y i s s t r o n g l y n e g a t i v e , d2e1
Therefore,
(6.3)
then t h e f i r s t v a r i a t i o n (6.17)
e n t r o p y i s e q u a l t o z e r o 6E' = 0
o f t h e p o p u l a t i o n s yi = Yi
..., Y I c o n d i t i o n
Y2,
..., hI
<0
.
f o r t h e same v a r i a t i o n s hi which imply t h e v a n i s h i n g o f t h e
v i r t u a l work of u r b a n growth due t o i n t e r - u r b a n
i n t e r a c t i o n t h e increment o f
t h e cumulative entropy
i s s t r o n g l y n e g a t i v e , which i m p l i e s a d e c r e a s e i n t h e c u m u l a t i v e e n t r o p y ,
q.0.d.
I n c o n c l u e i o n , a dynamic maximum e n t r o p y problem i s uncovered t o g e n e r a t e
the
r e l a t i v e dynamic u r b a n model
geographical substance (population).
of
s p a t i a l adaptation of
a homogeneous
This f i t n e s s function is equivalent t o a
l e a s t e f f o r t p r i n c i p l e found i n V o l t e r r a ' e e c o l o g y .
A f i n a l remark o n t h e p o s t u l a t e d i n t e g r a n d f o r s p a t i a l a s s o c i a t i o n s :
proposed
integrand
must
not
necessarily
be
unique.
Either
the
a c l a s s of
e q u i v a l e n t i n t e g r a n d s c a n p o s s i b l y e x i s t , t h e o n e proposed h e r e b e i n g m e r e l y
t h e i r c a n o n i c a l form;
o r q u i t e d i f f e r e n t o n e s might a l s o produce t h e same
result.
For a c a s e i n p o i n t
course,
would
imply t h a t
s e e C e l f a n d and Fomin [ 6 ] ,
p.
36.
This,
of
t h e r e a r e m u l t i p l e o b j e c t i v e f u n c t i o n s which c a n
produce a n a d a p t a t i o n path.
These a r e p o t e n t i a l l y i n t e r e s t i n g q u e s t i o n s f o r
f u t u r e t h e o r e t i c a l s p e c u l a t i o n and e m p i r i c a l r e s e a r c h .
D.
The H a m i l t o n i a n o f u r b a n c o n s e r v a t i v e r e l a t i v e dynamics.
We now t u r n t o t h e s e a r c h f o r a p o s s i b l e g o v e r n i n g H a m i l t o n i a n o f t h e
n o r m a l i z e d s p a t i a l dynamics.
Introducing the co-state variables:
ayi
a n d t h e new H a m i l t o n i a n :
one can obtain:
and t h e canonical system o f d i f f e r e n t i a l equations:
S t a t i n g t h e R a m i l t o n i a n H as a f u n c t i o n o f t h e c o - s t a t e
(7.1,
variables,
due t o
2) we o b t a i n :
e x t r e m a l s Yi,
dynamics.
1
Yi = 1 t h e v a l u e o f t h e H a m i l t o n i a n i s H = 2 o n t h e
i
and, t h e r e f o r e , i t i s a f i r s t i n t e g r a l o f t h e normalized s p a t i a l
Due t o t h e c o n d i t i o n
It
i s i n t e r e s t i n g t o n o t e t h a t one c a n e a s i l y o b t a i n from (2.17)
the
f i r s t i n t e g r a l f o r t h e normalized s p a t i a l dynamical system a s a Cobb-Douglas
f u n c t i o n II yiYi, where y
(production)
i
state
of
the
system.
This
*
,
i
i = 1,2,
expression i s
..., I ,
i s some e q u i l i b r i u m
a l s o similar
to the objective
f u n c t i o n of a g e o m e t r i c programming problem.
A s i t was mentioned e a r l i e r , V o l t e r r a found t h e s u f f i c i e n t c o n d i t i o n s f o r
t h e p r e s e n t a t i o n of t h e s o l u t i o n of system (2.1)
of
t h e normalized s p a t i a l dynamics,
obtain the explicit
f o r m u l a s of
i n quadratures.
I n the case
t h e a n a l o g i c a l c o n d i t i o n allows us t o
s o l u t i o n s i n elementary functions.
a l l o w s u s t o draw some l i n k s between s p a t i a l dynamics and t h e l o g i t model.
i s done i n Appendix C.
Here, one may n o t e t h e f o l l o w i n g : $
This
It
c o n d i t i o n (7.3)
c a n a l s o be w r i t t e n as
Yi = WP
1
((- I
pi + 1 )
1
+7
1 bijyj]
I
j
E x p r e s s i o n (7.6)
1 exp
((-
1
f
k
%+
1)
+ T1 1 bkj
j
yj)
(7.6)
c o r r e s p o n d s t o t h e l o g i t model o f random u t i l i t y c h o i c e w i t h
u t i l i t y functions
These U c o n t a i n s e p a r a b l e e f f e c t s ; o n e i s a n e g a t i v e a s s o c i a t i o n w i t h pi and
the other i s a positive association vith b
ij
and Y j .
One may v i s h t o i n t e r p r e t pi a s t h e r e l a t i v e m a r g i n a l c o s t of f i t n e s s f o r
urban
setting i's
economics).
population
A t equilibrium
(the
-
= U
Ui
equivalent
=
1
H
,
to
shadow p r i c e s
i n micro-
s o t h a t t h e Hamilitonian can be
$we t h a d c G i o r g i o L e o n a r d i from IIASA f o r p o i n t i n g t o u s t h i s r e w r i t i n g of
( 7 . 9 ) , s i n c e t h e denominator e q u a l s one by d e f i n i t i o n .
- 27 -
viewed a s a r e l a t i v e u t i l i t y f u n c t i o n .
cost
of
fitness
(1 b
interaction
i
J
relative
declines,
and
U t i l i t y thus i n c r e a s e s a s the marginal
increases
Y ) increases.
as
the
relative
inter-urban
I n view o f t h i s , t h e i n t e r p r e t a t i o n of t h e
ij j
O is insightful:
f i t t n e s s function
areas' current relative fitness level.
it is the
sum of
a l l urban
E ' i s t h e cumulative t o t a l f i t n e s s of
F i t n e s s i s such t h a t
t h e community of u r b a n a r e a s o v e r a t i m e h o r i z o n T.
i n t e r u r b a n i n t e r a c t i o n maximizes t h e c u m u l a t i v e e n t r o p y o f t h e a s s o c i a t i o n .
1
The term
(cumulative action.)
p y r e p r e s e n t s t h e v a l u e of e f f o r t t o adapt
i i
i
i n a l l s e t t i n g s , s o t h a t from (7.2) t h e c u r r e n t f i t n e s s l e v e l i s t h e n e t sum
of t h e u t i l i t y l e v e l e n j o y e d and t h e t o t a l c o s t of a d a p t a t i o n ( t h e e f f o r t t o
adapt)
.
One may ponder
planning
(decentralized
control)
w e l f a r e a s p e c t s of t h e proposed Hamiltonlan.
this
is a
r e l a t i v e growth model
comprehensive
economics.
(over
space,
and a g g r e g a t e s o c i a l
It must b e k e p t i n mind t h a t
and
the
i m p l i e d c o n t r o l s a r e much more
functions
and
agents)
than
those
i n welfare
Thus, a l t h o u g h t h e scheme may be similar t o w e l f a r e m a x i m i z a t i o n
problems, t h e e c o l o g i c a l b a s e o f t h e model proposed makes a n y p r a c t i c a l ( i . e . ,
specific)
u s e of
message
of
making,
a
this
it
line
restrictive.
of
s u b j e c t more
work,
E c o l o g i c a l models,
present
f u l l y addressed
a different
in
[5].
and
t h i s i s a main
perspective on policy
They p r o v i d e
the final
outcomes from a complex and b r o a d e r i n t e r p l a y o f a c t i o n s among a v e r y l a r g e
number of producers, consumers and g o v e r m e n t s .
Conclusions.
Volterra's
tional
o r i g i n a l a b s o l u t e growth c o n s e r v a t i o n c o n d i t i o n s and v a r i a -
p r i n c i p l e s were
reconsidered
f o r one and m u l t i p l e s p e c i e s ecology.
They were c o n t r a s t e d w i t h more s p e c i f i c c o n s e r v a t i o n c o n d i t i o n s f o r r e l a t i v e
growth and s p a t i a l d i s t r i b u t i o n i n urban systems and a p p r o p r i a t e e q u i v a l e n c e s
and
'
substantive
multiple
i n t e r p r e t a t i o n s were drawn.
1 bi Xi I n
integrand G =
species
Volterra's
stationary action
+-
Xi
i
c o n d i t i o n (3.2),
was found t o have a c o r r e s p o n d i n g one i n r e l a t i v e s p a t i a l urban
dynamics u n d e r p u r e c o m p e t i t i o n of t h e form:
1 Yi
(-
= 2
$I
+ -1
In Y
i
condition
(6.6).
Furthe.rmore,
(-
produces i n our c a s e E ' =
we
proved
1y i l n
that
t h e corresponding in;egral
y i ) d t , i.e.,
a s t a t i o n a r y cumulative
i
e n t r o p y measure, c o n d i t i o n (6.13)
of s p a t i a l dynamics.
The s t a t i o n a r y maximum c u m u l a t i v e e n t r o p y i n t e g r a l shown t o g o v e r n and
produce as i t s s o l u t i o n t h e u r b a n . r e l a t i v e s p a t i a l dynamics was viewed a s one
(among
possibly
equivalent
to
many)
a
adaptation
least
effort
(fitness)
principle
conservative ecological associations.
fifty
year
old
studies
involving
function
in
in
Volterra's
spatial
absolute
dynamics
grovth
The d e t a i l e d comparison of V o l t e r r a ' s
three
different
classes
of
ecological
problems w i t h r e c e n t developments i n e c o l o g i c a l u r b a n dynamics v a s shown t o
lend
new
insights
processes.
tovard
a
deeper
understanding
of
spatial
dynamic
I n p a r t i c u l a r , b a s i c c o n d i t i o n s r e s u l t i n g i n dynamic i n s t a b i l i t y
(competitive e x c l u s i o n i m p l y i n g c o n c e n t r a t i o n of p o p u l a t i o n i n a s i n g l e a r e a ) ,
o r n e u t r a l s t a b i l i t y i n r e l a t i v e u r b a n e v o l u t i o n were provided and c o n t r a s t e d
w i t h t h e s t a b l e motion of u r b a n r e l a t i v e dynamics viewed i n i s o l a t i o n w i t h i n
the nation's
interactions.
environment.
It
s e t t h e framework
f o r modeling multi-urban
T h i s t h e o r e t i c a l framework p r o v i d e s f o r e m p i r i c a l t e s t i n g , a t a s k which
o n s i n g l e r e l a t i v e g r o w t h dynamics was p a r t l y c a r r i e d o u t i n [ 5 ] ; m u l t i - u r b a n
interactions
i n a r e l a t i v e g r o w t h framework
a r e dealt with
i n forthcoming
p a p e r s , example [ 4 ] , w h e r e i t i s s h o v n t h a t e a c h p a r t i c u l a r form o f a n i n t e r u r b a n i n t e r a c t i o n m a t r i x g e n e r a t e s p a r t i c u l a r u n s t a b l e h i e r a r c h i c a l dynamics.
References
L. Curry, 1981, " D i v i s i o n of Labor from Geographical C o m p e t i t i o n , " Annals
of t h e A s s o c i a t i o n of American Geographers, Vol. 71, No. 2: 133-165.
D.S. Dendrinos, 1979, "A B a s i c Model of Urban Dynamics Expressed a s a S e t
of Volterra-Lotka E q u a t i o n s , " i n D.S. Dendrinos, C a t a s t r o p h e Theory i n
Urban and T r a n s p o r t a t i o n A n a l y s i s , Report # DOT/RSPA/DPB-25/80/2, U.S.
Department of T r a n s p o r t a t i o n , Washington, D.C.,
1980.
D.S.
Dendrinos, 8. M u l l a l l y , 1981, " E v o l u t i o n a r y P a t t e r n s
P o p u l a t i o n s , " G e o g r a p h i c a l A n a l y s i s , Vol. 13, No. 4: 328-344.
of
Urban
D.S. Dendrinos, 1984, "Nonlinear Dynamics of Urban H i e r a r c h i e s ; t h e U.S.
Mountain Region (1890-1980) "Paper t o be p r e s e n t e d a t t h e NATO Advanced
Summer I n s t i t u t e , 1985 Germany.
Report SES-821-6620 N a t i o n a l S c i e n c e
Foundation ( forthcoming).
D.S. Dendrinos ( w i t h H. M u l l a l l y ) , 1984 Urban E v o l u t i o n :
Studies i n the
Mathematical Ecology of Cities, Oxford U n i v e r s i t y P r e s s ( i n p r e s s ) .
1.M.
Gelfand, S.V. Fomin, 1963 C a l c u l u s of V a r i a t i o n s , r e v i s e d E n g l i s h
e d i t i o n , t r a n s l a t e d and e d i t e d by R.A. Silverman, P r e n t i c e - H a l l .
G.A.
Korn, T.M. Korn, 1961, M a t h m a t i c a l Handbook f o r S c i e n t i s t s and
E n g i n e e r s , McGraw-Hill, New York, Toronto, London.
F.M.
Scudo, J.R.
Z i e g l e r ( e d s ) , 1978 The Golden Age of T h e o r e t i c a l
Ecology:
192 3-1940, Springer-Verlag, L e c t u r e Notes i n Biomathematics,
Vol. 22.
C. Shannon, 1948, "A m a t h e n a t i c a l t h e o r y of caamunication",
T e c h n i c a l J o u r n e y , Vol. 2 7 , No.l:379-423;.No.4:623-656.
B e l l System
-
M. S o n i s , 1983, "Competition and Environment
A Theory of Temporal
I n n o v a t i o n D i f f u s i o n , " i n D. G r i f f i t h , A Lea ( e d s ) Evolving Geographic
S t r u c t u r e s , Martinus Nijhoff.
M. S o n i s , 1984, "Dynamic c h o i c e of a l t e r n a t i v e s , i n n o v a t i o n d i f f u s i o n and
e c o l o g i c a l dynamics o f Volterra-Lotka",
London P a p e r s i n Regional
S c i e n c e , Vol. 14 ( i n p r e s s ) .
APPENDIX A
I n o r d e r t o f u l l y c a p t u r e t h e r i c h n e s s of V o l t e r r a ' s
f o r m u l a t i o n , we go
back t o c e r t a i n b a s i c p r i n c i p l e s i n t h e c a l c u l u s of v a r i a t i o n s problems, f o r
example Gelfand and F m i n [ 6 ] .
The v a r i a t i o n h ( t ) of f u n c t i o n X ( t ) i s t h e
.
d i f f e r e n c e between a nev f u n c t i o n X(t) and X(t) s u c h t h a t :
We r e p l a c e i n t h e i n t e g r a l E a l l f u n c t i o n s X i ( t )
Xi(t)
+
hi(t)
increment
where
&(hi)
-
hi(0) = hi(T) = 0 ,
E(Xi
+
h i , X i + hi)
i = 2
- E(Xi,
hi)
by t h e " v a r i e d " f u n c t i o n s
,I
.
,
and
the
I n t h e c a s e of (3.1,2)
t h e increment is:
and i t i s p o s s i b l e t o f i n d , a p p l y i n g T a y l o r ' s f o r m u l a , t h a t :
We now form t h e f i r s t and second v a r i a t i o n s of t h e i n t e g r a l E:
The f i r s t v a r i a t i o n is:
construct
and t h e second v a r i a t i o n :
A necessary c o n d i t i o n f o r t h e i n t e g r a l E t o have a n extremum (maximum
of
a
minimum) i s t h a t i t s f i r s t v a r i a t i o n bE v a n i s h f o r a l l a d m i s s i b l e v a r i a t i o n s
hi ( i . e . ,
hi(0) = hi(T) = O).:
which i s e q u i v a l e n t t o t h e f a c t t h a t t h e f u n c t i o n s Xi(t)
s a t i s f y the Euler
equations
L
The above c o n d i t i o n s (A.8) a r e necessary but n o t s u f f i c i e n t f o r t h e e x i s t e n c e
of a n extremum; they d e f i n e t h e p r i n c i p l e of s t a t i o n a r y a c t i o n .
. I n other
words, t h e a c t u a l t r a j e c t o r y of V o l t e r r a ' s c o n s e r v a t i v e e c o l o g i c a l a s s o c i a t i o n
does not rninirnize.the cumulative a c t i o n E but o n l y c a u s e s i t s f i r s t v a r i a t i o n
t o vanish.
D i r e c t c a l c u l a t i o n provides immediately t h e e q u i v a l e n c e between
t h e system i n ( 3 . 7 ) and t h e system (2
The
antisymmetry
of
s u b s t i t u t i o n s prove
the
that
interaction
matrix
A
=
t h e E u l e r e q u a t i o n s (A.8)
(aij)
and
appropriate
a r e Volterra's
dynamic
equations (2.1).
A f i n a l a n a l y t i c a l n o t e on system
equations is a nou-linear
however,
aij
But
even
this
256)
.ki
+
aj%bj%
simple,
4Cai4rbi
and
very
= 0
s o l u t i o n f o r t h e system (2.1)
11, t h a t t h e c a s e of
restrictive,
conditions
(A. 10)
form
to construct
the
for
the
explicit
It i s shown, i n P a r t
r e l a t i v e growth a l l o w s f o r e x p l i c i t s o l u t i o n s t o b e
(Sonis
(A. 10)
.
analytical
i n elementary f u n c t i o n s .
obtained i n e l a n e n t a r y functions.
growth
t o give conditions f o r the
f o r each i , j, k
i n t e g r a b i l i t y i n q u a d r a t u r e s d i d n o t a l l o w him
logistic
Given t h e s p e c i a l form
i n q u a d r a t u r e s of t h e form:
'jk
+
a a b b
i j i j
systems t o q u a d r a t u r e s .
V o l t e r r a w a s a b l e ( [ a ] p.
i n t e g r a t i o n of (2.1)
the Volterra d i f f e r e n t i a l
one; therefore, i t is u s u a l l y impossible t o reduce
t h e i n t e g r a t i o n of n o n - l i n e a r
(3.1,
(2.1) :
are
[lo],
only
F u r t h e r , t h e y g i v e t h e model of g e n e r a l i z e d
p.
117).
sufficient
Moreover,
(but
not
it
is
be
necessary)
shown
that
for
the
i n t e g r a b i l i t y i n q u a d r a t u r e s , Appendix C.
( 2 ) ~ o t ea n e r r o r r e g a r d i n g t h e i n t e g r a n d on p. 240 of [ a ] :
e q u a t i o n C', l a s t term.
the parenthesis i n
APPENDIX B
The
system
(5.2)
with
condition
of
zero
aggregate
growth
(5.1)
is
e q u i v a l e n t t o t h e system
where
I t i s p o s s i b l e t o i n t e r p r e t t h e above system as a s p e c i a l c a s e of
s p a t i a l growth dynamics w i t h z e r o s e l f growth ai = 0 , i = 1,2,
weight" of
species equal t o unity:
coefficients b
ij
=
-
...,I,
relative
("average
bi = 1, i = 1 , 2 , . . . , I , and i n t e r a c t i o n
a i j i n a n a n a l o g i c a l t o V o l t e r r a ' s system ).
Thus, t h e
r e l a t i v e s p a t i a l growth model i n c l u d e s b o t h z e r o a g g r e g a t e ( r e g i o n a l ) growth
(V
-
C o n s t ) and z e r o i n d i v i d u a l c i t y growth ( a i , = 0 ) .
dynamics " z e r o growth dynamics."
The i n t r o d u c t i o n of new v a r i a b l e s
and nev i n t e r a c t i o n c o e f f i c i e n t s
r e s u l t i n a z e r o growth normalized dynamics:
We w i l l c a l l such
It
i s important
to
note
that
a z e r o growth
(normalized o r not)
relative
s p a t i a l dynamics can be l i n k e d d i r e c t l y t o V o l t e r r a ' s c o n s e r v a t i v e e c o l o g i c a l
t h e c o n d i t i o n of z e r o a g g r e g a t e growth (5.1) i s e q u i v a l e n t t o
associations:
i.e.',
t o Volterra's conservation condition.
T h i s f o l l o w s immediately from t h e
eq u a l i t y :
We n w f u r t h e r a n a l y z e t h e z e r o growth r e l a t i v e s p a t i a l dynamics:
We prove t h a t i n t h e above s y s t e m t h e m a t r i x B = ( b
fact
text.
with s i g n i f i c a n t
interpretational
implications
) is antisymmetric, a
i)
outlined
in
t h e main
By i n t r o d u c i n g t h e c o n d i t i o n :
(B. 10)
v a r i a b l e s yl,
~ ~ , . . . , y ~become
- ~
independent.
Due t o (B.8)
one c a n d e r i v e t h e
(B. 11)
identifyiq
yl,y2,...,y1-~.
a
polynanial
of
second
Condition (B.ll)
degree
in
the
independent
holds a s an I d e n t i t y ,
variables
thus r e q u i r i n g t h a t
a l l i t s c o e f f i c i e n t s must b e z e r o .
This, i n turn, implies that:
(B. 12)
V (biI
(bij
+
- 2bII)
+ bIi
b,,)
-
(biI
+
a
0
bIi)
,
-
i = 1,2
(bjI
+
,..., 1-1
bIj)
(B. 13)
- 2bII
i = 1,2,...,1-1.
= 0,
(B. 14)
Since V
>
0 , then:
which are t h e antisymmetry c o n d i t i o n s f o r m a t r i x B = (bij).
Inspite the
similarities between t h e above and V o l t e r r a ' s models, n o n e t h e l e s s , t h e r e i s a
big
difference
in
the
equilibrium
properties:
all
peculiar
conditions
a s s o c i a t e d w i t h t h e e x i s t e n c e of a n . e q u i l i b r i u m found i n a b s o l u t e e c o l o g i c a l
dynamics a r e a b s e n t i n t h e c a s e of r e l a t i v e s p a t i a l growth.
T h i s i s proven
below.
*
The e q u i l i b r i u m s t a t e s yi,
(B.l)
i = 1,2,.,1,
w i t h t h e antisymmetric i n t e r a c t i o n matrix
of
t h e dynamical system i n
B = (bij)
are the solutions
of t h e following system of e q u a t i o n s :
A
c o m p l e t e d e s c r i p t i o n of
analyzed next.
a l l possible
types
of
equilibrium s t a t e s ,
are
I t is easy t o s e e (by s u b s t i t u t i o n ) t h a t t h e s i m p l e s t s o l u t i o n s of
the
system a r e I d i f f e r e n t s o l u t i o n s of t h e type:
These e q u i l i b r i u m s t a t e s r e p r e s e n t t h e c o m p e t i t i v e e x c l u s i o n of s p e c i e s o r t h e
t o t a l c o n c e n t r a t i o n of whole g e o g r a p h i c a l s u b s t a n c e w i t h i n t h e r-th
( r = 1 , .1
region
F u r t h e r , l e t us c o n s i d e r t h e e q u i l i b r i u m s t a t e s with only
r (2 < r < I ) non-zero
For each r
coordinates.
r
(I) such e q u i l i b r i u m s t a t e s .
Without
l o s s of
t h e r e e x i s t n o t more t h a n
g e n e r a l i t y , one can assume
that:
For t h i s type of e q u i l i b r i u m t h e system (B.16) w i l l be:
The m a t r i x Br
of t h e l i n e a r system (B.20)
i s an antisymmetric r x r matrix.
I f i t s determinant i s non-zero,
t h e n t h e system has only a zero s o l u t i o n which
c o n t r a d i c t s c o n d i t i o n (B.2 1).
T h i s i s t h e d i f f e r e n c e between V o l t e r r a ' s and
our s p a t i a l r e l a t i v e growth c o n s e r v a t i v e dynamics.
This p o s s i b i l i t y h o l d s
But f o r even r i t
o n l y f o r even r , due t o t h e antisymmetry of t h e m a t r i x Br.
i s p o s s i b l e t h a t t h e determinant d e t Br w i l l d e g e n e r a t e t o z e r o , t h e n e c e s s a r y
c o n d i t i o n f o r e x i s t e n c e of e q u i l i b r i u m .
of
the
system
automatically.
(B.20),
due
to
I n t h e c a s e of odd r , t h e d e t e r m i n a n t
antisymmetry
or
Br,
is
equal
to
I n any c a s e , i f t h e d e t e r m i n a n t of t h e system e q u a l s zero:
zero
d e t Br = 0
,
t h e n t h e system (B.20,21)
where A
rj
has a s o l u t i o n
a r e t h e a l g e b r a i c complements ( o r c o - f a c t o r s ) ,
i n t h e r-th
row of t h e m a t r i x Br.
f o r t h e elements a
rj
T h i s s t a t e m e n t f o l l o w s from t h e well-known
p r o p e r t y of c o - f a c t o r s (Korn and Korn [ 7 ] , 1.5-21.:
r
i = r
d e t Br,
0
,
i f r
which i n o u r c a s e (B.22) i s
The
conservation
condition
F u r t h e r , f o r t h e same y
(B.21)
obviously
holds
*
f o r yi from
(B.23).
*
1
Now i t w i l l be shown t h a t o n l y t h e c o m p e t i t i v e e x c l u s i o n e q u i l i b r i u m
(B-18) c a n b e s t a b l e a s y m p t o t i c a l l y .
e q u a t i o n s (B.l)
The o r i g i n a l s y s t a a of
c a n be r e w r i t t e n , a f t e r t h e s u b s t i t u t i o n :
i s made, i n t h e form:
differential
I t i s well-known
(Korn and Korn 1 7 1 , 9.5-4),
t h a t t h e equilibrium s t a t e s f o r
such a system a r e s t a b l e a s y m p t o t i c a l l y i f , and o n l y i f , a l l e i g e n v a l u e s of
thenatrixL
-
) = (1
ij
) have n e g a t i v e r e a l p a r t s .
L n o u r case
(B. 30)
and, t h e r e f o r e , f o r t h e e q u i l i b r i u m s t a t e s (B.18)
a d i a g o n a l (1-1)
x (1-1)
t h e m a t r i x L h a s t h e form of
m a t r i x w i t h i t s d i a g o n a l e l e m e n t s Vbir
(1
#
r) :
The e i g e m a l u e s of t h e m a t r i x L are i t s d i a g o n a l e l e m e n t s , and, t h e r e f o r e , t h e
c o n d i t i o n f o r a s y m p t o t i c a l s t a b i l i t y of t h e c o m p e t i t i v e e x c l u s i o n e q u i l i b r i u m
a r e t h a t Vb
bir
ir
<
0,
<
0
,
i # r,
i = 1,2,..
or
':
., r-1,
rtl,..., I
.
T h i s means t h a t t h e c a n p e t i t i v e e x c l u s i o n e q u i l i b r i u m (B.18)
(B.32)
is asymptotically
s t a b l e i f , and o n l y i f , a l l non-diagonal
e l e m e n t s of t h e r-th
antisymmetric i n t e r a c t i o n matrix B = (bij)
are s t r o n g l y n e g a t i v e .
column of t h e
I t f o l l o w s immediately f r a n t h e antisymmetry of t h e i n t e r a c t i o n m a t r i x B
t h a t f r w I d i f f e r e n t c o m p e t i t i v e e x c l u s i o n e q u i l i b r i u m s t a t e s (B.18) o n l y o n e
c a n be a s y m p t o t i c a l l y s t a b l e s i n c e t h e n e g a t i v i t y of t h e r t h column of t h e
m a t r i x B i m p l i e s t h e p o s i t i v i t y of t h e r t h row,
i.e.,
t h e m a t r i x B can n o t
i n c l u d e t v o d i f f e r e n t n e g a t i v e columns.
Further,
f o r t h e e q u i l i b r i u m s t a t e s of t h e t y p e (B.19)
c o o r d i n a t e s (B.23) m a t r i x L h a s components from (B.29,
I
w i t h r non-zero
30):
i > r
O
i < r
The
sum of
trace
1
j*
lj
the
real
parts
of t h e m a t r i x L.
of
t h e eigenvalues
Due t o (B.25).
for
.
t h e matrix
(B. 33)
L
is
the
t h i s t r a c e i s equal t o zero:
T h i s means t h a t t h e sum of t h e r e a l p a r t s of t h e e i g e n v a l u e s i s e q u a l t o z e r o ,
which i s p o s s i b l e i f e i t h e r t h e r e a l p a r t of e a c h e i g e n v a l u e i s e q u a l t o z e r o ,
o r t h e r e e x i s t eigenvalues with strongly positive r e a l parts.
I n both c a s e s
t h e e q u i l i b r i u m states of t h e t y p e (B.23) a r e a s y m p t o t i c a l l y u n s t a b l e .
latter
case
the
equilibrium
is unstable,
i m a g i n a r y e i g e n v a l u e s i m p l i e s p e r i o d i c motion.
while
t h e f o r m e r c a s e of
In the
pure
APPENDIX C
The s u f f i c i e n t c o n d i t i o n s f o r t h e i n t e g r a t i o n of t h e normalized s p a t i a l
dynamical system i n e l e m e n t a r y f u n c t i o n s a r e :
bij
+
bjk
+
bki = 0
f o r each i , j , k
.
( c *1 )
These c o n d i t i o n s t o g e t h e r w i t h t h e antisymmetry of t h e i n t e r a c t i o n m a t r i x B =
(bij)
allow
the
construction
of
the
system's
f o l l o w i n g way ( S o n i s [ 101 , p. 116) :
then
where C1 i s a c o n s t a n t .
Therefore,
Thus,
One c a n o b t a i n , due t o t h e above c o n d i t i o n :
T h e r e f o r e , from (C.5),
f o r each f i x e d k
explicit
solution
in
the
yi = y i ( 0 ) exp bikt I L y j ( O ) exp b j k t
,
i = 1,2,.-*,I
(C.8)
j
The above e x p r e s s i o n d e s c r i b e s t h e g e n e r a l i z e d l o g i s t i c growth and r e p r e s e n t s
a
temporal
(Sonis
e x t e n s i o n of
[ll]).
This
t h e well-known
extension
logit
random u t i l i t y c h o i c e model
l i e s a t p r e s e n t .in t h e i n n e r
c o r e of
the
u n i f i c a t i o n of i d e a s of u r b a n dynamics, i n n o v a t i o n d i f f u s i o n and i n d i v i d u a l ' s
c h o i c e behavior.
Condition (C.l)
i s only a s u f f i c i e n t but not necessary c o n d i t i o n f o r t h e
integration i n quadratures.
For proving t h i s one c a n c o n s i d e r t h e f o l l o w i n g
normalized s p a t i a l dynamic systems
with t h e antisymmetric i n t e r a c t i o n matrix
(C. 10)
For t h i s m a t r i x c o n d i t i o n (C.l)
b12 + b23
+
b31
i s not t r u e because
= 1 - 1 - 1 = - 1 2 0 ,
(C. 11)
but t h e s u b s t i t u t i o n s :
y 2 + y 3 = 1 - Y1 B
c o n v e r t t h e system (C.9.1-4)
- y1 -
y 3 = y2
-
i n t o t h e system
1
(C. 12)
il =
Yl ( 1
i
- Y2
=
Y 3 = l -
(1
y1
yl)
-
(C. 13.2)
Y2)
- Y2
(C. 13.3)
with an obvious s o l u t i o n
y, = 1 / 1 + c,e-'
y2=1/1+C2e
y3
-
(c1c2
-
t
,
(C. 14.1)
,
1) / ( 1
(C. 14.2)
+ ~ ~ e (-1 +
~ c)2 e t )
.
(C.14.3)