Nonlinear spatial dynamics, equilibrium states, and ... - Science Direct

Nonlinear spatial dynamics, equilibrium states, and static optimization theory Dimitrios

S. Dendrinos

Department

of Urban

Planning,

University

of Kansas,

Lawrence,

Kansas,

USA

Are long-term deterministic sociospatial dynamics optimum? Can one find optimality criteria under which the dynamic equilibrium state of dynamic deterministic models could also be construed as best? These are the questions addressed in this paper, and the answers seem to be negative in the general case, at least for a number of optimization formulations examined. Keywords:

Volterra-Lotka dynamics, universal map of discrete relative dynamics, dynamic equilibrium states, linear programming, geometric programming

Introduction Questions regarding optimality of sociospatial configurations have always attracted the attention of social scientists. Sociospatial configurations, consisting of the spatial distribution of socioeconomic activities or stocks, result from individual deterministic behavior based on self-interest and/or from collective (social) actions based on selected and prevailing aggregate social preference function(s) at any point in space-time. Among others, the behavioral theory of welfare economics and social choice theory attempt to address these optimality issues. This paper is not, however, written in the mode of behaviorally driven sociospatial optima. Instead, it searches for optimality of the equilibrium states in a class of descriptive deterministic dynamic problems dealing with sociospatial configurations. These descriptive models bypass the complexity of the many social (individual and collective) agents’ motives and constraints in interacting with other agents and the environment. Instead, they focus on replicating the final outcome, that is, the resulting aggregate sociospatial configuration. In this paper, some optimum problems are stated in nonlinear spatial absolute and relative dynamics. They differ from prior efforts in that they do not attempt to derive governing potentials generating these dynamical forms, as for example those by Dendrinos and Sonis’ responsible for generating the Volterra-Lotka relative spatial dynamics or that by Dendrinos* for the Dendrinos-Sonis universal map of discrete relative spatial

Address reprint sas, Lawrence, Received

46

requests to Dr. Dendrinos Kansas 66045, U.S.A.

18 October

1988; accepted

Appl. Math. Modelling,

at the University

11 September

of Kan-

dynamics. The problems analyzed here also do not attempt to replicate optimum control and associated optimum management formulations of a dynamically adjusted environmental resource, as those by Goh,4 Clark,” and others or by Dendrinos and Mullally on the optimum aggregate urban growth management. Instead, the optimum set of problems derived here is associated with specific spatial static population allocations generated by the equilibrium states of nonlinear spatial dynamics. Two cases are examined as representatives of a broad spectrum of dynamic models. Initially, the absolute, continuous-time, one-species, multiple-location Volterra-Lotka dynamics model is reviewed. Then the relative, discrete-time, one-species, multiple-location Dendrinos-Sonis’ universal map is examined. In the first case a static linear programming problem is stated that is associated with any equilibrium spatial configuration derived from absolute spatial dynamics; in the second case a static geometric programming problem is tested against the equilibrium spatial population distribution state derived from the Dendrinos-Sonis universal map of relative dynamics. Interpretation of these optimum problems is supplied, together with a discussion of their prima1 and dual objective functions, constraints, and variables. Interpreting the primal and dual objective functions and their associated constraints, variables, and complementary slackness conditions affords new and fundamental insights into the dynamic equilibrium states to be obtained. In particular, the meaning of the dual variables is of interest in the discussion of sociospatial equilibrium states because it points to the presence of certain composite locational prices. These prices, in combination with the parameters of the primal and dual objective functions, provide the stage for the forces of concentration in and dispersal from a location by a stock (like human population) to be played out.

1989

1990, Vol. 14, January

0 1990 Butterworth

Publishers

Nonlinear

spatial dynamics,

The Volterra-Lotka programming

equilibrium

states, and static optimization

x,*2

system and linear

First, the standard Volterra-Lotka system is looked at under its multiple-location (rather than multispecies) formulation. Under conditions of absolute-size, onestock, multiple-location continuous dynamics the standard Volterra-Lotka system of quadratic equations is given by = Xi(t)

r5 lZi +

h=

Xi(t) ? 0

(1) (2)

1,2,..., Z) represents the locationspecific self-growth/decline rates of the (homogeneous) population, that is, - 0~5 ai 5 + c;c,and the square matrix 6,(i,j = 1,2,. . . , I) represents interlocational interaction coefficients, so - ~0i h, 5 + a. Subscript i is a location index covering the Z locations of the environment considered. Dynamic (stable or unstable) equilibrium x” (i = 1, 2,. . .) I) exists, that is, Xi(r) = 0 for all i’s, if for any location i, either the xj+ is zero, which is the trivial solution corresponding to competitive exclusion of the stock from that location, or the growth rate is zero, that is, _&(t)lxi(t)j, = 0. In case some of the x,?‘s are zero, say, xz (h = I, 2, . . . , H < I), then these locations can be taken out of the environment of I sites. The deletion does not affect the dynamics at equilibrium, since all interaction terms containing these locations would vanish from the right-hand side of the Volterra-Lotka specifications. This is demonstrated next. Isoclines corresponding to the dynamic equilibrium state are given by

x

bijXj

=

0

i=

1,2,...,1

(3)

or, in matrix notation, BX = A, where X and A are Z-element vectors and B is a square Z x Z matrix. A solution to condition (3) also satisfying (2) may or may not exist. Further, system (3) may or may not have a solution. The necessary condition for (3) to have at least one solution xi (it could have up to Z solutions) is that the determinant of the system, (B( = det B, be nonzero. This specification will be referred to as a consistency condition. If the vector Xialso satisfies the nonnegativity condition, then XT = X,, that is, the solution of (3), is a dynamic (stable or unstable) equilibrium. This will be referred to as an equilibrium condition. An equilibrium condition is necessarily a consistent one, but not vice versa. Consider a particular set of coefficients cii and b, such that a set of x7 exists satisfying both (3) and (2). Then one can assume without loss of generality that

cij +

c 6,x:

j=l

xh* = 0

2 0

i = 1,2,. . . ,I

(4)

1,2,...,H

h=

(6) (7)

whereas in the rest of the locations the following conditions prevail:

xj*> 0 ci,, +

j=H+l 2

1...,

6,,jxr = 0

Z

(8)

m = H + 1,. . . , Z

(9)

j=H+l

directly follows from conditions (7)-(9) that the locations h = 1, 2, . . . , H do not enter into the final state, and their interaction terms hjh and self-growth rates & are not found there either. The original Volterra-Lotka dynamical system of the larger environment of Z locations has now been reduced to a system containing conditions (7)-(9). The finding can be further amplified by the following slight modification of condition (6). If a slack quantity q, 2 0 is introduced into the remnant (and at equilibrium irrelevant) conditions (6), the equation can be rewritten as a strict equality:

It

I ah

-

qh

+

LhjTtT =0

2 J=H+

I

h = 1, 2, . . . , H

(10)

so that

4hG = 0

i=l

1,2,...,H

1

j=H+

whereai(i=

+

(5)

since some of the XT’S could be zero. Condition (4) is expressed in “greater than or equal to” terms by convenient choice of parameter signs. The system at the equilibrium state has two parts. In the particular subset in which there is competitive exclusion, that is, xh* = 0 (h = 1, 2, . . . ) H < I), the constraints have the form

1 i = 1, 2, . . . , Z

S. Dendrinos

1,2,...,1

bijXj(t)

j=I

f2(

i=

0

theory: Dimitrios

h = 1,2,...,H

(11)

Interpretation of the time invariant quantity qh is informative. It represents a repulsion force for the (homogeneous) stock at location h at the state of dynamic equilibrium. This repulsion is a force of complete spatial deconcentration from site h, that is, a force leading from the start to eventual abandonment. Volterra derived the governing potential for this ecology in a multispecies, single-location context. The reader is referred to Volterra’s classical paper found in Ref. 8 (pp. 65-236). Within a relative dynamics framework the governing potentials were derived by Dendrinos and Sonis.’ Here the emphasis is not on these governing potentials, but rather on an associated family of optimization problems hidden in the VolterraLotka formalism. The analysis from here on will deal with systems in which &*=A

(12)

x*>o

(13)



47

Nonlinear

spatial dynamics,

equilibrium


Constraint (12) states that the sum over all interlocational interactions for the population at a given location must equal the growth rate of the stock at that location, in absolute value because the two quantities have opposite signs, by condition (9). Assyme a vector of self (locational) growth coefftcients A anda square matrix of interspatial interaction coefficients B such that a unique (admissible and, even further, positive) dynamic equilibrium exists. Implied in the above specification is the fact that this system is consistent. This particular (and consequently quite restrictive) system at its dynamic equilibrium state has additional properties. They are revealed through the use of duality theory. Consider a (primal) linear objective function of the form, in matrix notation, minimizeg

(14)

$X=A

(1%

x>o

(16) an interior solution X such that

x=x*>0

(17)

then the ecologically optimum spatial population concentrations are also the dynamic (stable or unstable) equilibrium configurations. There could be an infinite number of ecological cost vectors C” resulting in (17). Identifying conditions restraining the C*‘s, which would make possible for the ecologically optimum to be further the dynamic equilibrium configurations, requires the use of duality theory. From classical duality theory the following always holds: X* is a solution to (15) and (16) if and only if each solution A* to the system JR* = 0

(18)

is also a solution to the system AA* = 0 The proof is straightforward. From (15), one has

48


S. Dendrinos

It directly follows that when (18) holds, then from (20), condition (19) holds. All the above is well known from the theorem regarding systems of linear equations (see, for example, Ref. 9, p. 16). Its converse, referred to as the Farkas lemma, and the above provide the building blocks of the Kuhn-Tucker conditions of duality theory. Next, this property is employed to derive the corresponding linear programming problem to a Volterra-Lotka system with a dynamic (stable or unstable) equilibrium, and to supply an interpretation of it and its dual. According to duality theory, the problem of minimizing g(X)- subject to L?X = A and the positivity constraint has a dual: maximize u(R) = iA

(21)

subject to the dual constraints

= CX

where the parameters C are nonnegative numbers. These parameters can be viewed as ecological, location-specific net composite costs associated with supporting on the average a unit of stock at that location. Quantity g is thus a spatially cumulative composite ecological cost associated with any spatial population distribution. Put differently, it represents the net cost of population concentration at the various locations of the environment considered. If a particular vector of costs, C*, is found such that minimizing g subject to the two constraints

produces

theory: Dimitrios

(19)

Assume that (19) holds.


;A = C*

(22)

A>0

(23)

Duality theory requires that if ,% is a solution to the primal problem, then the dual has also a solution li and further that g*(X) = u*(A)

(24)

In this case the complementary slackness conditions collapse to the two sets of constraints being met, that is, (15) and (22), to hold as strict equalities. Variables A are the Lagrange multipliers associated with the Lagrangian function: L = g(X) - A[A - ix]

(25)

Interpretation of the above dual variables, constraints, and objective function is of interest. The primal objective function can be construed as a composite ecological cost g of population concentration, as has already been noted. This cost may be due to the net effect of population agglomeration, and it can be thought of as a concentration force. Quantity u, on the other hand, represents the ecological net composite benefit of the (aggregate) spatial population dispersal. Each term in u identifies the location-specific composite ecological net benefit from the growth rate of the homogeneous stock, Xi, at that location at the dynamic equilibrium state. When-the optimum spatial population concentration pattern X prevails, that is, the minimum of the primal objective function is attained, the optimum population dispersal A is also reached. Put differently, the optimum ecological composite cost g* and benefit u* are obtained when the net ecological benefit/cost is zero so that the force of concentration equals that of dispersal. The Lagrange multipliers, that is, the dual variables, identify location-specific ecological values (or prices) of population dispersal, AT, at the equilibrium state, and they represent the marginal ecological cost of concentration. Whereas the primal constraints identify forces of co?centration restrained by the self-growth coefficients A, the dual constraints identify forces of

Nonlinear

spatial dynamics,

decentralization based interaction. The costs of detected from

AiC b,

equilibrium

the j are restrained C. This dual constraints:


of interthe net be di-

S. Dendrinos

stock sizes. There, it is demonstrated that the prospects for optimality are not so promising.

The Dendrinos-Sonis universal relative dynamics and geometric programming

= C;

indicating that the price of interspatial interaction or dispersal, given by the term on the left-hand side in (26), must equal at equilibrium the net cost of concentration. Since the problems have interior solutions, under the restrictions imposed, the ecological values (or prices) are always positive. Because there are an infinite number of vectors C* that would satisfy the primal problem restrictions, there are an infinite number of problems for which a particular dynamic equilibrium spatial population distribution is also optimal from a composite ecological standpoint. The particular vector for which the minimum minimorum_ of g*(X*), g**, and maximum maximor_um of u*(A), u**, is obtained will be designated as c. Searching for a particular set of 7’s that are most likely to determine the ecological costs as distributed in space, while there is no information regarding the specific locational advantages at each site, one might resort to an entropic approach in determining them. Natural resource abundance, topographical features, and climatic and other locational attributes are most appropriate in determining such costs. In the absence of such information, however, one might be interested in obtaining the most likely distribution of such costs. A new problem may be thus defined, one in which the following cost vector is picked so that the following entropic formulation is maximized: maxH = nC7 c, I

theory: Dimitrios

(27)

If, and under what conditions, a geometric programming problem can reproduce the equilibrium states of relative stock allocations is examined next. Specifically, the Dendrinos-Sonis universal map of relative dynamics is used as the testing ground. Geometric programming is chosen to carry out the test because it is the most likely candidate available from the known static operations research problems to date. Under conditions of relative-size, one-stock, multiple-location discrete dynamics the Dendrinos-Sonis’ universal algorithm has the form X;(t + 1) = F[(t)lx

i=

Fj(t)

1,2,...,1

(29)

j I;

= AjJJ#“Q

(30)

1 A;>0

(31) < 1

0 0

(34) 1

(35)

at all time periods t (0 5 I 5 T). Dynamic (stable or unstable) equilibrium is obtained when

xi*= Fit:(t)lC q.(t)

subject to the constraints

Solutions of the above ~i’s produce the most likely estimates of composite net ecological costs. The above concludes the analysis of optimality for the equilibrium state of the standard locational Volterra-Lotka dynamic formulation. It was shown that a rather restrictive consistent formulation is likely to contain a (potentially large) number of optimality criteria that would render it optimal. These criteria belong to a family of linear programming problems. It is of interest that a dynamic deterministic process can result in a spatial configuration reproduced by a search algorithm of the simplex (or any equivalent) type. The search for an optimum in a linear programming problem usually is not thought of as a dynamic process, although a dynamic process might be able to reproduce it from some arbitrary initial conditions. Next, the analysis switches to a more involved dynamic spatial distribution model dealing with relative

The algorithm has been extensively discussed and its parameters and behavior interpreted elsewhere in the mathematical literature, for example, in Refs. 3 and 7. Not much will be added here except to note that the vector A is a set of environmental (scale) parameters and that the matrix of exponents (Yis a set of interlocational comparative advantages elasticities. We will next examine whether, and under what conditions, a geometric programming problem can reproduce the expressions in (36). Geometric programming proves (Ref. 9, p. 4) that if Yi (i = 1, 2, . . . , I) are arbitrary nonnegative numbers and ai are arbitrary positive quantities satisfying the condition

&zij=

1

then the following inequality always holds:

csjYj'nYF



49

Nonlinear

spatial dynamics,

By making the substitution equality becomes

equilibrium


yi = SiYj the above in-

theory: Dimitrios

S. Dendrinos

so that Ing* - xxi(t)

lnA, = H(t)

(46)

or, put differently, which is referred to as the geometric inequality. The minimum at the left-hand side is the maximum of the right-hand side. This inequality is used next to formulate the primal and corresponding dual of a geometric programming problem associated with the Dendrinos-Sonis universal mapping. Consider the expressions g = x

where x(t) is a vector of relative population concentrations at the I locations and F(t) is a vector of locational disadvantages enjoyed by the stock at site i, given according to the universal map by (30). Then, if one wishes to minimize g (primal program) subject to the conservation condition of relative dynamics, which is the normality condition of geometric programming, one could find a dual function u:

U=n

A$ L

~x;(t)Q

(47)

If there are some problem specifications for which xi* = xi, that is, the optimum exists, is unique, and is equal to the dynamic equilibrium also, then In cxiy.F:’ (i

(40)

xi(f)Fi(f)

- ci E;(t) In Ai = H(t)

ln [F%(i,I;.cn]

>

- Cxi*ln& i

= H*

To an optimum there may or may not be a corresponding dynamic equilibrium vector of relative population concentrations. Thus any relative spatial population distribution may or may not be optimal: further, there may not be an optimal spatial population distribution that is dynamically feasible! Maximization of H* from (48) requires that ${ln(~rfF:)

- g.x!l,ar)

= 0

or

where ai = xi(t), such that u is maximized,

and

Di = C ‘YiXj = 0

--&ln J;

(42)

ExfFiy

= lnAyx

>

or

so that an optimum vector X, exists such that u*(X)

= IJ i

(A$I =

(51)

g*(X)

or

Of interest again is the interpretation of the dual variables of this geometric programming problem and its objective function. Whereas in the primal formulation the objective function contains the cumulative locational stock disadvantages (and is being minimized), in the dual formulation the objective function depicts positive effects of concentration. The dual variables Di identify the overall positive effects of scale upon location i from all other locations. System (42) may or may not possess solutions, and they may or may not be unique, depending on its determinant. If a solution X, exists, it may or may not be the dynamic equilibrium solution XT. Condition (42) is the orthogonality condition of geometric programming problems. From (43) one has lnu” = lng* = xf;(lnA; = xjS,lnA;

(44)

I

On the right-hand side of (44) the second term is the entropy of the system H: H(r) = -Cxi(t)ln?;(t)

50

Appl.

Math.

Modelling,

(45)

1990,

The Shannon maximum entropy H* of the relative spatial population distribution is obtained under particular environmental conditions, amax, proportional to the weighted comparative advantages at each location. Any incongruity between the environment and the locational advantages (advantages that are due to interspatial elasticities ati) appropriately scaled by relative population size at equilibrium is likely to prevent optimality according to condition (52).

Conclusions

- ln$

- FEilnyi

(52)

Vol. 14, January

For the first time, issues of optimality for deterministic dynamic equilibrium states have been raised. For the final long-term equilibrium configuration of two dynamic one-species multiple-location problems, two static optimum formulations were tested to see whether any equivalences existed. In the first case the linear programming formulation of the equilibrium state of the Volterra-Lotka problem was stated. In the second case

Nonlinear

spatial dynamics,

equilibrium


a geometric programming formulation was tested against the Dendrinos-Sonis dynamic equations. Conditions were sought under which the optimum is also the outcome of the dynamic equilibrium process. Under the absolute deterministic and descriptive Volterra-Lotka dynamics the linear programming problem was found capable of reproducing, in certain instances, the equilibrium state as its optimum. This did not prove to be the case under the relative discrete dynamics test with geometric programming problems. The finding seems to suggest that there might be a family of static optimization problems that can reproduce outcomes of equilibrium relative dynamic spatial stock concentrations (like the ones from the Dendrinos-Sonis equations) that have not yet been uncovered. On the other hand, the exercise might point to the fact that not all dynamic (stable or unstable) equilibrium states may necessarily be socially desirable, given our current knowledge of optimality criteria and problems. Acknowledgments The author wishes to thank Professor M. Cross and two anonymous referees for helpful comments and suggestions and his research assistant, Jian Zhang, for

theory: Dimitrios

S. Dendrinos

processing the final version, although the author retains full responsibility for any remaining errors. References Dendrinos, D. S. and Sonis, M. Variational principles and conservation conditions in Volterra’s ecology and in urban relative dynamics. J. Regional Sci. 1986, 26(2). 359-377 Dendrinos, D. S. Theoretical developments in discrete maps of relative spatial population dynamics. XXVIIEuropecrn Congress qf the Regionul Science Association, Athens, Aug. 25-28, 1987; published in Sistemi Urhani, No. 213, 1988 Dendrinos, D. S. and Sonis, M. The onset of turbulence in discrete relative multiple spatial dynamics. Appl. Muth. Comprct. 1987, 22, 25-44 Goh, B. S. Optimal control of a fish resource. Malayan Scientist 1969, 5, 65-70 Clark, C. W. Muthemcrtical Bioeconomics: The Optimal Munugement ofRenewuble Resources. Wiley Interscience, New York, 1976 Dendrinos, D. S. and Mullally, H. Optimum control in nonlinear ecological dynamics of metropolitan areas. Environment and Plunning A 1983. 15, 543-550 Dendrinos. D. S. and Sonis, M. Turbulence und Socio-Spatial Dynamics. Springer-Verlag, New York, 1990 Scudo, F. M. and Ziegler, J. R., eds. The Golden Age qf Theoretical Ewlogy: 1923-1940. Lecture Notes in Bio-Mathematits, Vol. 22. Springer-Verlag, New York, 1978 Duffin. R. J., Peterson, E. L.. and Zener, C. Geometric Programmin~: Theory undApplications. John Wiley and Sons, New York, 1967

Appl.

Math. Modelling,


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