Optimal Control and Spatial Heterogeneity: Pattern Formation in Economic-Ecological Models

Abstract

This paper extends Turing analysis to standard recursive optimal control frameworks in economics and applies it to dynamic bioeconomic problems where the interaction of coupled economic and ecological dynamics under optimal control over space creates (or destroys) spatial heterogeneity. We show how our approach reduces the analysis to a tractable extension of linearization methods applied to the spatial analog of the well known costate/state dynamics. We explicitly show the existence of a non-empty Turing space of diffusive instability by developing a linear-quadratic approximation of the original non-linear problem. We apply our method to a bioeconomic problem, but the method has more general economic applications where spatial considerations and pattern formation are important. We believe that the extension of Turing analysis and the theory associated with the dispersion relationship to recursive infinite horizon optimal control settings is new. JEL Classification Q2, C6

The Economics of Spatial-Dynamic Processes: Applications to Renewable Resources

“…Brock and Xepapadeas [10], hereafter B/X, characterize the solution to this problem. An important conclusion is that when the underlying system is a continuous spatial-dynamic system, the fundamental state and co-state equations of the optimized Hamiltonian are diffusion equations, expressed as partial differential equations (PDEs) in both time and space.…”

“…9 Any of these may be appropriate in a particular setting, depending upon the nature of the geometry and other features of the setting. The important point is that B/X [10] derive spatial transversality conditions that are analogous to more familiar conditions for non-spatial intertemporal optimization problems. These spatial transversality conditions can be chosen to fit the particular geometry and boundary conditions appropriate to the problem at hand, and they will influence the nature of the optimal solution accordingly.…”

“…10 The dispersal coefficients, d jk , reflect the rate at which biomass moves from one patch to another. The own dispersal rate (rate of emigration) is assumed negative (d jk <0 with j=k) and the cross patch dispersal rates are positive when the patches are connected to each other (d jk >0 with j≠k).…”

“…9 In the new economic geography, the circle setting is consistent with the geometry of the "racetrack" economy [28]. 10 See [64] for a discrete model of a fish population that includes other forms of dispersal processes. 11 The assumption of no mortality in the dispersal process is not a requirement and one can easily assume that there is some loss in the dispersal process.…”

Biodiversity and geography

“…Rauscher (2000) and Pflüger (2001) address environmental issues in a genuine new-economic-geography setting, but do not address issues related to biodiversity issues. Some ecology-economic models have analyzed the spatial interdependence of landscapes (Bockstael 1996; Koskela and Ollikainen (2001) or renewable resource models that include a spatial diffusion process (Brock and Xepapadeas 2006;Smith et al 2007; Sanchirico and Wilen 2005) in a wide class of environmental problems, from biological invasions to natural reserve creation to provision of ecological services. But these models typically do not consider the centrifugal-centripetal effects of new economic geography to analyze the optimal spatial allocation of economic activity as opposed to biodiversity conservation.…”