Application of Distributed Parameter Control Model in Wildlife Damage Management

Abstract

A bioeconomic model for optimal control of wildlife damage by migratory small mammal populations is developed under the framework of a nonlinear distributed parameter control problem. The model first simulates the spatio-temporal dynamics of dispersal population by parabolic diffusive Volterra-Lotka partial differential equation and then optimizes a criterion function of present value combined costs of wildlife damage and harvesting. The existence of a unique optimal solution for a finite time problem is proved. An iterative procedure for numerical solution of the Optimality System with parabolic equations of opposite orientations is developed. The theoretical model is applied to a real life problem using biological and economic data for beaver populations under certain simplistic assumptions.

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

“…13 While the concept of a metapopulation is the cornerstone of the main paradigms in terrestrial conservation biology [62], its application to the marine context is relative new. As Joan an unmanaged patch to a managed patch [6,7,8,39,45] and a model of the spatial movements of elephants that generate value from wildlife viewing but also create damages by destroying crops [70,75]. Clark presents an optimized two patch model of a biological population dispersing between inshore and offshore areas [13], 14 Brown and Roughgarden illustrate the optimal value of a larval pool to system wide fishery profits [11], and Janmaat investigates the implications of different governance regimes across a patchy system [42].…”

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

“…We start by replacing problem (20) - (23) with its linear quadratic approximation. In doing so we extend the method developed by Fleming (1971), and Magill (1977) 12 -by which a non-linear optimal stochastic control problem is replaced by a simpler linear quadratic optimal stochastic control problem -to the case in which a deterministic control problem (such as a resource management problem), where the transition of the system is described by a partial differential equation with a diffusion term, and not by an ordinary differential equation, is replaced by a linear quadratic approximation.…”

“…Biological resources tend to disperse in space under forces promoting "spreading", or "concentrating" (Okubo, 2001); these processes along with intra and inter species interactions induce the formation of spatial patterns for species. In the management of economic-ecological problems, the importance of introducing the spatial dimension can be associated with a few attempts to incorporate spatial issues, such as resource management in patchy environments (Sanchirico andWilen, 1999, 2001;Sanchirico, 2004;Brock and Xepapadeas, 2002), the study of control models for interacting species (Lenhart and Bhat, 1992; Lenhart et al, 1999), the control of surface contamination in water bodies (Bhat et al 1999), or the creation of marine reserves (Neubert, 2003). 1 See for example Alfred Weber (1909), Harold Hotelling (1929), Walter Christaller (1933), and August Löcsh (1940) for early analysis.…”

“…A new -to our knowledge -characteristic of our continuous spacetime approach is that we are able to embed Turing analysis in an optimal control recursive infinite horizon approach in a way that allows us to 7 See for example Lenhart and Bhat (1992); Lenhart et al (1999); Bhat et al (1999); Raymond andZidani (1998, 1999). locate sufficient conditions on parameters of the system (for example, the discount rate on the future, and interaction terms in the dynamics) for diffusive instability to emerge even in systems that are being optimally controlled.…”

“…10 Similar conditions have been derived for other cases. such as the control of parabolic equations (Raymond andZidani,1998, 1999), boundary control , or distributed parameter control (Dean Carlson et al, 1991; Lenhart and Bhat, 1992). 11 In some cases in order to simplify notation, and when no confusion arises, sub-…”