Abstract
We study the economic management of a renewable resource, the stock of which is spatially distributed and moves over a discrete or continuous spatial domain. In contrast to standard harvesting models where the agent can control the take-out from the stock, we consider the case of optimal stock enhancement. In other words, we model an agent who is, either because of ecological concerns or because of economic incentives, interested in the conservation and enhancement of the abundance of the resource, and who may foster its growth by some costly stock–enhancement activity (e.g., cultivation, breeding, fertilizing, or nourishment). By investigating the optimal control problem with infinite time horizon in both spatially discrete and spatially continuous (1D and 2D) domains, we show that the optimal stock–enhancement policy may feature spatially heterogeneous (or patterned) steady states. We numerically compute the global bifurcation structure and optimal time-dependent paths to govern the system from some initial state to a patterned optimal steady state. Our findings extend the previous results on patterned optimal control to a class of ecological systems with important ecological applications, such as the optimal design of restoration areas.
Highlights
The ongoing decline of natural resources means that the problem of the optimal management of ecological systems is of ever increasing importance
We investigate the management of a spatially distributed renewable natural resource where an agent or a policy maker is directly interested in the abundance of a resource, rather than in its yield or depletion, and is able to contribute to its growth and prosperity by means of some spatially targeted stock–enhancement policy
We show that the interest of the agent is different from the familiar harvesting model, the presence of spatial couplings may induce an optimal diffusion–induced instability, which leads to spatially heterogeneous optimal stock–enhancement policies
Summary
The ongoing decline of natural resources means that the problem of the optimal management of ecological systems is of ever increasing importance. Neubert (2003), Neubert and Herrera (2008) and Ding and Lenhart (2009a) find spatially heterogeneous distributions of the fishing effort to be optimal, and show that no-take regions (i.e., the concentration of fishing activities outside those localized regions) are typically part of an optimal fishery management strategy.1 These authors assume that the stocks equal zero at the boundary (Dirichlet boundary conditions), postulating that the fish are instantly killed as soon as they touch the shore (i.e., totally hostile boundary), which is a questionable assumption from an ecological perspective. Brock and Xepapadeas (2008, 2010) systematically investigate the emergence of heterogeneous solutions in spatial optimal control problems with an infinite time horizon These authors show that heterogeneity may arise in a Turing like bifurcation where a spatially homogeneous steady state becomes unstable in the presence of diffusion. While our main result is that in a large parameter domain the optimal policy is to govern the system to a POSS, we show how this steady state can be reached in an optimal way when we start at some historically given non-optimal situation
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