Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time-horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward-backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time-dependent paths, and their dependence on parameters. We start with some one-species fishing models, and then extend the analysis to a predator-prey model of the Lotka-Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers Just like ordinary differential equation-constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs). These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states. The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low-stock initial state, and nonmonotonic (in time) harvesting efforts. Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators.