Spatial ecology, optimal control and game theoretical fishing problems

Of paramount importance in both ecological systems and economic policies are the problems of harvesting of natural resources. A paradigmatic situation where this question is raised is that of fishing strategies. Indeed, overfishing is a well-known problem in the management of live-stocks, as being too greedy may lead to an overall dramatic depletion of the population we are harvesting. A closely related topic is that of Nash equilibria in the context of fishing policies. Namely, two players being in competition for the same pool of resources, is it possible for them to find an equilibrium situation? The goal of this paper is to provide a detailed analysis of these two queries (i.e optimal fishing strategies for single-player models and study of Nash equilibria for multiple players games) by using a basic yet instructive mathematical model, the logistic-diffusive equation. In this framework, the underlying model simply reads [Formula: see text] where K accounts for natural resources, [Formula: see text] for the density of the population that is being harvested and [Formula: see text] encodes either the single player fishing strategy or, when dealing with Nash equilibria, a combination of the fishing strategies of both players. This article consists of two main parts. The first one gives a very fine characterisation of the optimisers for the single-player game where one aims at solving [Formula: see text], under [Formula: see text] and [Formula: see text] constraints on the fishing strategies [Formula: see text]. In particular, we show that, depending on the value of these constraints, this optimal control problem may behave like a convex or, conversely, concave problem. We also provide a detailed analysis of the large diffusivity limit of this problem. In the case where two players are involved, we rather write [Formula: see text] as [Formula: see text] where [Formula: see text], the fishing strategy of the i-th player, also satisfies [Formula: see text] and [Formula: see text] constraints. Defining [Formula: see text] we aim at finding a Nash equilibrium. We prove the existence of Nash equilibria in several different regimes and investigate several related qualitative queries, for instance providing examples of the well-known tragedy of commons. Our study is completed by a variety of numerical simulations that illustrate our results and allow us to formulate open questions and conjectures.