  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
(\emph{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 $-\mu\Delta \theta=\theta(K(x)-\alpha(x)-\theta)$
where $K$ accounts for natural resources, $\theta$ for the density of the
population that is being harvested and $\alpha=\alpha(x)$ 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. In the case where two players are
involved, we aim at finding a Nash equilibrium. We prove the existence of Nash
equilibria in several different regimes \textcolor{black}{and investigate
several related qualitative queries}.Our study is completed by a variety of
numerical simulations that illustrate our results and allow us to formulate
open questions and conjectures.
