论文标题
空间生态学，最佳控制和游戏理论捕鱼问题
Spatial ecology, optimal control and game theoretical fishing problems
论文作者
论文摘要
在生态系统和经济政策中，至关重要的是收获自然资源的问题。提出这个问题的范式是捕鱼策略。的确，过度捕捞是现场储备管理中的一个众所周知的问题，因为太贪婪可能会导致我们正在收获的人口的总体巨大耗尽。一个密切相关的话题是纳什均衡在捕鱼政策的背景下。也就是说，两个玩家正在争夺同一资源池，他们是否有可能找到平衡情况？本文的目的是通过使用基本但具有启发性的数学模型，逻辑 - 散文方程式，对这两个查询（\ emph {i.e}最佳捕捞策略以及对多个玩家游戏的NASH Equilibria研究）进行详细分析。 In this framework, the underlying model simply reads $-μΔθ=θ(K(x)-α(x)-θ)$ where $K$ accounts for natural resources, $θ$ for the density of the population that is being harvested and $α=α(x)$ encodes either the single player fishing strategy or, when dealing with Nash equilibria, a combination of the fishing strategies of both players.本文由两个主要部分组成。第一个给出了单人游戏的优化器的表征非常出色。在涉及两名球员的情况下，我们旨在寻找NASH均衡。我们证明了纳什平衡在几个不同的方案\ textcolor {black} {{并研究几个相关的定性查询}中。
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 $-μΔθ=θ(K(x)-α(x)-θ)$ where $K$ accounts for natural resources, $θ$ for the density of the population that is being harvested and $α=α(x)$ 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.
