Spatially Distributed Differential Game Theoretic Model of Fisheries

Abstract

We consider a differential game of fisheries in a fan-like control structure of the type "supervisor-several agents". The dynamics of the controlled system is described by a non-linear differential equation model which is identified on the Azov Sea data. An averaging by two spatial coordinates is conducted. Different information structures of the game are generated by the control methods of compulsion (supervisor restricts the feasible strategies of agents) and impulsion (she exerts an impact to their payoff functionals). Both Stackerlberg and inverse Stackelberg games are considered. For the numerical investigation we use a discretization of the initial model and the method of qualitatively representative scenarios in simulation modeling.Mathematics 2019, 7, 732 2 of 13 the Stackelberg games to fisheries can be found in [16,[18][19][20]. Game theoretic models of biological resource management with asymmetric players and different planning horizons are studied in [21,22]. Game problems of harvesting in a biological community are considered also in [23].The papers [24][25][26][27] present non-linear differential equation models of the shallow bodies of water (on the example of Azov Sea). The works [28,29] develop these models by the consideration of fish capture as a control impact: in [28] an optimal control problem is solved, and [29] treats a conflict control model which is formalized as a hierarchical differential game (a Stackelberg game).The differential Stackelberg games are described in the monograph [30], and the inverse Stackelberg games (with a feedback by control) are reviewed in [31]. An original approach to the analysis of differential Stackelberg games with consideration of different information structures which determine the order of moves of all players, their knowledge about the strategies of other players, and the state of the controlled dynamical system is presented in [32]. In that paper, the structure of optimal strategies is described and a method of solution of Stackelberg games and inverse Stackelberg games is proposed. The method is based on building of a mutually advantageous program of actions and punishment of the defectors [32].This paper develops the results presented in [28,29]. Unlike them, we consider here a fan-like control system (a differential game of a supervisor and several agents) and closed-loop strategies of the players. In the case of several agents, the set of their best responses to the supervisor's strategy is the set of Nash equilibria in the normal-form game of the agents. Besides, we consider a spatial coordinate [33].The authors' approach to the modeling of fisheries is based on the concept of sustainable management [34][35][36]. In this frame, the hierarchical control mechanisms (compulsion and impulsion methods) are presented as solutions of the Stackelberg games with phase constraints which reflect the requirements to the state of a controlled system (the conditions of sustainable development). Compulsion means the supervisor's impact to the sets of feasible ac...