A Stochastic Differential Fishery Game for a Two Species Fish Population with Ecological Interaction

Abstract

We combine and extend two existing lines of research in game theoretic studies of fisheries. The first line of research is the inclusion of the aspect of predation and the consideration of multi-species fisheries within classical game theoretic models of fisheries and goes back to Quirk and Smith (1977), Anderson (1975) and most recently Sumaila (1996). The models developed in this line are either static or discrete time and do not include ecological uncertainty. The second line of research includes continuous time and uncertainty, but focuses on single species models and does not capture any features of ecological interaction, see for example Jorgensen and Yeung (1996). In this article we develop a continuous time framework, where ecological interaction is described by a stochastic dynamics, including the cases of predator prey and competition. We obtain a stochastic differential game and derive feedback Nash-equilibrium strategies in semi-analytic form. Furthermore we compare the results with the case where fisheries regulations restrict each fishery as to only being allowed to fish one particular species and study the inefficiencies which arise from this. In addition to that, we also consider the case where fisheries cooperate. Here we observe quite different effects on the ecosystem, depending on whether the system is competitive or predator prey.