A stochastic differential Fishery game for a two species fish population with ecological interaction

We combine and extend two existing lines of research in game theoretic studies of fisheries, building up on Quirk and Smith (1977), Anderson (1975), Fisher and Mirman (1996), Sumaila (1997) and most recently Datta and Mirman (1999) who developed either static or discrete time models, not including ecological uncertainty and Jorgensen and Yeung (1996) who do include uncertainty but do not capture any features of ecological interaction. 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 Markov feedback Nash-equilibrium strategies in semi-analytic form. Furthermore we compare the results with the case where fisheries regulations restrict each fishery to harvest only one 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.