Discounted supply curves and optimal harvesting in a multi-species fishery

In this paper, the problem of exploiting two ecologically interdependent fish stocks is considered. It is supposed that there exists a predator–prey relationship between both species. To model this kind of interaction, Leslie's model is used, which assumes that survivorship of the predators is dependent on the prey abundance. An optimal control problem is formulated by maximizing a strictly convex cost function representing the discounted net value of future revenues when the old prices are taken as given. Supply–demand curves of equilibrium values are determined from the canonical equations that are obtained from the first order optimality conditions of the optimal control problem. It is shown that, under some assumptions, curves display a backward bending, typical in supply curves of single species fishery, but driving to extinction or leaving out of the fishery are other possibilities. Finally, the stability analysis reveals that saddlepoints or instability is found.