Necessary optimality conditions for three species reaction–diffusion system

An optimal control problem is studied for an ecosystem composed by one predator and two prey populations. Its dynamics is modelled by a reaction–diffusion system of Volterra type. Two control variables are introduced in the system; their meaning is the mixture rates between predator and each prey population. The goal of this paper is to maximize the total density of the three populations at a fixed time moment. The existence of the optimal control is established and necessary optimality conditions are found with the aid of a maximum principle.