An optimal control problem for a two-prey and one-predator model with diffusion

An optimal control problem is studied for a reaction-diffusion system that models an ecosystem composed by one predator and two prey populations. One proposed to maximize the total density of the three populations. To do this, one proves the existence of an optimal solution and one establishes first and second order optimality conditions. Several numerical simulations performed in both one-dimensional and two-dimensional isolated environments and using different Holling type functional responses support the theoretical results.