Optimal control problem for an ecosystem with two competing preys and one predator

This work is devoted to the optimal control problem for an ecosystem composed of one predator and two competing preys, and our goal is to maximize the total density of the three populations. We first investigate the existence and uniqueness of the positive strong solution for the controlled system, and find an optimal solution under given initial conditions. Then we establish the first order necessary optimality condition, and point out that the optimal control has a bang-bang form. Moreover, the second order necessary and sufficient optimality conditions are established.