An optimal control for a nonlinear system is considered. The existence of an optimal control pair, the characterization of the optimal control in terms of the optimality system, and the uniqueness of solutions for the control problem are established. The uniqueness requires smallness assumptions on parameters \(\lambda,\mu\), in the payoff-functional defined by \(J: C\subset L^\infty_+ (\Omega) \times L^\infty_+ (\Omega)\to\mathbb{R}\), as: \[ J(d_1, d_2)= \int_\Omega\biggl\{ \lambda u_{d_1,d_2} (x)d_1(x)-\bigl(d_1(x) \bigr)^2+ \mu v_{d_1,d_2}(x) d_2(x)-\bigl(d_2(x) \bigr)^2\biggr\} dx, \] where \(C\) is a convenient convex subset of \(L^\infty_+(\Omega)\times L^\infty_+(\Omega)\) and \((u_{d_1,d_2}, v_{d_1,d_2})\) is the unique coexistence state of the system \[ \begin{aligned} -\Delta u=\sigma(x) v-d_1(x)u-c_2u(u+v)\quad & \text{in }\Omega,\\ -\Delta v=b (x)u- d_2(x)v-c_2 v(u+v)\quad & \text{in }\Omega,\\ {\partial u\over \partial \nu}= {\partial v\over\partial\nu}=0\quad & \text{on }\partial \Omega. \end{aligned} \] {}.