Abstract: We described  two competing species  model  by a reactiondiffusion equation with advection term. We hoped that when resources were allocated, the weighted density of two species in the ecosystem reached the maximum during the whole time period and at the final time. By proving the existence of the solution of the equation, the existence of an optimal control and the necessary condition  of the optimal control problem, and the results show  that the population density reaches the maximum when the resource allocation satisfies a bangbang control.

Key words: reactiondiffusionadvection equation, optimal control, population model, adjoint equation