Dynamic analysis and optimal control of a toxicant-population model with reaction-diffusion

In this paper, we study the threshold dynamics and optimal control of a toxicant-population model with reaction-diffusion to understand how toxicant affect populations. In order to obtain the extinction and persistent conditions of the toxicant, the basic reproduction number of the model is considered, when R-0 < 1, the toxicant-free equilibrium is globally attractive, when R-0 > 1, the solution to the system is uniformly persistent. We also introduce the optimal control strategy, with the method of dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation is constructed and the optimal control is obtained. Finally, we conduct numerical simulations to verify the theoretical analysis.