Optimal contraception control for a nonlinear competitive vermin population model with spatial diffusion and size structure in a polluted environment

This study examines an optimal contraception control problem for a nonlinear competitive vermin population model that is dependent on size structure and spatial diffusion in a polluted environment. The control variables are the average number of female sterilant and the input rate of exogenous toxicant. It has a good guiding effect on curbing environmental pollution and controlling the number of vermin. These results provide a theoretical basis for controlling and preventing vermin and curbing environmental pollution. The hybrid system belongs to the class of separable models, and its solution is separable in terms of size and spatial location. Therefore, we transform the system into two subsystems and prove the existence and uniqueness of non-negative solutions via the comparison principle and fixed point theorem, respectively. The necessary conditions for the optimal contraception strategy are derived by constructing an adjoint system and using tangent-normal cones. The method of characteristics and the finite difference method approximate the non-negative solutions of the two subsystems, and some numerical examples illustrate the theoretical results.