Optimal Control of a Class of Reaction‐Diffusion Predator‐Prey Models With Allee Effect and Migration Behavior

ABSTRACT In this paper, we discuss the optimal control problem for a reaction‐diffusion predator‐prey model with Allee effects and migration behavior. First, using methods such as operator semigroup theory, the Ascoli‐Arzel theorem, and embedding theorems, the existence and uniqueness of the strong solution of the state system, as well as the existence of the optimal control, are rigorously proven. Next, the Lagrange multiplier method is employed to construct the adjoint state system and the first‐order optimality conditions are derived. Furthermore, a fully discrete finite element method is employed to numerically solve both the state and adjoint systems. A discrete iterative scheme, integrated with a forward‐backward algorithm, is developed to perform accurate numerical simulations of the optimal control strategy. Finally, analysis of the numerical results confirms the effectiveness and reliability of the proposed control approach.