A new optimal cross-diffusive control for a class of spatio-temporal predator-prey models

We investigate a new optimal control problem for a class of quasilinear parabolic predator-prey models encompassing spatio-temporal aspects and featuring a bounded class of nonlinear functional responses. The considered predator-prey population resides in a two-dimensional habitat and is characterized by intense predatory behavior and costly acquisition of growth-promoting stimuli for the prey. Given that the standard control approach of acting on reaction terms cannot be used in the presence of such characteristics, we lay the ground-work for a new approach allowing to balance the concentrations of predators and prey. Namely, we add a cross-diffusion term to the prey equation, such that the associated time-dependent cross-diffusivity acts as control variable, incorporating various ways to direct prey towards lower concentrations of predators. Based on Schaefer's fixed point theorem and compactness arguments, we establish that the state system admits a unique essentially-bounded positive weak solution. Thereon, by means of an adequate adjoint system, we derive a necessary optimality condition for the optimization problem at hand. Finally, we demonstrate the effectiveness of our new-developed control strategy through illustrative numerical examples.