Internal stabilizability for a reaction-diffusion problem modeling a predator-prey system

In this work we consider a 2× 2 system of semilinear partial differential equations of parabolic-type describing interactions between a prey population and a predator population, featuring a Holling-type II functional response to predation. We address the question of stabilizing the predator population to zero, upon using a suitable internal control supported on a small subdomain ω of the whole spatial domain Ω, and acting on predators. We give necessary and sufficient conditions for this stabilizability result to hold.