Zero-stabilization for some diffusive models with state constraints (Q2874972)

The paper is devoted to the zero-controllability and the zero-stabilizability for the nonnegative solutions to some Fisher-like models with nonlocal terms describing the dynamics of biological populations with diffusion, logistic term and migration. A necessary and sufficient condition for the nonnegative zero-stabilizability for a linear integro-partial differential equation is obtained. For a related semilinear problem a necessary and sufficient condition for the local nonnegative zero-stabilizability are also derived in terms of the magnitude of the above mentioned principal eigenvalue. The rate of stabilization corresponding to a simple feedback stabilizing control is dictated by the principal eigenvalue. A necessary and sufficient condition for the stabilization to zero of the predator population in a predator-prey system is also established.