Internal null stabilization for some diffusive models of population dynamics

We investigate the large-time behavior of the solutions to some Fisher-type models with nonlocal terms describing the dynamics of biological populations with diffusion, logistic term and migration. Two types of logistic terms are taken into account. A necessary condition and a sufficient condition for the internal null stabilizability of the solution to a Fisher model with nonlocal term are provided. In case of null stabilizability (with state constraints) a feedback stabilizing control of harvesting type is proposed. The rate of stabilization corresponding to the feedback stabilizing control is dictated by the principal eigenvalue to a certain linear but not selfadjoint operator. A large principal eigenvalue leads to a fast stabilization to zero.Another goal is to approximate this principal eigenvalue using a method suggested by the theoretical result concerning the large time behavior of the solution to a certain Fisher model with a special logistic term. An iterative method to improve the position (by translations) of the support of the feedback stabilizing control in order to get a larger principal eigenvalue, and consequently a faster stabilization to zero is derived. Numerical tests illustrating the effectiveness of the theoretical results are given.