Propagation and Blocking in Periodically Hostile Environments

We study the persistence and propagation (or blocking) phenomena for a species in periodically hostile environments. The problem is described by a reaction-diffusion equation with the zero Dirichlet boundary condition. We first derive the existence of a minimal nonnegative nontrivial stationary solution and study the large-time behavior of the solution of the initial boundary value problem. In addition to the main goal, we then study a sequence of approximated problems in the whole space with reaction terms with very negative growth rates which are outside the domain under investigation. Finally, for a given unit vector, by using the information of the minimal speeds of approximated problems, we provide a simple geometric condition for the blocking of propagation and we derive the asymptotic behavior of the approximated pulsating travelling fronts. Moreover, for the case of the constant diffusion matrix, we provide two conditions for which the limit of approximated minimal speeds is positive.