The effect of an unfavorable region on the invasion process of a species with Allee effect

To model a propagating phenomena through the environment with an unfavorable region, we consider a reaction–diffusion equation with negative growth rate in the unfavorable region and bistable reaction outside of it. We study rigorously the influence of $L$, the width of the unfavorable region, on the propagation of solutions. It turns out that there exists a critical value $L^{\ast }$ depending only on the reaction term such that, when $L<L^{\ast }$, spreading happens for any solution in the sense that it passes through the unfavorable region successfully and establish with minor defect in the region; when $L=L^{\ast }$, spreading happens only for a species with large initial population, while residue happens for a population with small initial data, in the sense that the solution converges to a small steady state; when $L>L^{\ast }$ we have a trichotomy result: spreading/residue happens for a species with large/small initial population, but, for a species with medium-sized initial data, it cannot pass through the region either and converges to a transition steady state.