Existence of stable localized structures in population dynamics through the Allee effect

We consider a one-dimensional chain of sites, appropriate for colonization by a biological species. The dynamics at each site is subjected to the demographic Allee effect. We consider non-zero probability p of dispersal to the nearby sites and we prove for small values of p, the existence of asymptotically stable localized solutions, such that the population of the central site almost equals the carrying capacity, while the populations at nearby sites attain small values, which drop exponentially with the distance from the central site. We study numerically a chain of 101 sites. Three different cases of behavior are observed, corresponding to the source-sink effect, the rescue effect and extinction. We study the bifurcations leading to transition from one behavior to the other. The motion of the invasion wavefront and the effect of heterogeneity of sites are also studied