Allee effect in population dynamics: Existence of breather-like behavior and control of chaos through dispersal

We consider a one-dimensional chain of identical sites, appropriate for colonization by a biological species. The dynamics at each site is subjected to the demographic Allee effect. We consider nonzero probability p of dispersal to the nearby sites and we prove, for small values of p, the existence of asymptotically stable time-periodic and space-localized solutions, such that the central site carries the vast majority of the metapopulation,, while the populations at nearby sites attain very small values. We study numerically a chain of three sites, both for the case of open ends or periodic boundary conditions. We study the bifurcations leading to transition from chaotic to periodic behavior and vice-versa and note that the increase of the dispersal probability in both cases controls the chaotic behavior of the metapopulation.