Importance of dispersal adaptations of two competitive populations between patches

A model of two competitive populations with dispersals between two patches is proposed. It is assumed that the dispersal ability is adaptive because of the attraction from a better patch with higher fitness, and that the changing rate of dispersal probability of an individual is proportional to its fitness difference in the two patches. It is proved that adaptations of dispersal can change the spatial distribution of the two populations. When the intraspecific competition is stronger than the interspecific competition in every patch, it is shown that the two populations achieve an ideal spatial distribution through weak dispersal adaptations and exhibit periodic fluctuations for strong dispersal adaptations. It is also demonstrated that co-adaptations of dispersal can change the patterns of persistence of two competitive populations and increase their survival probabilities from the extinction in both patches to the persistence in one or two patches.