Migration Of The Dynamical Behavior Of The Population Of The Competition Between Populations Plaque

The aim of this work is to construct several mathematical models of popula-tion dispersal and to analyze the dynamic behavior of these models. We mainly focus on two competitive populations with dispersals between two patches.In the first part, we mainly study the dispersals produced by the local com-petitive pressures from both intraspecific competition and interspecific compe-tition. Linear functions of local population density are adopted to describe the migration rates of the two competitive populations. We obtain theoretically the sufficient and necessary condition under which the two competitive populations persist in the two patches. Based on this conclusion, we compare the results of persistence led by the constant dispersal with the result of dispersal from local competitive pressure and find that, no matter what kind of persistence pattern it is initially with the constant dispersals, dispersals led by the local competitive pressures can make the two populations persist in the two patches. As the circum-stance changes, the density-dependent dispersals can alter thoroughly the fate of the weaker population and increase its survival probability. Lastly, through computer simulations, we get some important results about the bifurcation led by the dispersals from competitive pressures and find the existence of seven pos-itive steady states which changes the distribution of the two populations in the two patches and induce more complex outcomes of competition. At the same time, the density-dependent dispersals may also lead to more stable coexistence regions of interacting populations and increase their survival possibilities.In the second part of this paper, we adopt two dynamical variables to de-scribe the density-dependent migrations, where patch fitness is considered as the key point for biological individuals to decide whether to emigrate or not since bi-ological individuals may find important information in a long distance in a patch environment through all kinds of methods, which directs their dispersals. On this base, we study the important influences led by the adaptive dispersals on the spa-tial distribution and persistence of the two competitive populations. We find that under some conditions, the two populations achieve an ideal spatial distribution through weak dispersal adaptations and exhibit periodic fluctuations for strong dispersal adaptations. We have also found the extreme changes of the spatial distribution from one dominative population to another under the influences of dispersal adaptations. In addition, we get the important conclusion that popu-lations can survive only in the better patch and desert the worse one through dispersal adaptations, which breaks through the conclusion with the constant dispersal and embodies the superiority and rationality of adaptive dispersals.