The Study of Population Dynamics Behavior with Diffusion

Species dispersal and migration are one of the most prevalent phenomena of nature,and therefore become one of the most interested subjects for researchers no matter inhome and abroad. In this paper, based on the study of other authors, we give a moredepth and abroad research about the species dispersal and migration. The main contentsof this paper include: two species predator-prey Lotka-Volterra type continuous dispersalsystem with delay and Holling type II response function, two patches periodic singlespecies impulsive diffusion logistic system and two patches periodic predator-prey systemwith prey impulsive diffusion.The main contents in this paper can be summarized as follows:1. In section 1, firstly, we introduce the ecological background of this paper. Then,it is given that some research conclusion recently about continuous dispersal system andimpulsive dispersal system. We investigate our research system finally.2. In section 2, it is studied that two species predator-prey Lotka-Volterra typecontinuous dispersal system with delay and Holling type II response function. By usingcomparison theory and some analysis methods, suffcient conditions of integrable form forthe boundedness, permanence, extinction and existence of positive periodic solution areestablished, respectively.3. In section 3, we study a periodic single species logistic system with impulsivediffusion in two patches, suffcient and necessary conditions on the existence, uniquenessand global asymptotically stable of periodic solution and the extinction of species for thissystem are established by applying some analysis methods and constructing appropriateLyapunov function. Moreover, for the system with n patches, we obtained analogousresults using the same methods.4. Finally, in section 4, we study a periodic predator-prey system with prey impulsivediffusion in two patches, by using comparison theorem of impulsive differential equationand other analysis methods, suffcient and necessary conditions on the permanence and extinction of prey population for the predator-prey system with predator have other foodsource are established and, considering the predator-prey system with predator have notother food source, we show the suffcient and necessary conditions that under which thepredator population will be permanent or extinct exactly.The results show that our conclusions are perfect, and it is meaningful since there arefew studies about nonautonomous impulsive diffusion single species system and predator-prey system.