Dynamical Behaviors Of Dispersal Species With Impulses

Species dispersal in two or more patches is one of the most prevalent phenomenaof nature. Many empirical works and monographies on population dynamics with dif-fusion have been done. Meanwhile, species are subjected to perturbations of man-madeor natural at fixed times, which can often be characterized mathematically in the form ofimpulses, have been also studied extensively. However, in all of these studies till now, it israrely to consider dispersal species with impulses, which is also one of the most prevalentphenomena of nature. Hence, four kinds of dispersal systems with impulses are presentedand studied in this dissertation.The backgrounds and research status on population dynamics with diffusion andwith impulses are presented. A simple introduction on dynamics of dispersal specieswith impulses is done. Four population dynamical systems are studied in the presentdissertation: an impulsive periodic predator-prey system with diffusion, an impulsivereaction-diffusion predator-prey system, a two-patches impulsive migration periodicLotka-Valterra competitive system and species impulsive migration systems with Marko-vian switching.Two species periodic predator-prey Lotka-Volterra type dispersal system withHolling type III functional response and impulses in a patchy environment is studied.On the basis of comparison theory of impulsive differential equations, analytical tech-niques and constructing appropriate auxiliary functions, conditions for the permanenceand extinction of the predator-prey system, and for the existence of a unique globally sta-ble periodic solution are established. Numerical examples are shown to verify the validityof our results.Based on the upper and lower solution method and comparison theory of differentialequations, sufficient conditions for the ultimate boundedness and permanence of an im-pulsive reaction-diffusion periodic predator-prey system with ratio-dependent functionalresponse are established. By constructing appropriate auxiliary function, the conditionsfor the existence of a unique globally stable positive periodic solution of this predator-preysystem are also obtained. Some numerical examples are presented to verify our results.By using analytical techniques and constructing auxiliary functions, a two-patches impulsive migration periodic N-species Lotka-Valterra competitive system is investi-gated. Sufficient conditions for the permanence, extinction and existence of a uniqueglobally stable positive periodic solution of the system are established. Some numericalexamples are shown to verify our results and discuss the model further.A single species stochastic impulsive migration logistic system with Markovianswitching and an N species stochastic impulsive migration Lotka-Volterra model withMarkovian switching in n different patches are presented and studied. By constructingappropriate auxiliary functions, some sufficient conditions on the global positivity, theultimate boundedness in mean and the extinction in mean of these two systems are estab-lished. A real world example is provided to illustrate the validity of our results.