Research On Asymptotic Properties Of Several Stochastic Population Systems

The survival problems of biological population have become one of the most impor-tant problems in the society. In recent years, researches on survival problems of popula-tion have received great attention, and many results on deterministic population systemshave been reported. Unfortunately, in the real world, the population is affected by theenvironmental noise and impulsive perturbations more or less. Therefore it is of key im-portance to study the stochastic population systems in order to explore the effects of en-vironmental noise and impulse on population systems. This paper will focus on studyingthe dynamics of the following stochastic population models:1. Asymptotic properties of a stochastic non-autonomous predator-prey model withimpulsive perturbations. In the model, the impulsive effects and the white noise are takeninto account to study the effects of them on the asymptotic properties. We first provethe existence and uniqueness of the global positive solutions by using equivalent method,and then present sucient conditions for persistence and extinction. Our results illustratethat the negative impulse and the white noise are able to accelerate the extinction of thespecies, and that the positive impulse is able to contribute to the survival of the species.2. Asymptotic properties of a stochastic Lotka-Volterra cooperative system with im-pulsive perturbations. We establish sucient conditions for extinction and stability in thesense of time average for every species. Our results show that the impulse has importantimpacts on the species: the existence of the positive impulse is good for the survival ofthe species while the existence of the negative impulse contributes to the extinction of thespecies.3. Dynamics of Logistic system with jumping noise under regime switching. Weobtain the critical value for the persistence and extinction of the species, and we givesucient conditions for stochastic permanence. Our conclusions demonstrate that theasymptotic properties of the population have close relations with the jumping noise andthe stationary distribution of the Markov chain. Especially, the Markov switching canregulate the survival of the species.4. Asymptotic properties of stochastic hybrid Gilpin-Ayala system with jumps underthe disturbance for two parameters. We first present sucient conditions for persistenceand extinction, and give the threshold for persistence and extinction. Then we establishsucient conditions for stochastic permanence by using M-matrix theory and Lyapunovmethod. Finally, we show the impacts of the jumping noise and Markov switching onthe species using examples and numerical simulations. Our results and examples illus-trate that the noise has important effects on the long time behaviour of the populations,especially, the Markov switching can control the survival of the population by means ofchanging its stationary distribution.5. Stochastic Lotka-Volterra systems under regime switching with jumps. In themodel, the white noise, telegraph noise and jumping noise are taken into account at thesame time. We first prove the existence and uniqueness of the global positive solutions;then we obtain the stochastic permanence by using the stochastic analysis techniques andM-matrix theory; finally, we discuss the extinction of the species, moment average in timeand asymptotic pathwise estimate of the system.