Asymptotic Properties Of Some Stochastic Biological Models With Jumps

In this thesis, several classes of stochastic biological models perturbed by Levy noises are investigated. Some asymptotic properties of the solutions of the models are obtained. In addition, a neutral stochastic functional differential equation is considered, the existence and the sufficient conditions for stability in distribution of the solution to the equation are established. The thesis will be written in six chapters.Chapter1briefly introduce the background and the practical significance of the problem which will be studied in the following chapters, and give a review of the devel-opment for the researches on stochastic biological models. Meanwhile, the main results and the framework of this thesis are presented.Chapter2begins with some preliminaries. It is devoted to the basic theories and definitions about stochastic analysis and stochastic differential equations needed in the studies of this thesis. The important inequalities which are related are also presented.In chapter3, a stochastic predator-prey model with Beddington-DeAngelis func-tional response and perturbed by Levy noises is studied. First, we give the existence and uniqueness of the global positive solution of the model from the point of biology. Next, the important properties including the moment estimation from above and the growth rate of the solution are obtained. At last, some sufficient conditions on the extinction of each species are established.Chapter4is devoted to the investigation of a non-autonomous stochastic Gilpin-Ayala competition model with jumps. The existence and uniqueness of the global positive solution of the model are obtained, the asymptotic properties of the p-moments and the asymptotic pathwise properties of the solution are investigated. Furthermore, the extinction and the non-persistence of each species are considered.In chapter5, we consider a stochastic SIRS epidemic model with jumps. The existence of the global positive solution is first obtained. The stochastic ultimate bound-edness of the solution is then studied. Finally, according to the value of Ro, if R0<1, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, if R0>1, there is a stationary distribution, which means that the disease will prevail.In chapter6, we consider a neutral stochastic differential delay equation driven by a- stable processes. First, by the Banach fixed point theorem, the existence and uniqueness of the mild solution of the model is obtained. Also, by the infinite dimensional stochastic analysis trick and the semigroup method, the stability in distribution of the mild solution is investigated, some sufficient conditions for the stability in distribution of the mild solution are established.