The Asymptotic Behavior Of Stchastic Biologival Models And Epidemic Models

Population ecology is a science to study the development law of the species, which usually pays attention on the quantity and quality of the species. Mathematical models can help us to understand, explain and predict the change of each species, so as to manage the species more well. Epidemiology is to predict the development trend of the disease by studying internal rules of the disease, and then give the optimal strategy to control it. Therefore, a lot of scholars have built up deterministic mathematical models of ecological system and epidemic system and studied their dynamic behaviors. However, there always exists white noise in the environment, so stochastic differential equations can reflect the reality more accurately. In this paper, we consider the parameters in the population models and epidemic models with stochastic perturbation, and study their dynamic behaviors.For the population models, we mainly discuss the dynamics of the Lotka-Volterra multi-mutualism system, multi-competitive system and predator-prey systems with stochas-tic perturbation. First, we show there exists a unique positive solution of the stochastic systems by stochastic comparison theorem or Lyapunov analysis method, which is very important to study the dynamics of the systems. Then we are going to investigate the persistence and non-persistence of the systems. If there are equilibria, we give the sta-bility of them by stochastic Lyapunov functions. If there is none of equilibrium, we show that they are persistence in the time average by stochastic comparison theorem and the martingale theory etc. Besides, we obtain there exist stationary distributions and they are ergodic, which also means the systems are persistent. On the other hand, we investi-gate there is one species or some species to die out. Specially, the large white noise may bring the extinction of the species.For the epidemic models, we mainly study the dynamics of the stochastic multigroup SIR model. It is well know that in the deterministic epidemic models, the basic productive number R0 is a threshold value, which determines whether the epidemic dies out or prevail. If R0≤1, the disease will die out; while if R0>1, the disease will be prevalent. Studying the stochastic epidemic models, we are also interested in this. In this paper, we point out that the disease will die out when R0<1 and it is prevalent when R0>1, if the white noise is small. But there is an interesting result. When the white noise becomes large, the disease will also die out even if R0>1, i.e., the large white noise suppresses the prevalence of the disease.All in all, in this paper, we point out that the stochastic systems imiate the cor-responding deterministic systems if the white noise is small; while if the white noise is large, the stochastic systems have more different properties. In the reality, the large white noise can be consider as the bad weather and serious disease etc.