The Dynamics Of Stochastic Dispersal Species Models

Population ecology originated from the population statistics, it is a science to study the development law of the species, and its research methods is using math-ematical models to understand, explain and predict the change of each species, so as to manage and protect the species more well. In the beginning, scholars built up deterministic mathematical models of ecological system and studied their dynamic behaviors. Lotka-Volterra model is a milestone in theoretical ecology. Various ecological model have been proposed later, and many scholars systematic studied their dynamic properties. Dispersal is a life history trait that has profound effects on both species persistence and evolution and it is prevalent in almost all species. Many scholars introduce diffusion phenomena to the deterministic model. However, there always exists white noise in the environment, which will lead to var-ious species in the ecosystem are subject to various forms of random interference. Therefore, stochastic differential equations can reflect the reality more accurately. In this paper, we consider the dynamic behaviors when the intrinsic rate is stochas-tic perturbed in single population models and predator-prey model with diffusion, respectively.In this paper, we consider the dynamical behavior of diffusion single-species models and diffusion predator-prey model when the intrinsic rate of increase is disturbed by environmental white noise, and study the asymptotical properties of the diffusion single-species models under the combined effect of environmental white noise and color noise. First, we show there exists a unique positive solution of the stochastic systems by Lyapunov analysis method, which is the base to study the dynamics of the systems. Then we study the p-th moment boundedness, and based on it, we study that whether the system has random persistence and the mean time persistence. In the diffusion deterministic single population models and predator-prey model, they have positive equilibrium points under the certain conditions, but the introduction of random perturbations cause the random systems have no equilibrium points. However, the solutions of the random system is in a neighborhood of the deterministic system when the intensity of the white noise is small, The performance of solutions in a fluctuations, that is the random system has a stationary distribution and has ergodicity. Specially, the large white noise may bring the extinction of the species.All in all, in this paper, we point out that the stochastic systems imiate the corresponding deterministic systems if the white noise is small; while if the white noise is large, the stochastic systems have more different properties, such as unper-sistence and extinction. In the reality, the large white noise can be considered as the bad weather and rapidly changing environment etc.