Research On A Few Kinds Of Random Biological Models

In the real world,any object will be interfered by other objects or self interference caused by the vibration of the object itself.We call these disturbances "noise disturbance",it can exist in a variety of forms in the environment,and it has a different influence on the movement and the form of things.The essence of noise is a kind of random variable,some mathematical models,such as Gauss noise,time delay system,sinusoidal noise,and Levi noise,have been established to simulate the effects of noise disturbance on the ecosystem in nature.In the model research,we use stochastic process to characterize the noise disturbance,it is of both theoretical and practical significance to consider the various dynamic properties of the model solution under the disturbance of the noise.This paper mainly studies several kinds of stochastic biological models,considers some properties of the solution under the interference of noise.Firstly,the random Belousov-Zhabotinskii model(B-Z model)is studied,we establish the existence and uniqueness of the solution to the B-Z model,and then obtain that the solution has ergodic properties;Secondly,it is proved that the existence and uniqueness the solution of a predator-prey system with Allee effect is obtained under stochastic condition,and the existence of positive periodic solution is also obtained;In the end,the stochastic predator-prey model of Lotka-Volterra(L-V)is studied.We prove that the model has a nontrivial positive periodic solution,which has global attractivity.The method used in this paper is constructing Lyapunov function,which is also the difficulty of this paper.By analyzing these models,we use some basic theories of stochastic differential equations and construct appropriate Lyapunov functions on the basis of existing experience and methods,so that the global existence and uniqueness,the existence and ergodicity of periodic solutions and the global attractiveness of solutions of correlated stochastic models are obtained.