Dynamic Behavior Of Stochastic Dispersal Models Perturbed By White Noise

There exists a kind of constrained relationship among different populations in nature: populations rely on abundant natural resources for its growth,for example,population B depends on predation of population A,and population A depends on others for their existence.This constrained relationship constitutes a system called prey-predator system.In recent years,the deterministic predator-prey system has been extensively studied by a large number of scientific researchers who have obtained many very good results.The mathematical model has become an important tool for analyzing the constrained relationship between prey and predator,and the obtained results are now widely used in the formulation of animal protection policies.In the real world,noise exists in our surrounding environment,hence it is of practical significance to study the dynamic behavior of deterministic prey-predator dynamic systems under different environmental noise disturbances.The content of this article is arranged as follow:1.The long-time asymptotic behaviors of a single-species stochastic reaction-diffusion model with Markov Switching in a 9)patch environment have been studied.The conditions required for the system of two patches are appropriately reinforced.The two-dimensional system is extended to the 9)dimension case.The method is mainly using constructing different Lyapunov functions,and it has been proved the result that the positive solution of the random system exists and is unique with probability 1 provided any given positive initial value of the system.Then,we analyze the asymptotic properties of the solution when the time goes to infinity,which are the random boundedness and stochastic stability.Finally,a sufficient condition for the extinction of this single-species stochastic reaction-diffusion model system is given,that is,when the noise intensity is high,species will become extinct.2.Because of the complexity of the dispersal of biological populations,studying the linear diffusion prey-predator system has some certain limitations.We will study the model with direct diffusion based on the existing literature model,that is,considering the dynamic properties of stochastic predator-prey models with direct diffusion in 9)different patch environments.We use the method of constructing different Lyapunov functions,then the result about existence and uniqueness of positive solutions of stochastic systems are obtained,and then the stochastic boundedness of positive solution are analyzed.After that the sufficient conditions for the extinction of the prey and the whole system are given.Finally,the conclusions of all the theorems are verified by numerical simulation.The conclusions obtained in this paper improve the existing conclusions and make us have a deeper understanding of the random predator-prey system.From this point of view,the conclusions obtained in this paper are of some significance.