Geometric Method For Asymptotic Properties Of Stochastic Lotka-Volterra Model

In recent years,biological mathematics has been paid much attention by many scholars.Scholars describe the interaction among populations through the population model.but in real life,population model is influenced by the interaction of the population,and is also affected by many other factors in the population living environment.Therefore,in order to describe the population model more accurately,it is necessary to establish stochastic population model by adding random errorterm in the population model.Many scholars have studied the persistence of stochastic population models,including stochastic permanence,weak persistence and persistence in mean,but there are few results on almost sure permanence.This paper is a study of almost sure permanence and uniformly ultimate boundedness of stochastic Lotka-Volterra model.It is practical significance to analyze the persistence of population models in order to analyze the law of population development in ecosystems.In this paper,the asymptotic properties of the three kinds of stochastic Lotka-Volterra model are analyzed by geometric methods.The asymptotic properties mainly include two aspects: uniformly ultimate boundedness and almost sure permanence.In this paper,the asymptotic property of the stochastic Lotka-Volterra model is studied by the geometric shape of the stationary distribution.That is,it shows that the stationary distribution of stochastic Lotka-Volterra model is located within the first quadrant and far from the coordinate axis by proving that the solution from the outside of a region in the first quadrant enters the interior of the region in finite time with probability 1,and the solution from the interior of a region in the first quadrant stays in this region in finite time with probability 1,thus demonstrates that the stochastic Lotka-Volterra model has uniformly ultimate boundedness and almost sure permanence.The main content of this paper is divided into three parts.In the first part,the operator trajectories of three stochastic Lotka-Volterra models are analyzed.It is proved that the trajectories of LV(x,y)=0 of the three stochastic Lotka-Volterra models lie the interior of the first quadrant.The second part uses mainly the geometric method to prove that the stochastic Lotka-Volterra model has uniformly ultimate boundedness and almost sure permanence.The third part simulates mainly three kinds of stochastic Lotka-Volterra models and verifies the asymptotic properties of the stochastic Lotka-Volterra model.