Two Types Of Stochastic/Random Lotka-Volterra Systems' Dynamics

Lotka-volterra system has been widely used in biology,chemistry,game theory,economics,0)(8..Deterministic lotka-volterra system is usually studied under different conditions of competitive type,conservative type and dissipative type in literature,and a great deal of results have been obtained.However,the interference of random factors is ubiquitous in real life,so we considered that the random Lotka-Volterra system is closer to the real situation.In the recent two decades,much attention has been paid to the research of stochastic dynamical systems.A lot of literature has studied the existence of random attractor of the various forms of random dynamic system,the internal structure of random attractor,stochastic stability,stochastic periodic solutions,persistence,extinguishing,ergodicity,coexistence and exclusion conditions,and good results have been achieved.Since the importance of stochastic dynamical systems,this paper studies the sufficient conditions for the existence of random attractors in finite dimensional random Lotka-Volterra systems,and the sufficient conditions for the existence of random periodic solutions of a three dimensional stable dissipative Lotka-Volterra system under perturbation of white noise are also studied.The first part of this paper is to study the sufficient conditions for the existence of random attractors in a finite-dimensional random Lotka-Volterra system perturbed by real noise(article reviews about the real noise will introduce),using the method of looking for bounded closed set of random absorption under certain conditions.Because bounded closed set is a compact set in finite dimensional space,the existence of system attractor is proved.The second part is to study the sufficient conditions for the existence of the stochastic periodic solution of the three-dimensional stable dissipative Lotka-Volterra system under the influence of white noise on the natural growth rate.The method adopted here is to construct a proper Lyapunov function which satisfies the conditions of Theorem 3.8 in the book of Khasminskii R(2012).The third part is to use Matlab to numerically simulate the problem studied in this paper.Through theoretical proof and numerical simulation,we prove that when the parameters of a finite-dimensional random Lotka-Volterra system meet certain conditions,there must be a random attractor in the random system.It is also proved that the random periodic solution exists in a three dimensional stable dissipative Lotka-Volterra systemaffected by environmental noise when the parameters meet certain conditions.Finally,numerical simulation shows that stochastic attractors and stochastic periodic solutions do exist,and it is also found that the larger the noise is,the greater the difference between the images of the stochastic system and the corresponding deterministic system will be,and when the noise is large enough to a certain extent,the dynamic properties of the solutions of the stochastic system will change.