Coexistence And Extinction Of Stochastic Lotka-Volterra Competition Models

The classical deterministic Lotka-Volterra competitive model has been widely used in many fields.There are abundant results on the study of the dynamical behavior of the solution of the system in different environments.By introducing two constants determined by the system parameters,the dynamic behavior of the Lotka-Volterra com-petition model is classified.Recently,this idea was also implemented in the stochastic Lotka-Volterra competition model.With the help of ergodicity of the solution distribu-tion of the equation on the boundary,the researchers introduce two constants related to the Laypunov exponent and determined only by the parameters of the system.The co-existence and extinction of two-dimensional stochastic Lotka-Volterra competition sys-tem are classified completely according to the sign of constants.In this dissertation,the above-mentioned ideas have been fully implemented in the stochastic Lotka-Volterra competition model with toxin effect,and have been extended to the three-dimensional stochastic Lotka-Volterra competition model.The results are as follows.Firstly,we discuss a class of stochastic Lotka-Volterra competition model with toxin effect.Two constants??1,?2?are introduced,which are only related to parameters.By means of ergodicity of the solution distribution of the equation on the boundary and Markov s inequality,it is proved that these constants are related to the Lyapunov exponent.The coexistence and extinction of these models are classified by the sign of these two values and a few numerical simulations are given to support the results.Secondly,the following three-dimensional stochastic Lotka-Volterra competition model is discussed.Three constants(?_3z,?_3xand ?_3y) are introduced,which can be calculated from the parameters.By means of ergodicity of the solution distribution of the equation on the boundary and Markov's inequality,it is proved that these constants are related to the Lyapunov exponent.The coexistence of three-dimensional competitive stochastic Lotka-Volterra model is classified according to the sign of these three constants,and then theoretical results are verified by numerical simulation.