Dynamical Properties Of Two Kinds Of Stochastic Population Models

The dynamic relationship between predator and prey has important biological significance,so it has been an important research topic of biological mathematics.Many scholars have carried out research in this field,and have produced a great deal of achievements.The existence of biological populations in nature must be affected by environmental noise.Therefore,it is necessary to introduce the influence of environmental noise on parameters in the model and apply the knowledge of stochastic differential equations to study the dynamic properties of stochastic models after stochastic perturbation.The corresponding numerical examples are given to verify the correctness of the theoretical results.Firstly,we study a class of predator-prey model with functional function,get the existence and uniqueness of solution,the random persistence of model,the asymptotic behavior of model,the time average persistence of model,the stationary distribution and ergodicity of model.The existence and uniqueness of the solution can be proved,and the random persistence of the solution is proved by constructing the auxiliary function and applying the It? formula.Then we take some appropriate inequality techniques and use the knowledge of random differential equations such as constructing the appropriate Lyapunov function,It? formula and exponential martingale inequality to study the asymptotic behavior of the research model.The time average persistence of the model under the hypothetical conditions can also be studied by means of constructive auxiliary function.The Lyapunov function and It? formula are used to study the stationary distribution and ergodicity.The corresponding numerical examples are given to verify the correctness of the theoretical results.Secondly,we study another kind of predator-prey model which has positive equilibrium solution,get the existence and uniqueness of the solution,the global asymptotic stability of the solution.The existence and uniqueness of the solution are studied by using the appropriate Lyapunov function,It? formula,stop time and index function.Then we select the Lyapunov function and use the It? formula,some inequality techniques and classification discussion to study the global asymptotic stability of the solution.The corresponding numerical examples are given to verify the correctness of the theoretical results.