Qualitative Analysis Of Two Stochastic Mutualism Population Models With Lévy Jumps

In nature,there are many complex interactions among species in biological commu-nities,such as competition,predation and mutualism.Moreover,populations are closely related to their living environment and are easily affected by stochastic environmental fac-tors.Therefore,the establishment and analysis of population dynamics model in stochastic environments has become an important subject in biomathematics research.In recent years,a large number of stochastic models with Brownian motion have been used to describe the relationship between species and their living environment.However,in the real world,pop-ulation systems may suffer sudden environmental shocks such as earthquakes,epidemics,tsunamis and hurricanes.These natural disasters can change population size in a short time.These phenomena can not be accurately described by white noise.Therefore,it is necessary to establish and analyze stochastic population model with Lévy jumps.In this paper,we will discuss the qualitative properties of two stochastic mutualism population models with Lévy jumps.Chapter 1 describes the research status of stochastic population models and gives some theoretical knowledge needed in this paper.Chapter 2 considers permanence of a stochastic mutualism population model with Lévy jumps First,existence and uniqueness of global positive solution of the model are proved by con-structing Lyapunov function.Then,sufficient conditions for stochastic permanence of the model are obtained by using Ito formula and Chebyshev's inequality.Finally,the numerical simulations are given.The corresponding results in the existing literature are generalized.Chapter 3 studies asymptotic behavior of a stochastic mutualism population models with two-parameter perturbations and Lévy jumps First,existence and uniqueness of global positive solution of the model are proved by con-structing Lyapunov function.Then,stochastic ultimate boundedness and extinction of the model are proved by using Ito formula and exponential martingale inequality.Furthermore,the stationary distribution of the corresponding model is given without Lévy jumps.Finally,the numerical simulations are given.