On Stationary Distribution And Periodic Solution Of Stochastic Population Models

This paper characterizes the dynamics of stochastic multi-species mutualismmodel, two kinds of stochastic predator-prey models and stochastic competitivemodel with disease by constructing suitable Lyapunov functions and using the the-ory of stationary distribution and periodicity developed by Has’ minskii. Firstly,We consider the existence and uniqueness of the positive solution to the systems.Moreover, suffcient conditions which guarantee the existence of stationary distri-bution or periodic solution of the systems are established. Non-persistence and longtime behavior around the equilibrium of the deterministic model are investigated.Finally, the results are illustrated by computer simulations. It splits naturally intothree parts.We investigate stochastic Lotka-Volterra multi-species mutualism model inPart I. Using the method of Lyapunov analysis, we study the existence and unique-ness of the positive solution to the system provided the coeffcients of the systemare continuous periodic functions. Moreover, suffcient conditions which guaranteethe existence of periodic solution of the system are established by the theory of M-matrix and periodicity of Has’ minskii. We also obtain conditions for the globallyattractiveness of the periodic solution. Furthermore, we prove that all species willbe extinct if the intensity of noise is large enough. The results are illustrated bycomputer simulations. In addition, We discuss the dynamics of stochastic multi-species mutualism model with constant coeffcients. Asymptotic property of thesystem is obtained, and there exist stationary distributions and they are ergodic.Part II is to study the dynamics of two kinds of stochastic predator-preymodels. One is stochastic predator-prey model with modified Lesile-Gower andHolling-type II schemes. For the stochastic model with constant coeffcients, exis-tence and uniqueness of the positive solution, existence of stationary distribution,non-persistence are investigated. For the stochastic model with periodic coeff-cients, we obtain the suffcient conditions for the existence of periodic solution.Moreover, we also study stochastic predator-prey model with modified Holling-Tanner and Beddington-De Angelis type. We obtain that there exist stationarydistributions and they are ergodic.Part III is to study the stochastic competitive model subject to an additionalfactor, a disease spreading among one of the populations. If there are equilibriafor the stochastic system, we study the stochastic stability around the equilibria.Otherwise, we study the persistence of the system. In particular, we obtain thatthere exist stationary distributions and they are ergodic.