Dynamic Behaviors Of Commensalism System

This paper consists of four parts as follows:Firstly,a two species discrete commensalism system is proposed and studied.By using the Jury's conditions of the stability in second order discrete system,the local stability property of positive equilibrium is investigated;After that,sufficient conditions which ensure the global asymptotic stability of the interior equilibrium are obtained.Secondly,a nonautonomous discrete two species Lotka-Volterra commensalism system with delays is considered.Base on the comparison theorem of difference equation,the permanence of the system is obtained.Then,by constructing a new discrete Lyapunov functional,a set of sufficient conditions which guarantee the system global attractivity are obtained.For the almost periodic cases,by using the Hull theory of the discrete almost periodic system,sufficient condition which ensure the existence of a unique almost periodic solution is obtained.Thirdly,a nonautonomous commensal symbiosis model of two populations is proposed and studied.By using the comparison theorem of differential equations,we showed that the system is permanent;After that,by using a continuation theorem base on Gaines and Mawhin's coincidence degree,we study the global existence of positive periodic solutions of a nonautonomous commensal symbiosis model of two species.Finally,by constructing a suitable Lyapunov function,sufficient conditions which ensure the global attractivity of the positive periodic solution is obtained.Finally,a commensalism system with distributed delays is proposed and studied.By using the comparison principle of ODE and constructing a suitable Lyapunov functional,sufficient conditions are derived for the global asymptotic stability of the positive solution of the system.