Dynamic Behaviors Of Three Kinds Of Predator-prey-mutualist System

This paper consists of three parts as follows:Firstly,we consider a periodic predator-prey-mutualist system with the density of the predator.By using the Brouwer fixed point theorem and constructing a suitable Lyapunov function,some sufficient conditions which ensure the existence of a unique globally asymptotically stable periodic solution of the system are obtained.An example together with its numerical simulation is presented to verify the feasibility of our results.Secondly,a discrete predator-prey-mutualist system is considered.First of all,through the analysis of the right side of the system,a set of sufficient conditions which guarantee the permanence of the system is obtained.Secondly,by using the Brouwer fixed pointed theorem,the mean-value theorem and the properties of exponential functions,some sufficient conditions which ensure the existence of a unique globally asymptotically stable periodic solution of the system are obtained.Then,the sufficient conditions which ensure the extinction of predator species is obtained.On this basis,a set of conditions which ensure the globally asymptotically stable of prey-mutualist are obtained.Two examples together with their numerical simulations are presented to verify the feasibility of our results.Thirdly,a periodic predator-prey-mutualist system without the density of the predator is investigated.Since the estimate of the bounds of the solution of the system can not be simple scaling of the coefficients,we need to analyze the right side of the system in detail,a set of sufficient conditions for the permanence of the system is obtained.An example together with its numerical simulation is presented to verify the feasibility of our results.