Study On The Permanence And Stability Of Three Kinds Of Predator-prey Systems

This paper consists of three parts：Firstly, we propose and study a discrete Holling II type predator-prey model withmodified Leslie-Gower and time delay. With the help of the theory of differenceequation and some subtle analysis, sufficient conditions which ensure the permanenceof the system are obtained. By constructing a suitable Lyapunov functional, sufficientconditions which ensure the global attractivity of the system are obtained.Secondly, we propose and study a discrete ratio-dependent n-species differencepredator-prey system with Holling II type functional response. By applying thecomparison theorem and some new analysis technique, some sufficient conditionswhich guarantee the permanence of the system are obtained. After that, by constructinga suitable Lyapunov function, sufficient conditions which ensure the global attractivityof the system are obtained. Example together with its numerical simulation lendscredence to the plausibility of the main results..Finally, we study a discrete two-prey one-predator model with infinite delays. Byapplying extremum principle of function, sufficient conditions which ensure thepermanence of the system are obtained; Also, by contructing a suitable Lyapunovfunctional, sufficient conditions are established to ensure the global stability of positivesolutions of the system. The plausibility of the main results is demonstrated by annumerical simulation.