Propagation Dynamics Of A Predator-Prey System With Three Species

In this paper we consider a three species predator-prey system consisting of a single predator and a pair of strong-weak competing preys.Under certain parameter conditions,the system can exist two semi-coexistence equilibrium points E*and E*,as well as one positive coexistence equilibrium point Ec,where E*means that the system has no strong competing prey at this time,E*means that the system now has no weak competing prey,and Ec represents that the three species co-exist.First we prove the existence of traveling wave solutions using generalized upper-lower solutions,and show that such traveling wave solutions connect two equilibrium points,where the unstable tail limit is the semi-coexistent equilibrium point E*or E*.When the unstable tail limit is E*and E*is stable,the stable tail limit is proved to be E*using the reduced dimensional contracting rectangle method.When the unstable tail limit is E*and Ec is stable,another equilibrium point is proved to be Ec by constructing an appropriate Lyapunov function.This suggests that the strong competing prey can successfully invade the environment when the predator and weak competing prey are aboriginal species.At this point,either the strong competing prey will eventually replace the weak competing prey,or the three species will eventually reach the coexistence state.When the unstable tail limit is E*and Ec is stable,another equilibrium point is proved to be Ec,again using the Lyapunov function.From the point of view of biological mathematics,when the predator and the strong competing prey are aboriginal species,we can prove that the weak competing prey can also complete the invasion under certain conditions on the parameters,and the three species eventually reach the coexistence state.For all the cases,the expression of the minimum wave speeds are also given.