Traveling Waves And Asymptotic Spreading Speed For A Time Periodic Lotka-volterra Predator-prey System

Predator-prey system is a type of classical model, which formulates the interactions between different species. Studying the traveling waves and asymptotic spreading speed of this system can illustrate and predict a series of ecological phenomena in nature. There-fore, it has been widely investigated in recent decades. But much attention has been paid to traveling waves and asymptotic spreading of autonomous systems. The aim of this paper is to study the time periodic traveling wave solutions and asymptotic spreading speeds for a periodic Lotka-Volterra predator-prey system with diffusion.Firstly, we study the existence and asymptotic behavior of periodic traveling wave solutions for this system. Precisely, we construct an admissible pair of upper and lower solutions to obtain an invariant convex set, then define nonlinear operators on the set,and apply Schauder's fixed point theorem to get the existence of periodic traveling wave solutions. With the help of asymptotic spreading speed theory and some convergence results, we obtain the precise behavior of traveling wave solutions. Additionally, we pay attention to the nonexistence of traveling wave solutions, which is established by the comparison principle and spreading speed theory.We then discuss the asymptotic spreading speed of this periodic system under two different conditions, which bases on comparison principle, auxiliary equations and theory of spreading speed. In first case, when the predator has stronger spreading ability than the prey, the precise spreading speed of the predator and upper and lower bounds of that for the prey are obtained, which implies that the spreading speed of the predator does not change by introducing the prey, and that of the prey is slowed. So the predator has a negative effect on the evolution of the prey. In second case, we give the spreading speed of the prey and a lower bound of that for the predator when the prey has stronger spreading ability. These results show that the interaction does not change the spreading speed of the prey, but accelerates the spreading speed of the predator, and the prey contributes to the evolution of the predator.