Asymptotic Speeds Of Spread And Traveling Wave Solutions Of A Second Order Integrodifference Equation Without Monotonicity

This paper is concerned with the propagation modes of the following second order integrodifference equation The equation arises from delayed effect in population dynamics, and is derived from the study of delayed equation with piecewise constant arguments. In literature, this equation cannot generate monotone semifolws and does not satisfy the local monotonicity.By investigating the solution of integrodifference equation of one order, we construct some auxiliary equations of one order, which was satisfied by the solution of equation of second order. Applying some known results, we establish the asymptotic speeds of spread of un(x) is c* if the initial value admits nonempty compact support.We further study the traveling wave solutions for any positive wave speed. More precisely, we present the existence of traveling wave solutions if the wave speed is c* by Schauder fixed point theorem and upper and lower solutions. If the wave speed is c*, the existence of traveling wave solutions is confirmed by passing to a limit function. With the help of theory of asymptotic spreading, we show the nonexistence of nontrivial traveling wave solutions if the wave speed is smaller than c*.In particular, our results imply that if the initial value admits nonempty compact support, then the asymptotic speed of spread equals to the minimal wave speed of non-trivial traveling wave solutions.