Monostable Traveling Wave Solutions Of Nonlocal Dispersal Equations

In recent years, a lot, of nonlocal dispersal equations have been derived from the research in many disciplines, such as material science, biology, epidemiology and neural network. Although the nonlocal dispersal represented by the integral operator is closer to the reality, it leads to the many new mathematical difficulties and the essential change of dynamics. For example, the solution semi-flows are not usually compact. And the solutions do not have a priori regularity. In the study of nonlocal dispersal equations, one important topic is their traveling wave solutions, which can well model the oscillatory phenomenon and the propagation with finite speed of nature.Firstly, we study the monotonicity, uniqueness and stability of traveling wave solutions, and spreading speed for a nonlocal dispersal equation with degenerate monostable nonlinearity. By considering the corresponding linear equation, we dis-cuss the exactly exponentially asymptotic behavior of traveling wave solutions at infinity. We then apply the sliding method to obtain the monotonicity and unique-ness of traveling wave solutions. By the squeezing technique, the asymptotic stability of traveling wave solutions with minimal speed is established. Furthermore, we con-sider the spreading speed of the solution of the initial problem with the compact initial value. The result implies that the spreading speed is coincident with the minimal wave speed.Secondly, we investigate the asymptotic stability of traveling wave solutions for a delayed nonlocal dispersal equation with age structure. By appealing to the weighted energy method together with the comparison principle, we prove the stabil-ity of traveling wave solutions with large speed. The result shows that the solution of the initial problem converges to the corresponding traveling wave solution with an exponential decay in time. In particular, the time delay does not affect the stability of traveling wave solutions.Finally, we consider the traveling wave solutions for a predator-prey system with nonlocal dispersal and stage structure. By introducing partially quasi-monotone conditions and partially exponentially quasi-monotone conditions for the nonlinear-ity, we establish the existence for a general nonlocal dispersal system. Our methods are to use the cross-iteration scheme together with upper and lower solutions and the Schauder's fixed point theorem. Then we apply the results to the predator-prey system and obtain the existence of traveling wave solutions connecting 0 with coexistence equilibrium.