Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction

<p style='text-indent:20px;'>This paper is concerned with the traveling wave fronts for a lattice dynamical system with global interaction, which arises in a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. We prove that all non-critical traveling wave fronts are globally exponentially stable in time, and the critical traveling wave fronts are globally algebraically stable by the weighted energy method combined with the comparison principle and the discrete Fourier transform.</p>

Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

This paper is concerned with nonlocal diffusion systems of three species with delays. By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. In order to obtain more information of the asymptotic behavior of such solutions at positive infinity, for the special kernels, we discuss the asymptotic behavior of such solutions of such system without delays, via the stable manifold theorem. In addition, by using the sliding method, the strict monotonicity and uniqueness of traveling wave fronts are also obtained.

Stability of Traveling Wave Fronts for a Three Species Predator-Prey Model with Nonlocal Dispersals

In this paper, we consider a predator-prey model with nonlocal dispersals of two cooperative preys and one predator. We prove that the traveling wave fronts with the relatively large wave speed are exponentially stable as perturbation in some exponentially weighted spaces, when the difference between initial data and traveling wave fronts decay exponentially at negative infinity, but in other locations, the initial data can be very large. The adopted method is to use the weighted energy method and the squeezing technique with some new flavors to handle the nonlocal dispersals.