Traveling waves for a nonlocal diffusion system with asymmetric kernels and delays

Abstract

This paper mainly deals with the (non)existence, asymptotic behaviors and uniqueness of traveling waves to a nonlocal diffusion system with asymmetric kernels and delays for quasi-monotone case. The difference from some previous works is the asymmetry reflected in both diffusion and reaction terms, and this not only has an impact on the positivity of minimal wave speed and the wave profiles of traveling waves with the same speed spreading from the left and right of the x-axis, but also leads to some difficulties for the nonexistence and asymptotic behaviors of traveling waves, which are overcome by using new techniques. Thereby, the results for traveling waves of nonlocal diffusion equations with symmetric kernels and with (or without) delays are improved to equations with asymmetric kernels, and those conclusions for scalar equations and systems with Laplace diffusion and local nonlinearities are also generalized to the nonlocal case. Finally, some concrete applications and numerical simulations are shown to confirm our theoretical results.