This paper is mainly concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIRS model with nonlocal delayed transmissions. We find that the existence and nonexistence of traveling wave solutions are determined by the critical wave speed [Formula: see text]. More specifically, we establish the existence of traveling wave solutions for every wave speed [Formula: see text] and [Formula: see text] by means of upper-lower solutions and Schauder’s fixed point theorem. Nonexistence of traveling wave solutions is obtained by Laplace transform for any wave speed [Formula: see text] and [Formula: see text].
In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in n-dimensional space. More precisely, we prove that all planar traveling waves with speed c > c * are exponentially stable in L ∞ (R n) in the form of t − n 2α e −ετ σt for some constants σ > 0 and ετ ∈ (0, 1), where ετ = ε(τ) is a decreasing function refer to the time delay τ > 0. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed c = c * , we show that they are algebraically stable in the form of t − n 2α. The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.
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