Traveling wave solutions in a nonlocal reaction-diffusion population model

Abstract

This paper is concerned with a nonlocal reaction-diffusion equation with the form ∂u ∂t = ∂ 2 u ∂x 2 + u 1 + αu − βu 2 − (1 + α − β)(φ * u) , (t, x) ∈ (0, ∞) × R, where α and β are positive constants, 0 < β < 1 + α. We prove that there exists a number c * ≥ 2 such that the equation admits a positive traveling wave solution connecting the zero equilibrium to an unknown positive steady state for each speed c > c *. At the same time, we show that there is no such traveling wave solutions for speed c < 2. For sufficiently large speed c > c * , we further show that the steady state is the unique positive equilibrium. Using the lower and upper solutions method, we also establish the existence of monotone traveling wave fronts connecting the zero equilibrium and the positive equilibrium. Finally, for a specific kernel function φ(x) := 1 2σ e − |x| σ (σ > 0), by numerical simulations we show that the traveling wave solutions may connects the zero equilibrium to a periodic steady state as σ is increased. Furthermore, by the stability analysis we explain why and when a periodic steady state can appear.

Traveling waves for nonlocal Lotka-Volterra competition systems

“…It should be emphasized that the work of Berestycki et al [3] did not require that the nonlocality is weak( that is, the nonlocal delay is small). After that, many important results about traveling wave solution and the spreading speed of equation (7) have been obtained, see [1,2,11,8,7,20,16,17,18,19,39] and the references therein. In addition, traveling wave soltuions of delayed Fisher-KPP equation was also studied deeply, see [4,11,21,30] and the references therein.…”

“…In addition, traveling wave soltuions of delayed Fisher-KPP equation was also studied deeply, see [4,11,21,30] and the references therein. Inspired by [3,17,18], in this paper we study traveling wave solutions of system (2) when the nonlocality without any restriction. In addition, throughout this paper, we will assume that a 1 and a 2 satisfies the following hypotheses.…”