Traveling fronts in diffusive and cooperative Lotka–Volterra system with nonlocal delays

Abstract

This paper is concerned with a diffusive and cooperative Lotka-Volterra model with distributed delays and nonlocal spatial effect. By using an iterative technique recently developed by Wang, Li and Ruan (Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006), 185-232), sufficient conditions are established for the existence of traveling wave front solutions connecting the zero and the positive equilibria by choosing different kernels. The result is an extension of an existing result for Fisher-KPP equation with nonlocal delay and is somewhat parallel to the existing result for diffusive and cooperative Lotka-Volterra system with discrete delays.

Stability of planar waves in reaction-diffusion system

“…The existence of traveling wave fronts of system (34) and (35) was obtained by [6] and [10]. The asymptotic behavior of traveling wave fronts of system (34) was proved by Lü-Wang [13].…”

“…Lü-Wang [12] obtained the existence of traveling wave fronts of system (10) with g 1 (x, t) = g 2 (x, t) = δ(x)δ(t), where δ(x) and δ(t) are the Dirac δ-function. In this paper, we are interested in the stability of planar wave, which connects (0, 1) and (1, 0) for system (10). For this we need to introduce some new variables.…”

“…The nonlocal delays in reaction-diffusion equations were first introduced by Britton [1,2], where the delay term involves a weighted spatio-temporal average over the whole of the infinite spatial domain and the whole of the previous times. Traveling wave solutions of reaction-diffusion equations with nonlocal delays have been studied by many authors, see [5,10,23,24] for details. Lü-Wang [12] obtained the existence of traveling wave fronts of system (10) with g 1 (x, t) = g 2 (x, t) = δ(x)δ(t), where δ(x) and δ(t) are the Dirac δ-function.…”

Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity

“…Since time delay and nonlocality play very important roles in biological and epidemiological models (see Britton [5] and Ruan [34]), they have a crucial effect on the dynamics of the equation (1.1); see Gourley et al [19], Li et al [23,24], Wang and Li [40] and Wu [45]. There has been significant progress in the study of traveling wave solutions for both bistable and monostable equations; see, for example, Ai [1], Ashwin et al [2], Billingham [4], Faria et al [14,15], Gourley and Kuang [17,18], Liang and Wu [25], Ou and Wu [29], Ruan and Xiao [35], Wang et al [41,43], Wu and Zou [46], Zou [48], and the references cites therein.…”

Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications

“…Since then, the theory of reaction-diffusion equations attracts much attention due to its significant nature in mathematical theory and practical fields, see, e.g., Ashwin et al [1], Britton [5], Li and Wang [24], Murray [28], Smoller [33], Wang et al [37,38], Wu [39], Ye and Li [42]. In fact, (1.1) is an approximate description of the original problem in the work of Fisher [17].…”