Entire solutions of a diffusive and competitive Lotka–Volterra type system with nonlocal delays

Abstract

This paper is concerned with the entire solution of a diffusive and competitive Lotka-Volterra type system with nonlocal delays. The existence of the entire solution is proved by transforming the system with nonlocal delays to a fourdimensional system without delay and using the comparing argument and the sub-super-solution method. Here an entire solution means a classical solution defined for all space and time variables, which behaves as two wave fronts coming from both sides of the x-axis.

Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case

“…These solutions can not only describe the interaction of traveling waves but also characterize new dynamics of diffusion equations. For the study of such entire solutions, we refer to [4,[14][15][16]19,23,27] for reaction-diffusion equations without delay, [22,35,37,42] for reaction-diffusion equations with nonlocal delay, [38,39] for delayed lattice differential equations with nonlocal interaction, [21,34] for nonlocal dispersal equations without delay ((1.9) below), [28,40,44] for reaction-diffusion systems, and [43] for periodic lattice dynamical systems. However, to the best of our knowledge, the issues on entire solutions for nonlocal dispersal equations with spatio-temporal delay have not been addressed, especially for infinite delay equations.…”

Stability of planar waves in reaction-diffusion system

“…Now, we consider a function η(x, t) which satisfies (22). Lemma 2.4 implies that there exists a constant T 2 0 such that η(x, t) ε for t T 2 .…”

“…[22] Let (φ(z), ψ(z), ρ(z), ϕ(z)) be a monotone solution of (14) with c > c min (c min is the critical speed ), then these hold that…”

“…It follows from Lemma 2.1 and the proof of Theorem 2.1 in [22] thatΛ(c) Λ 3 (c) and Λ 3 (c) Λ(c) Λ (c). By using Lemma 3.1, similar to the proof of Lemma 2.2, we obtain the following result.…”

Entire Solutions with Annihilating Fronts to a Nonlocal Dispersal Equation with Bistable Nonlinearity and Spatio-Temporal Delay

“…Recently, many types of front-like entire solutions have been observed for various evolution equations by mixing the traveling wave solutions and some spatially independent solutions, see [11][12][13][15][16][17]20,21,25,[27][28][29][30][31][32][33]. For examples, Hamel and Nadirashvili [12] established three-, four-and five-dimensional manifolds of entire solutions for the Fisher-KPP equation.…”