Entire solutions of Lotka-Volterra competition systems with nonlocal dispersal

This paper is mainly concerned with entire solutions of the following two-species Lotka-Volterra competition system with nonlocal (convolution) dispersals: (0.1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left\{{\matrix{{{u_t} = k * u - u + u(1 - u - av),} \hfill & {x \in \mathbb{R},\,\,t \in \mathbb{R},} \hfill \cr {{v_t} = d(k * v - v) + rv(1 - v - bu),} \hfill & {x \in \mathbb{R},\,\,t \in \mathbb{R}.} \hfill \cr}} \right.$\end{document} Here a ≠ 1, b ≠ 1, d, and r are positive constants. By studying the eigenvalue problem of (0.1) linearized at (ϕc(ξ), 0), we construct a pair of super- and sub-solutions for (0.1), and then establish the existence of entire solutions originating from (ϕc(ξ), 0) as t → −∞, where ϕc denotes the traveling wave solution of the nonlocal Fisher-KPP equation ut = k * u − u + u (1 − u). Moreover, we give a detailed description on the long-time behavior of such entire solutions as t → ∞. Compared to the known works on the Lotka-Volterra competition system with classical diffusions, this paper overcomes many difficulties due to the appearance of nonlocal dispersal operators.