Invasion entire solutions in a competition system with nonlocal dispersal (Q484390)

Consider the following Lotka-Volterra competition system with nonlocal dispersal \[ \begin{cases} {\partial u(x,t)\over \partial t}=J_1*u(x,t)-u(x,t)+u(x,t)(1-u(x,t)-a_1v(x,t)), \\ {\partial v(x,t)\over \partial t}=d(J_1*u(x,t)-u(x,t))+rv(x,t)(1-v(x,t)-a_2u(x,t)), \end{cases} \] where \((x,t)\in \mathbb{R}\times \mathbb{R}\) and \(d,r,a_1,a_2\) are positive constants. The kernels \(J_i\) for \(i=1,2\) are probability functions representing the dispersal of the two species respectively and satisfy the conditions (J1): \(J_i\in C^1(\mathbb{R}), J_i\geq 0, J_i(x)=J_i(-x)\), and \(\displaystyle \int_{\mathbb{R}}J_i(x)dx=1\), (J2): \(J_i\) is compactly supported, and \(M_i:=\sup\{|y|:y\in \text{supp}(J_i)\}>0\). Assume \(a_1<1<a_2\), so that \(u\) represents the stronger competitor. Then, an invading phenomenon between the resident and the invader is observed in terms of the traveling solutions \(\phi(\xi)=u(x,t), \psi(\xi)=v(x,t)\) with \(\xi=x+ct\), that is, \[ \lim_{\xi\rightarrow -\infty}(\phi(\xi),\psi(\xi))=(0,1),\quad \lim_{\xi\rightarrow +\infty}(\phi(\xi),\psi(\xi))=(1,0). \] The main purpose of this paper is to show that the existence of invading traveling wave solutions implies the existence of invading entire solutions \(u(x,t), v(x,t)\). Under the technical condition that there exists a positive number \(\eta_0\) such that \[ {\phi(\xi)\over 1-\psi(\xi)}\geq \eta_0\text{ for }\xi\leq 0, \] the authors show, among others, that for any fixed \(x\in \mathbb{R}\), \[ \lim_{t\rightarrow -\infty}(u(x,t),v(x,t))=(0,1),\lim_{t\rightarrow +\infty}(u(x,t),v(x,t))=(1,0), 0\leq u,v\leq 1. \]