Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays
This article is concerned with a diffusive and cooperative Lotka-Volterra model with distributed delays and nonlocal spatial effect: \[ \begin{aligned} \frac{\partial u_1(t,x)}{\partial t}=d_1\Delta u_1(t,x)+r_1u_1(t,x)[1-a_1(g_1*u_1)(t,x)+b_1(g_2*u_2)(t,x)],\\ \frac{\partial u_2(t,x)}{\partial t}=d_2\Delta u_2(t,x)+r_2u_2(t,x)[1+b_2(g_3*u_1)(t,x)-a_2(g_4*u_2)(t,x)],\end{aligned} \] where \((g_j*u)(t,x)=\int_{-\infty}^t\int_{-\infty}^{\infty}g_j(t-s,x-y)u(s,y)dy\;ds, j=1,2,3,4\), \(u=(u_1,u_2)^{T}\). By using an iterative technique developed recently by \textit{Z.-C. Wang, W.-T. Li} and \textit{S. Ruan} [J. Differ. Equations 222, 185--232 (2006; Zbl 1100.35050)], sufficient conditions are established for the existence of traveling wave fronts connecting the two equilibria by choosing different kernels.
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