Asymptotic Behavior of Traveling Fronts and Entire Solutions for a Periodic Bistable Competition-Diffusion System

Abstract

This paper is concerned with a time periodic competition-diffusion system \begin{equation*} \begin{cases} {u_t}={u_{xx}}+u(r_1(t)-a_1(t)u-b_1(t)v),\quad t>0,~x\in \mathbb R, {v_t}=d{v_{xx}}+v(r_2(t)-a_2(t)u-b_2(t)v),\quad t>0,~x\in \mathbb R, \end{cases} \end{equation*} where $u(t,x)$ and $v(t,x)$ denote the densities of two competing species, $d>0$ is some constant, $r_i(t),a_i(t)$ and $b_i(t)$ are $T-$periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang [J. Differential Equations, 255 (2013), 2402-2435] that this system admits a periodic traveling front connecting two \textbf{stable} semi-trivial $T-$periodic solutions $(p(t),0)$ and $(0,q(t))$ associated to the corresponding kinetic system. Assume further that the wave speed is non-zero, we investigate the asymptotic behavior of the periodic \textbf{bistable} traveling front at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then give some key estimates. Finally, by applying super- and subsolutions technique as well as the comparison principle, we establish the existence and various qualitative properties of \emph{entire solutions} defined for all time and whole space.