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

TL;DR

This paper studies the long-term behavior of traveling fronts in a periodic competition-diffusion system, establishing the existence and properties of entire solutions using advanced analytical methods.

Contribution

It introduces a dynamical approach combined with Laplace transforms to analyze asymptotics and proves the existence of entire solutions for the system.

Findings

Asymptotic behavior of bistable traveling fronts characterized.

Existence of entire solutions established for all time and space.

Key estimates derived for the solutions' properties.

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.