Pulsating Fronts for a Bistable Lotka-Volterra Competition System with Advection in a Periodic Habitat

Abstract

This paper is concerned with the following Lotka-Volterra competition system with advection in a periodic habitat \begin{equation*} \begin{cases} \frac{\partial u_1}{\partial t} =d_1(x)\frac{\partial^2 u_1}{\partial x^2}-a_1(x)\frac{\partial u_1}{\partial x}+u_1\left(b_1(x)-a_{11}(x)u_1-a_{12}(x)u_2\right),\\ \frac{\partial u_2}{\partial t} =d_2(x)\frac{\partial^2 u_2}{\partial x^2}-a_2(x)\frac{\partial u_2}{\partial x}+u_2\left(b_2(x)-a_{21}(x)u_1-a_{22}(x)u_2\right), \end{cases} t>0,~x\in\Bbb R, \end{equation*} where $d_i(\cdot )$, $a_i(\cdot )$, $b_i(\cdot )$, $a_{ij}(\cdot )$ $(i,j=1,2)$ are $L$-periodic functions in $C^{\nu }(\mathbb{R})$ with some $\nu \in (0,1)$. Under certain assumptions, the system admits two periodic locally stable steady states $(u_1^{\ast }(x),0)$ and $(0,u_2^{\ast }(x))$. In this work, we first establish the existence of the pulsating front $U(x,x+ct)=(U_1(x,x+ct),U_2(x,x+ct))$ connecting two periodic solutions $(0,u_2^{\ast }(x))$ and $(u_1^{\ast }(x),0)$ at infinities. By using a dynamical method, we confirm further that the pulsating front is asymptotically stable for front-like initial values. As a consequence of the global asymptotically stability, we finally show that the pulsating front is unique up to translation.