Existence and stability of time periodic traveling waves for a periodic bistable Lotka–Volterra competition system

Abstract

This paper is concerned with the time periodic Lotka–Volterra competition–diffusion system{ut=d1(t)uxx+u(r1(t)−a1(t)u−b1(t)v),vt=d2(t)vxx+v(r2(t)−a2(t)u−b2(t)v),x∈R,t>0, where di(t),ri(t),ai(t),bi(t)∈Cθ2(R) are T-periodic functions, 0<θ<1, i=1,2. Under certain conditions, the system admits two stable semi-trivial periodic solutions (p(t),0) and (0,q(t)) and a unique coexistence periodic solution (u0(t),v0(t)), which is unstable and satisfies 0<u0(t)<p(t) and 0<v0(t)<q(t) for t∈[0,T]. In this paper we prove that the system admits a time periodic traveling wave solution (u(x,t),v(x,t))=(U(x+ct,t),V(x+ct,t)) connecting two periodic solutions (p(t),0) and (0,q(t)) as x→±∞, where c is the wave speed. By using a dynamical method, we show that the time periodic traveling wave solution (U(x+ct,t),V(x+ct,t)) is asymptotically stable and unique modulo translation for front-like initial values.