Propagation phenomena for a bistable Lotka–Volterra competition system with advection in a periodic habitat

This paper is concerned with a Lotka–Volterra competition system with advection in a periodic habitat ∂ u 1 ∂ t = d 1 ( x ) ∂ 2 u 1 ∂ x 2 - a 1 ( x ) ∂ u 1 ∂ x + u 1 b 1 ( x ) - a 11 ( x ) u 1 - a 12 ( x ) u 2 , ∂ u 2 ∂ t = d 2 ( x ) ∂ 2 u 2 ∂ x 2 - a 2 ( x ) ∂ u 2 ∂ x + u 2 b 2 ( x ) - a 21 ( x ) u 1 - a 22 ( x ) u 2 , t > 0 , x ∈ R , where d i ( · ) , a i ( · ) , b i ( · ) , a ij ( · ) ( i , j = 1 , 2 ) are L -periodic functions in C ν ( R ) for some L > 0 . Under certain assumptions, the system is bistable between two (linearly) stable periodic steady state solutions ( 0 , u 2 ∗ ( x ) ) and ( u 1 ∗ ( x ) , 0 ) . We establish the existence of pulsating traveling front ( U 1 ( x , x + c t ) , U 2 ( x , x + c t ) ) connecting ( 0 , u 2 ∗ ( x ) ) to ( u 1 ∗ ( x ) , 0 ) . Furthermore, we confirm that the pulsating traveling front is globally asymptotically stable for wave-like initial values. We finally show that such pulsating traveling front is unique (up to translation). The methods involve the sub- and supersolutions, spreading speeds of monostable systems, and the monotone semiflows approach. From the biological point of view, this kind of pulsating traveling front provides a spreading way for two strongly competing species interacting in a heterogeneous habitat.