Propagation Phenomena for A Reaction–Advection–Diffusion Competition Model in A Periodic Habitat

Abstract

This paper is devoted to the study of propagation phenomena for a Lotka-Volterra reaction-advection-diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trival steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reaction-diffusion model.

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

“…In terms of (B1), there holds λ 0 (d 2 , a * 2 , −a * 22 ) < λ 0 (d 1 , a * 1 , a * 11 − a * 12 ). The same argument as in the proof of [38,Proposition 4.2] implies that there exists a unique positive periodic function…”

“…We would like to mention here that, under the assumptions that λ 0 (d 1 , a 1 , b 1 − a 12 u * 2 ) > 0 (that is, (0, u * 2 (x)) is a linearly unstable steady state) and that the system has no steady state in Int(P + ), it follows that (u * 1 (x), 0) is globally asymptotic stable for all initial values (φ 1 , φ 2 ) ∈ P + with φ 1 ≡ 0 (see e.g., [38, Theorem 2.1]), namely, system (1.1) admits a monostable structure. Yu and Zhao [38] then considered the propagation phenomena of system (1.1) with this monostable structure and established the existence of spatially periodic traveling waves connecting the two semitrivial periodic solutions. In the present work, we focus system (1.1) on the standard bistable structure.…”

Propagation dynamics for a spatially periodic integrodifference competition model

“…is an L-periodic function in x for any fixed ξ ∈ R. Moreover, we say that V (ξ, x) connects β to 0 if lim ξ→−∞ |V (ξ, x) − β(x)| = 0 and lim ξ→+∞ |V (ξ, x)| = 0 uniformly for x ∈ R. According to [36], we need the following assumptions:…”

“…A general theory of travelling waves and spreading speeds in a periodic habitat was developed by Weinberger [33], Liang and Zhao [22], and Fang and Zhao [5]. Recently, Yu and Zhao [36] studied the propagation phenomena of a two species reaction-advection-diffusion competition model in a periodic habitat by appealing to the abstract results in [5,22].…”

Competition in periodic media:Ⅰ-Existence of pulsating fronts

“…Even though both articles were mostly concerned by scalar equations, they were careful enough to include monotone systems, such as two-species competitive ones, in their framework. Notice that Yu and Zhao [30] used a similar framework to prove, in the weak competition case, the existence of monostable pulsating fronts connecting two extinction states despite the presence of an intermediate coextinction state (Weinberger's framework requires no intermediate stationary state) (see also for a similar work in space-time periodic media).…”