Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system

Abstract

This paper is concerned with a time periodic competition-diffusion systemwhere 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 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 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 entire solutions defined for all time and whole space.

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

“…One can also see [2,23] for spreading speeds and traveling waves for systems with nonlocal dispersal. erogeneous environment, one can see [1,31,[40][41][42] for time periodic traveling waves, [7,8] for other types of entire solutions, and [26] for spreading speeds and traveling waves of timeperiodic semiflows in one dimensional environment. For time-space periodic environment, one can see a more recent paper [11] for the existence of traveling waves and spreading speeds of time-space periodic monotone semiflows.…”

Traveling waves and spreading properties for a reaction-diffusion competition model with seasonal succession*

“…It is worthy mentioning some works on time-periodic competition models. For example, Zhao and Ruan [30] established the existence and non-existence of traveling wave solutions in the monostable case; and the authors of [1,2,6,27] obtained the existence of bistable traveling waves. The propagation dynamics and monostable waves were also studied in [7] for a timespace periodic competition model.…”

A Lotka-Volterra competition model with nonlocal diffusion and free boundaries

“…See e.g. Bao et al [1,2], Pan et al [24], Zhao and Ruan [42] for the existence of traveling wave fronts, and Du et al [8][9][10], Li et al [22] and Wang and Lv [33] for the existence of invasion entire solutions. Also, there are many works concerned with the spectral theory of nonlocal dispersal operators and entire solutions of nonlocal dispersal equations, see Coville [7], Hetzer and Shen [16], Li et al [19][20][21], Shen and Zhang [29,30], Sun et al [25,26], Sun et al [28], Yang et al [38], Zhang et al [39] and Zhang et al [40,41].…”