Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

Abstract

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka-Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c* such that for each wave speed c ≤ c*, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c < c* are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c > c*.

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

“…which implies that ν 1 is a simple pole of (λI − A) −1 (see [31,Remark A.2.4]). Moreover, a similar argument to [45,Proposition 3.6] shows that there exists some ε ′ > 0 such that…”

“…Recently, there have been quite a few works focusing on the nonautonomous Lotka-Volterra competition-diffusion system. Among others, Zhao and Ruan [45] established the existence, uniqueness and stability of time periodic traveling wave fronts for system (1.1) under monostable assumptions. Later, they extended these results to a class of periodic advection-reaction-diffusion systems in [46].…”

“…One of the main difficulties arises in obtaining the exact exponential decay rate of the periodic traveling wave front as it approaches its limiting states. Inspired by [45], we apply the two-sided Laplace transform method to obtain the exact exponential decay rate of the periodic traveling front as it approaches its stable limiting states, which is essentially based on the a priori exponential estimates of periodic traveling wave tails at infinity. Here we would like to emphasize that, unlike the a priori exponential estimates of periodic traveling wave tails at the unstable limiting state characterized by the principle eigenvalue associated with the linearized system (see [45,Lemma 3.3]), the a priori exponential estimates at the stable limiting state can only be characterized by a perturbation of the corresponding principle eigenvalues ν ± i,ǫ (i = 2, 3) with some small ǫ > 0 and ν ± i (i = 1, 4) (see Lemmas 2.1 and 2.2), which causes additional difficulties and makes it vastly more complicated to use the two-sided Laplace transform method.…”

“…Since (U (·, z), V (·, z)) is periodic in t ∈ R, and (U, V )(t, z) is positive and bounded for any (t, x) ∈ R × R, the Harnack inequality for cooperative parabolic systems (see [2,11,45]) implies that there exists a positive constant N such that…”

Spatial invasion dynamics for a time period predator‐prey system

“…For nonmonotone semiflows, Yi et al investigated the traveling wave solutions and asymptotic spreading by constructing two monotone semiflows. Moreover, when the coupled systems with time periodic parameters are concerned, Bao and Wang and Zhao and Ruan studied the exclusive traveling wave solutions in a competitive system, Wang et al investigated the traveling wave solutions of an SIR model, and Bo et al and Wang and Lin studied the coinvasion‐coexistence traveling wave solutions of a competitive system and a predator‐prey system, respectively.…”