Traveling wave solutions for time periodic reaction-diffusion systems

Abstract

This paper deals with traveling wave solutions for time periodic reaction-diffusion systems. The existence of traveling wave solutions is established by combining the fixed point theorem with super-and sub-solutions, which reduces the existence of traveling wave solutions to the existence of superand sub-solutions. The asymptotic behavior is determined by the stability of periodic solutions of the corresponding initial value problems. To illustrate the abstract results, we investigate a time periodic Lotka-Volterra system with two species by presenting the existence and nonexistence of traveling wave solutions, which connect the trivial steady state to the unique positive periodic solution of the corresponding kinetic system.2010 Mathematics Subject Classification. Primary: 45C05, 45M05; Secondary: 92D40.

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.…”

“…By (A), this is not a cooperative system such that the theory of monotone semiflows cannot be applied to it, and the current manuscript will be investigated by the ideas in previous works . Firstly, we estimate the asymptotic speed of spreading of if the initial value of the predator admits nonempty compact support.…”

“…3,8 In the literature, they have been widely studied. In particular, there are some important conclusions for monotone semiflows 9 (we refer to see the works of Fang et al, 10 Liang et al, 11 Liang and Zhao, 12 Weinberger, 13 20 and Wang and Lin 21 studied the coinvasion-coexistence traveling wave solutions of a competitive system and a predator-prey system, respectively.…”

“…By (A), this is not a cooperative system such that the theory of monotone semiflows cannot be applied to it, and the current manuscript will be investigated by the ideas in previous works. 19,20,22 Firstly, we estimate the asymptotic speed of spreading of (1) if the initial value of the predator admits nonempty compact support. By constructing auxiliary equation and applying the classical theory of asymptotic spreading, we confirm the invasion speed of the prey by showing two items of the asymptotic speed of spreading.…”

“…19 Furthermore, by the stability of positive periodic solution of the corresponding kinetic systems, we confirm the asymptotic behavior of traveling wave solutions. 20 Moreover, the nonexistence of traveling wave solutions is established by constructing auxiliary equations and applying the theory of asymptotic spreading.…”