Traveling fronts in monostable equations with nonlocal delayed effects

Zhi Cheng Wang, Wan Tong Li, Shigui Ruan

Research output: Contribution to journalArticle

134 Scopus citations

Abstract

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction-diffusion equation with delay ∂ u(x, t)/∂ t= dΔu(x,t)+f(u(x, t),∫ -∞∞ h(x - y) u(y, t - τ) dy). Under the monostable assumption, we show that there exists a minimal wave speed c> 0, such that the equation has no traveling wave front for 0 < c < cand a traveling wave front for each c ≥ c. Furthermore, we show that for c > c, such a traveling wave front is unique up to translation and is globally asymptotically stable. When applied to some population models, these results cover, complement and/or improve a number of existing ones. In particular, our results show that (i) if ∂2 f (0, 0) > 0, then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if ∂2 f (0, 0) = 0, then the delay and nonlocality do not affect the spreading speed.

Original languageEnglish (US)
Pages (from-to)573-607
Number of pages35
JournalJournal of Dynamics and Differential Equations
Volume20
Issue number3
DOIs
StatePublished - Sep 1 2008

Keywords

  • Asymptotic stability
  • Delay
  • Existence
  • Monostable equation
  • Nonlocal reaction-diffusion equation
  • Traveling wave front
  • Uniqueness

ASJC Scopus subject areas

  • Analysis

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