Journal of Dynamics and Differential Equations

, Volume 20, Issue 3, pp 573–607 | Cite as

Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

  • Zhi-Cheng Wang
  • Wan-Tong Li
  • Shigui RuanEmail author


In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion equation with delay
$$\frac{\partial u\left(x, t\right)}{\partial t}= d\Delta u\left(x, t\right)+f\left(u\left(x, t\right),\int\limits_{-\infty }^\infty h\left(x - y\right) u\left(y, t - \tau\right) dy\right)\!.$$
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 < cc* and a traveling wave front for each c ≥ c*. Furthermore, we show that for cc*, 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.


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

Mathematics Subject Classification (2000)

35K57 35R10 35B40 34K30 58D25 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of MiamiCoral GablesUSA

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