Journal of Mathematical Sciences

, Volume 124, Issue 4, pp 5119-5153

First online:

Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics

  • S. A. GourleyAffiliated withDepartment of Mathematics and Statistics, University of Surrey, Guildford
  • , J. W.-H. SoAffiliated withDepartment of Mathematical and Statistical Sciences, University of Alberta, Edmonton
  • , J. H. WuAffiliated withDepartment of Mathematics and Statistics, York University

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


We present a short survey on the biological modeling, dynamics analysis, and numerical simulation of nonlocal spatial effects, induced by time delays, in diffusion models for a single species confined to either a finite or an infinite domain. The nonlocality, a weighted average in space, arises when account is taken of the fact that individuals have been at different points in space at previous times. We discuss and compare two existing approaches to correctly derive the spatial averaging kernels, and we summarize some of the recent developments in both qualitative and numerical analysis of the nonlinear dynamics, including the existence, uniqueness (up to a translation), and stability of traveling wave fronts and periodic spatio-temporal patterns of the model equations in unbounded domains and the linear stability, boundedness, global convergence of solutions and bifurcations of the model equations in finite domains.