Journal of Mathematical Sciences

, Volume 124, Issue 4, pp 5119–5153

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

  • S. A. Gourley
  • J. W.-H. So
  • J. H. Wu
Article

DOI: 10.1023/B:JOTH.0000047249.39572.6d

Cite this article as:
Gourley, S.A., So, J.W. & Wu, J.H. Journal of Mathematical Sciences (2004) 124: 5119. doi:10.1023/B:JOTH.0000047249.39572.6d

Abstract

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.

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • S. A. Gourley
    • 1
  • J. W.-H. So
    • 2
  • J. H. Wu
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of Surrey, GuildfordSurreyEngland
  2. 2.Department of Mathematical and Statistical SciencesUniversity of Alberta, EdmontonAlbertaCanada
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada