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

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey, GU2 7XH, England

    S. A. Gourley

  2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

    J. W.-H. So

  3. Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

    J. H. Wu

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  3. J. H. Wu
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Gourley, S.A., So, J.WH. & Wu, J.H. Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics. Journal of Mathematical Sciences 124, 5119–5153 (2004). https://doi.org/10.1023/B:JOTH.0000047249.39572.6d

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