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A Traveling Wave Model for Invasion by Precursor and Differentiated Cells

Bulletin of Mathematical Biology volume 71, pages 291–317 (2009)Cite this article

Abstract

We develop and investigate a continuum model for invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric nervous system, but is presented in general terms and is of broader applicability. Two cell populations are identified and modeled explicitly; a population of precursor cells that migrate and proliferate, and a population of differentiated cells derived from the precursors which have impaired migration and proliferation. The equation describing the precursor cells is based on Fisher’s equation with the addition of a carrying-capacity limited differentiation term. Two variations of the proliferation term are considered and compared. For most parameter values, the model admits a traveling wave solution for each population, both traveling at the same speed. The traveling wave solutions are investigated using perturbation analysis, phase plane methods, and numerical techniques. Analytical and numerical results suggest the existence of two wavespeed selection regimes. Regions of the parameter space are characterized according to existence, shape, and speed of traveling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, our results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells.

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Affiliations

  1. Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia

    Abbey J. Trewenack & Kerry A. Landman

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  1. Abbey J. Trewenack
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  2. Kerry A. Landman
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Correspondence to Kerry A. Landman.

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Trewenack, A.J., Landman, K.A. A Traveling Wave Model for Invasion by Precursor and Differentiated Cells. Bull. Math. Biol. 71, 291–317 (2009). https://doi.org/10.1007/s11538-008-9362-x

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  • DOI: https://doi.org/10.1007/s11538-008-9362-x

Keywords

  • Traveling wave
  • Cell migration
  • Proliferation
  • Differentiation