A Traveling Wave Model for Invasion by Precursor and Differentiated Cells
 Abbey J. Trewenack,
 Kerry A. Landman
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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 carryingcapacity 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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 Title
 A Traveling Wave Model for Invasion by Precursor and Differentiated Cells
 Journal

Bulletin of Mathematical Biology
Volume 71, Issue 2 , pp 291317
 Cover Date
 20090201
 DOI
 10.1007/s115380089362x
 Print ISSN
 00928240
 Online ISSN
 15229602
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Traveling wave
 Cell migration
 Proliferation
 Differentiation
 Industry Sectors
 Authors

 Abbey J. Trewenack ^{(1)}
 Kerry A. Landman ^{(1)}
 Author Affiliations

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