Exact solutions of linear reaction-diffusion on a uniformly growing domain: criteria for successful colonization

Simpson, Matthew (2015) Exact solutions of linear reaction-diffusion on a uniformly growing domain: criteria for successful colonization. PLOS ONE, 10(2), e0117949..

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Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction-diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction-diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction—diffusion process on 0<x<L(t), where L(t) is the length of the growing domain. Comparing our exact solutions with numerical approximations
confirms the veracity of the method. Furthermore, our examples illustrate a delicate interplay between: (i) the rate at which the domain elongates, (ii) the diffusivity associated with the spreading density profile, (iii) the reaction rate, and (iv) the initial condition. Altering the balance between these four features leads to different outcomes in terms of whether an initial profile, located near x = 0, eventually overcomes the domain growth and
colonizes the entire length of the domain by reaching the boundary where x = L(t).

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3 citations in Scopus
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4 citations in Web of Science®

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ID Code: 91502
Item Type: Journal Article
Refereed: Yes
Keywords: Reaction diffusion, growing tissues, cell diffusion, cell proliferation, development
DOI: 10.1371/journal.pone.0117949
ISSN: 1932-6203
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200)
Divisions: Current > Institutes > Institute of Health and Biomedical Innovation
Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Copyright Owner: Copyright © 2015 the Author
Copyright Statement: This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Deposited On: 24 Dec 2015 01:40
Last Modified: 03 Jan 2016 20:55

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