Survival probability for a diffusive process on a growing domain

Simpson, Matthew, Sharp, Jesse, & Baker, Ruth (2015) Survival probability for a diffusive process on a growing domain. Physical Review E, 91, 042701.

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Abstract

We consider the motion of a diffusive population on a growing domain, 0 < x < L(t ), which is motivated by various applications in developmental biology. Individuals in the diffusing population, which could represent molecules or cells in a developmental scenario, undergo two different kinds of motion: (i) undirected movement, characterized by a diffusion coefficient, D, and (ii) directed movement, associated with the underlying domain growth. For a general class of problems with a reflecting boundary at x = 0, and an absorbing boundary at x = L(t ), we provide an exact solution to the partial differential equation describing the evolution of the population density function, C(x,t ). Using this solution, we derive an exact expression for the survival probability, S(t ), and an accurate approximation for the long-time limit, S = limt→∞ S(t ). Unlike traditional analyses on a nongrowing domain, where S ≡ 0, we show that domain growth leads to a very different situation where S can be positive. The theoretical tools developed and validated in this study allow us to distinguish between situations where the diffusive population reaches the moving boundary at x = L(t ) from other situations where the diffusive population never reaches the moving boundary at x = L(t ). Making this distinction is relevant to certain applications in developmental biology, such as the development of the enteric nervous system (ENS). All theoretical predictions are verified by implementing a discrete stochastic model.

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3 citations in Scopus
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4 citations in Web of Science®

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ID Code: 91500
Item Type: Journal Article
Refereed: Yes
Keywords: reaction diffusion, growing tissue, survival probability, partial differential equation
DOI: 10.1103/PhysRevE.91.042701
ISSN: 1095-3787
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Biological Mathematics (010202)
Divisions: Current > Institutes > Institute of Health and Biomedical Innovation
Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Funding:
Copyright Owner: Copyright 2015 American Physical Society
Deposited On: 04 Jan 2016 03:48
Last Modified: 25 Oct 2016 23:41

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