Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T00:49:16.585Z Has data issue: false hasContentIssue false
  • English
  • Français

CRITICAL TIMESCALES AND TIME INTERVALS FOR COUPLED LINEAR PROCESSES

Part of: Partial differential equations

Published online by Cambridge University Press:  22 April 2013

MATTHEW J. SIMPSON ,
ADAM J. ELLERY ,
SCOTT W. MCCUE  and
RUTH E. BAKER
MATTHEW J. SIMPSON*
Affiliation:
Mathematical Sciences, Queensland University of Technology, Brisbane 4001, Australia email a.ellery@qut.edu.auscott.mccue@qut.edu.au
ADAM J. ELLERY
Affiliation:
Mathematical Sciences, Queensland University of Technology, Brisbane 4001, Australia email a.ellery@qut.edu.auscott.mccue@qut.edu.au
SCOTT W. MCCUE
Affiliation:
Mathematical Sciences, Queensland University of Technology, Brisbane 4001, Australia email a.ellery@qut.edu.auscott.mccue@qut.edu.au
RUTH E. BAKER
Affiliation:
Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford OX1 3LB, UK email ruth.baker@maths.ox.ac.uk
*
✉matthew.simpson@qut.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. Sample and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for example the analysis of models describing coupled chemical decay and cell differentiation processes.

Keywords

reaction–diffusion equationcoupled reaction–diffusion equationssteady statecritical time

MSC classification

Secondary: 35A09: Classical solutions
Type
Research Article
Information
The ANZIAM Journal , Volume 54 , Issue 3 , January 2013 , pp. 127 - 142
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Barenblatt, G. I., Scaling (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Berezhkovskii, A. M., Sample, C. and Shvartsman, S. Y., “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61; doi:10.1016/j.bpj.2010.07.045.CrossRefGoogle ScholarPubMed
Berezhkovskii, A. M., Sample, C. and Shvartsman, S. Y., “Formation of morphogen gradients: Local accumulation time”, Phys. Rev. E 83 (2011) 051906; doi:10.1103/PhysRevE.83.051906.CrossRefGoogle ScholarPubMed
Berezhkovskii, A. M. and Shvartsman, S. Y., “Physical interpretation of mean local accumulation time of morphogen gradient formation”, J. Chem. Phys. 135 (2011) 154115; doi:10.1063/1.3654159.CrossRefGoogle ScholarPubMed
Cho, C. M., “Convective transport of ammonium with nitrification in soil”, Canad. J. Soil Sci. 51 (1970) 339–350; doi:10.4141/cjss71-047.CrossRefGoogle Scholar
Clement, T. P., Sun, Y., Hooker, B. S. and Petersen, J. N., “Modelling multispecies reactive transport in ground water”, Ground Water Modeling and Remediation 18 (1998) 79–92; doi:10.1111/j.1745-6592.1998.tb00618.x.CrossRefGoogle Scholar
Denman, P. K., McElwain, D. L. S., Harkin, D. G. and Upton, Z., “Mathematical modelling of aerosolised skin grafts incorporating keratinocyte clonal subtypes”, Bull. Math. Biol. 69 (2007) 157–179; doi:10.1007/S11538-006-9082-z.CrossRefGoogle ScholarPubMed
Ellery, A. J., Simpson, M. J., McCue, S. W. and Baker, R. E., “Critical timescales for advection–diffusion–reaction processes”, Phys. Rev. E 85 (2012) 041135; doi:10.1103/PhysRevE.85.041135.CrossRefGoogle Scholar
Ellery, A. J., Simpson, M. J., McCue, S. W. and Baker, R. E., “Moments of action provide insight into critical times for advection–diffusion–reaction processes”, Phys. Rev. E 86 (2012) 031136; doi:10.1103/PhysRevE.86.031136.CrossRefGoogle ScholarPubMed
Fernando, A. E., Landman, K. A. and Simpson, M. J., “Nonlinear diffusion and exclusion processes with contact interactions”, Phys. Rev. E 81 (2010) 011903; doi:10.1103/PhysRevE.81.011903.CrossRefGoogle ScholarPubMed
Gordon, P. V., Sample, C., Berezhkovskii, A. M., Muratov, C. V. and Shvartsman, S. Y., “Local kinetics of morphogen gradients”, Proc. Natl Acad. Sci. 108 (2011) 6157–6162; doi:10.1073/pnas.1019245108.CrossRefGoogle ScholarPubMed
Hickson, R. I., Barry, S. I. and Mercer, G. N., “Critical times in multilayer diffusion. Part 1: Exact solutions”, Int. J. Heat Mass Transfer 52 (2011) 5776–5783; doi:10.1016/j.ijheatmasstransfer.2009.08.013.CrossRefGoogle Scholar
Hickson, R. I., Barry, S. I. and Mercer, G. N., “Critical times in multilayer diffusion. Part 2: Approximate solutions”, Int. J. Heat Mass Transfer 52 (2011) 5784–5791; doi:10.1016/j.ijheatmasstransfer.2009.08.012.CrossRefGoogle Scholar
Hickson, R. I., Barry, S. I., Sidhu, H. S. and Mercer, G. N., “Critical times in single-layer reaction diffusion”, Int. J. Heat Mass Transfer 54 (2011) 2642–2650; doi:10.1016/j.ijheatmasstransfer.2011.01.019.CrossRefGoogle Scholar
Kolomeisky, A. B., “Formation of a morphogen gradient: acceleration by degradation”, J. Phys. Chem. Lett. 2 (2011) 1502–1505; doi:10.1021/jz2004914.CrossRefGoogle Scholar
Landman, K. A., Cai, A. Q. and Hughes, B. D., “Travelling waves of attached and detached cells in a wound-healing cell migration assay”, Bull. Math. Biol. 69 (2007) 2119–2138; doi:10.1007/S11538-007-9206-0.CrossRefGoogle Scholar
Landman, K. A. and McGuinness, M. J., “Mean action time for diffusive processes”, J. Appl. Math. Decision Sci. 4 (2000) 125–141; doi:10.1155/S1173912600000092.CrossRefGoogle Scholar
Landman, K. A. and White, L. R., “Predicting filtration time and maximizing throughput in a pressure filter”, AIChE J. 43 (1997) 3147–3160; doi:10.1002/aic.690431204.CrossRefGoogle Scholar
Lunn, M., Lunn, R. J. and Mackay, R., “Determining analytic solutions of multiple species contaminant transport, with sorption and decay”, J. Hydrol. 180 (1996) 195–210; doi:10.1016/0022-1694(95)02891-9.CrossRefGoogle Scholar
Montgomery, J. H., Groundwater chemicals desk reference, 4th edn. (CRC Taylor and Francis, Boca Raton, FL, 2007).CrossRefGoogle Scholar
McNabb, A., “Mean action times, time lags, and mean first passage times for some diffusion problems”, Math. Comput. Modell. 18 (1993) 123–129; doi:10.1016/0895-7177(93)90221-J.CrossRefGoogle Scholar
McNabb, A. and Wake, G. C., “Heat conduction and finite measures for transition times between steady states”, IMA J. Appl. Math. 47 (1991) 193–206; doi:10.1093/imamat/47.2.193.CrossRefGoogle Scholar
Simpson, M. J. and Landman, K. A., “Analysis of split operator methods applied to reactive transport with Monod kinetics”, Adv. Water Resour. 30 (2007) 2026–2033; doi:10.1016/j.advwatres.2007.04.005.CrossRefGoogle Scholar
Simpson, M. J., Landman, K. A. and Clement, T. P., “Assessment of a nontraditional operator split algorithm for simulation of reactive transport”, Math. Comput. Simul. 70 (2005) 44–60; doi:10.1016/j.matcom.2005.03.019.CrossRefGoogle Scholar
Simpson, M. J., Landman, K. A. and Hughes, B. D., “Cell invasion with proliferation mechanisms motivated by time-lapse data”, Phys. A 389 (2010) 3779–3790; doi:10.1016/j.physa.2010.05.020.CrossRefGoogle Scholar
Simpson, M. J., Towne, C., McElwain, D. L. S. and Upton, Z., “Migration of breast cancer cells: Understanding the roles of volume exclusion and cell-to-cell adhesion”, Phys. Rev. E 82 (2010) 041901; doi:10.1103/PhysRevE.82.041901.CrossRefGoogle ScholarPubMed
van Genuchten, M. Th., “Convective-dispersive transport of solutes involved in sequential first-order decay reactions”, Comput. Geosci. 11 (1985) 129–147; doi:10.1016/0098-3004(85)90003-2.CrossRefGoogle Scholar
Vogel, T. M. and McCarty, P. L., “Biotransformation of tetrachloroethylene to trichloroethylene, dichloroethylene, vinyl chloride, and carbon dioxide under methanogenic conditions”, Appl. Environ. Microbiol. 49 (1985) 1080–1083; http://www.ncbi.nlm.nih.gov/pmc/articles/PMC238509/.CrossRefGoogle ScholarPubMed
Zheng, C. Z. and Bennett, G. D., Applied contaminant transport modeling, 2nd edn. (John Wiley, New York, 2002).Google Scholar