Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-2qt69 Total loading time: 0.28 Render date: 2022-08-08T04:21:02.796Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Travelling-wave analysis of a model describing tissue degradation by bacteria

Published online by Cambridge University Press:  01 October 2007

D. HILHORST
Affiliation:
CNRS and Laboratoire de Mathématiques, Université de Paris-Sud, 91405 Orsay Cedex, France (email: Danielle.Hilhorst@math.u-psud.fr)
J. R. KING
Affiliation:
Centre for Mathematical Medicine, Theoretical Mechanics Section, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK (email: John.King@nottingham.ac.uk)
M. RÖGER
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany (email: roeger@mis.mpg.de)

Abstract

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the ‘degradation-rate parameter’ solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates, the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, non-linear selection of the minimal speed occurs.

Type
Papers
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aronson, D. G. & Weinberger, H. F. (1975) Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In Partial Differential Equations and Related Topics, Program, 1974, Tulane Univ., New Orleans, LA, Lecture Notes in Math., Vol. 446. Springer, Berlin pp. 5–49.Google Scholar
[2]Benguria, R. D. & Depassier, M. C. (1994) Validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation. Phys. Rev. Lett. 73, 22722274.CrossRefGoogle ScholarPubMed
[3]Coddington, E. A. & Levinson, N. (1955) Theory of Ordinary Differential Equations, McGraw-Hill, New York-Toronto-London.Google Scholar
[4]Diekmann, O. (1976–1977) Limiting behaviour in an epidemic model. Nonlinear Anal. 1, 459470.CrossRefGoogle Scholar
[5]Ebert, U. & van Saarloos, W. (2000) Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts. Physica D 146, 199.CrossRefGoogle Scholar
[6]Hadeler, K. P. & Rothe, F. (1975) Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251263.CrossRefGoogle Scholar
[7]Hilhorst, D., King, J. R. & Röger, M. (2007) Mathematical analysis of a model describing the invasion of bacteria in burn wounds. Nonlinear Anal. (TMA) 66, 11181140.CrossRefGoogle Scholar
[8]King, J. R., Koerber, A. J., Croft, J. M., Ward, J. P., Sockett, R. E. & Williams, P. (2003) Modelling host tissue degradation by extracellular bacterial pathogens. Math. Med. Biol. 20, 227260.CrossRefGoogle ScholarPubMed
[9]Kolmogorov, A. N., Petrovsky, I. G. & Piskunov, N. S. (1937) Etude de l'equation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moskow Univ. Math. Bull. 1, 125.Google Scholar
[10]Lucia, M., Muratov, C. B. & Novaga, M. (2004) Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium. Comm. Pure Appl. Math. 57, 616636.CrossRefGoogle Scholar
[11]Muratov, C. B. (2004) A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. Discrete Contin. Dyn. Syst. Ser. B 4, 867892.CrossRefGoogle Scholar
[12]Perko, L. (2001) Differential Equations and Dynamical Systems, Vol. 7 of Texts in Applied Mathematics, Springer-Verlag, New York.Google Scholar
[13]Stokes, A. N. (1976) On two types of moving front in quasilinear diffusion. Math. Biosci. 31, 307315.CrossRefGoogle Scholar
[14]van Saarloos, W. (1989) Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence. Phys. Rev. A 39, 63676390.CrossRefGoogle ScholarPubMed
[15]van Saarloos, W. (2003) Front propagation into unstable states. Phys. Rep. 386, 29222.CrossRefGoogle Scholar
[16]Volpert, A. I., Volpert, V. A. & Volpert, V. A. (1994) Traveling Wave Solutions of Parabolic Systems, Vol. 140 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI.Google Scholar
[17]Ward, J. P., King, J. R., Koerber, A. J., Croft, J. M., Sockett, R. E. & Williams, P. (2004) Cell-signalling repression in bacterial quorum sensing. Math. Med. Biol. 21, 169204.CrossRefGoogle ScholarPubMed
5
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Travelling-wave analysis of a model describing tissue degradation by bacteria
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Travelling-wave analysis of a model describing tissue degradation by bacteria
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Travelling-wave analysis of a model describing tissue degradation by bacteria
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *