Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-30T19:25:14.917Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Form of Smooth-Front Travelling Waves in aReaction-Diffusion Equation with Degenerate Nonlinear Diffusion

Published online by Cambridge University Press:  27 July 2010

J.A. Sherratt
J.A. Sherratt*
Affiliation:
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK
*
* Corresponding author: E-mail:jas@ma.hw.ac.uk
Get access

Abstract

Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use asmodels of biological phenomena. This paper begins with a survey of applications toecology, cell biology and bacterial colony patterns. The author then reviews mathematicalresults on the existence of travelling wave front solutions of these equations, and theirgeneration from given initial data. A detailed study is then presented of the form ofsmooth-front waves with speeds close to that of the (unique) sharp-front solution, for theparticular equation ut =(uux)x + u(1− u). Using singular perturbation theory, the author derives anasymptotic approximation to the wave, which gives valuable information about the structureof smooth-front solutions. The approximation compares well with numerical results.

Keywords

nonlinear diffusiontravelling wavessharp frontsmooth front

Information

Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 5: Reaction-diffusion waves , 2010 , pp. 64 - 79
Copyright
© EDP Sciences, 2010

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.)

Article purchase

Temporarily unavailable

References

D.G. Aronson. Density dependent interaction systems. In: W.H. Steward et al. (eds.) Dynamics and Modelling of Reactive Systems, pp. 1161-176. Academic Press, New York, 1980.
Ayati, B.P., Webb, G.F., Anderson, A.R.A.. Computational methods and results for structured multiscale models of tumor invasion. Multiscale Modeling & Simulation, 5 (2006), 1-20.CrossRefGoogle Scholar
Benguria, R.D., Depassier, M.C.. Speed of fronts of the reaction-diffusion equation. Phys. Rev. Lett., 77 (1996), 1171-1173.CrossRefGoogle ScholarPubMed
Biró, Z.. Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type. Adv. Nonlinear Stud., 2 (2002), 357-371.CrossRefGoogle Scholar
Cai, A.Q., Landman, K.A., Hughes, B.D.. Multi-scale modelling of a wound healing cell migration assay. J. Theor. Biol., 245 (2007), 576-594. CrossRefGoogle Scholar
Cohen, I., Golding, I., Kozlovsky, Y., Ben-Jacob, E., Ron, I.G.. Continuous and discrete models of cooperation in complex bacterial colonies. Fractals, 7 (1999), 235-247.CrossRefGoogle Scholar
Feng, P., Zhou, Z.. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Comm. Pure Appl. Anal., 6 (2007), 1145-1165.Google Scholar
Fisher, R.A.. The wave of advance of advantageous genes. Ann. Eugenics, 7 (1937), 353-369.CrossRefGoogle Scholar
García-Ramos, G., Sànchez-Garduño, F., Maini, P.K.. Dispersal can sharpen parapatric boundaries on a spatially varying environment. Ecology, 81 (2000), 749-760.CrossRefGoogle Scholar
Gatenby, R.A., Gawlinski, E.T.. A reaction-diffusion model of cancer invasion. Cancer Res., 56 (1996), 4740-4743.Google ScholarPubMed
Gilding, B.H., Kersner, R.. A Fisher/KPP-type equation with density-dependent diffusion and convection: travelling-wave solutions. J. Phys. A: Math. Gen., 38 (2005), 3367-3379.CrossRefGoogle Scholar
Gurney, W.S.C., Nisbet, R.M.. The regulation of inhomogeneous population. J. Theor. Biol., 52 (1975), 441-457.CrossRefGoogle Scholar
Gurney, W.S.C., Nisbet, R.M.. A note on nonlinear population transport. J. Theor. Biol., 56 (1976), 249-251.CrossRefGoogle Scholar
Hastings, A., Cuddington, K., Davies, K.F., Dugaw, C.J., Elmendorf, S., Freestone, A., Harrison, S., Holland, M., Lambrinos, J., Malvadkar, U., Melbourne, B.A., Moore, K., Taylor, C., Thomson, D.. The spatial spread of invasions: new developments in theory and evidence. Ecol. Lett., 8 (2005), 91-101. Google Scholar
Hellmann, J.J., Byers, J.E., Bierwagen, B.G., Dukes, J.S.. Five potential consequences of climate change for invasive species. Conserv. Biol., 22 (2008), 534-543.CrossRefGoogle ScholarPubMed
Hilhorst, D., Kersner, R., Logak, E., Mimura, M.. Interface dynamics of the Fisher equation with degenerate diffusion. J. Differential Equations, 244 (2008), 2870-2889.CrossRefGoogle Scholar
E.J. Hinch. Perturbation Methods. Cambridge University Press, 1991.
Kamin, S., Rosenau, P.. Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation. Rend. Mat. Acc. Lincei, 15 (2004a), 271-280.Google Scholar
Kamin, S., Rosenau, P.. Emergence of waves in a nonlinear convection-reaction-diffusion equation. Adv. Nonlinear Stud., 4 (2004b), 251-272.CrossRefGoogle Scholar
Kawasaki, K., Mochizuki, A., Matsushita, M., Umeda, T., Shigesada, N.. Modeling spatio-temporal patterns generated by Bacillus subtilis. J. Theor. Biol., 188 (1997), 177-185.CrossRefGoogle ScholarPubMed
J. Kevorkian, J.D. Cole. Multiple Scale and Singular Perturbation Methods. Springer-Verlag, New York, 1996.
Kim, S.Y., Torres, R., Drummond, H.. Simultaneous positive and negative density-dependent dispersal in a colonial bird species. Ecology, 90 (2009), 230-239.CrossRefGoogle Scholar
Lutscher, F.. Density-dependent dispersal in integrodifference equations. J. Math. Biol., 56 (2008), 499-524.CrossRefGoogle ScholarPubMed
Maini, P.K., McElwain, S., Leavesley, D.. A travelling wave model to interpret a wound healing migration assay for human peritoneal mesothelial cells. Tissue Eng., 10 (2004), 475-482. CrossRefGoogle Scholar
Malaguti, L., Marcelli, C.. Finite speed of propagation in monostable degenerate reaction-diffusion-convection equations. Advanced Nonlinear Studies, 5 (2005), 223-252.CrossRefGoogle Scholar
Malaguti, L., Marcelli, C.. Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations. J. Differential Equations, 195 (2003), 471-496.CrossRefGoogle Scholar
Mansour, M.B.A.. Traveling wave solutions of a nonlinear reaction-diffusion-chemotaxis model for bacterial pattern formation. Applied Mathematical Modelling, 32 (2008), 240-247.CrossRefGoogle Scholar
Matthysen, E.. Density-dependent dispersal in birds and mammals. Ecography, 28 (2005), 403-416.CrossRefGoogle Scholar
Medvedev, G.S., Ono, K., Holmes, P.J.. Travelling wave solutions of the degenerate Kolmogorov-Petrovski-Piskunov equation. Eur. J. Appl. Math., 14 (2003) 343-367. CrossRefGoogle Scholar
Newman, W.I.. Some exact solutions to a nonlinear diffusion problem in population genetics and combustion. J. Theor. Biol., 85 (1980), 325-334.CrossRefGoogle Scholar
A. Okubo, A. Hastings, T. Powell. Population dynamics in temporal and spatial domains. In: A. Okubo, S.A. Levin (eds.) Diffusion and ecological problems: modern perspectives, pp. 298-373. Springer, New York, 2001.
Painter, K.J., Sherratt, J.A.. Modelling the movement of interacting cell populations. J. Theor. Biol., 225 (2003), 327-339.CrossRefGoogle ScholarPubMed
Saether, B.-E., Engen, S., Lande, R.. Finite metapopulation models with density-dependent migration and stochastic local dynamics. Proc. R. Soc. Lond. B, 266 (1999), 113-118.CrossRefGoogle Scholar
Sànchez-Garduño, F., Maini, P.K.. Existence and uniqueness of a sharp front travelling wave in degenerate nonlinear diffusion Fisher-KPP equations. J. Math. Biol., 33 (1994a), 163-192.CrossRefGoogle Scholar
Sànchez-Garduño, F., Maini, P.K.. An approximation to a sharp front type solution of a density dependent reaction-diffusion equation. Appl. Math. Lett., 7 (1994b), 47-51. CrossRefGoogle Scholar
Sànchez-Garduño, F., Maini, P.K.. Travelling wave phenomena in some degenerate reaction-diffusion equations. J. Differential Equations, 117 (1995), 281-319.CrossRefGoogle Scholar
Sànchez-Garduño, F., Maini, P.K., Kappos, E.. A review on travelling wave solutions of one-dimensional reaction diffusion equations with non-linear diffusion term. FORMA, 11 (1996a), 45-59.Google Scholar
Sànchez-Garduño, F., Kappos, E., Maini, P.K.. A shooting argument approach to a sharp type solution for nonlinear degenerate Fisher-KPP equations. IMA J. Appl. Math. 57 (1996b), 211-221.CrossRefGoogle Scholar
Satnoianu, R.A., Maini, P.K., Sànchez-Garduño, F., Armitage, J.P.. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete and Continuous Dynamical Systems B, 1 (2001), 339-362.Google Scholar
Sengers, B.G., Please, C.P., Oreffo, R.O.C.. Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration. J. R. Soc. Interface, 4 (2007), 1107-1117. CrossRefGoogle ScholarPubMed
Sherratt, J.A.. Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations. Proc. R. Soc. Lond. A 456 (2000), 2365-2386.CrossRefGoogle Scholar
Sherratt, J.A., Murray, J.D.. Models of epidermal wound healing. Proc. R. Soc. Lond. B, 241 (1990), 29-36. CrossRefGoogle ScholarPubMed
Sherratt, J.A., Marchant, B.P.. Non-sharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion. Appl. Math. Lett. 9 (1996), 33-38.CrossRefGoogle Scholar
Sherratt, J.A., Chaplain, M.A.J.. A new mathematical model for avascular tumour growth. J. Math. Biol., 43 (2001), 291-312.CrossRefGoogle ScholarPubMed
Shigesada, N., Kawasaki, K., Teramoto, E.. Spatial segregation of interacting species. J. Math. Biol., 79 (1979), 83-99.Google ScholarPubMed
Simpson, M.J., Landman, K.A., Bhaganagarapu, K.. Coalescence of interacting cell populations. J. Theor. Biol. 247 (2007), 525-543.CrossRefGoogle ScholarPubMed
Simpson, M.J., Landman, K.A., Hughes, B.D., Newgreen, D.F.. Looking inside an invasion wave of cells using continuum models: proliferation is the key. J. Theor. Biol., 243 (2006), 343-360. CrossRefGoogle ScholarPubMed
Skellam, J.G.. Random dispersal in theoretical populations. Biometrika, 38 (1951), 196-218.CrossRefGoogle ScholarPubMed
Smith, M.J., Sherratt, J.A., Lambin, X.. The effects of density-dependent dispersal on the spatiotemporal dynamics of cyclic populations. J. Theor. Biol., 254 (2008), 264-274.CrossRefGoogle ScholarPubMed
Tremel, A., Cai, A., Tirtaatmadja, N., Hughes, B.D., Stevens, G.W., Landman, K.A., OConnor, A.J.. Cell migration and proliferation during monolayer formation and wound healing. Chem. Eng. Sci., 64 (2009), 247-253.CrossRefGoogle Scholar
Veit, R.R., Lewis, M.A.. Integrodifference models for persistence in fragmented habitats. Bull. Math. Biol., 59 (1997), 107-137.Google Scholar
Wu, Y.. Existence of stationary solutions with transition layers for a class of cross-diffusion systems. Proc. R. Soc. Ed., 132A (2002), 1493-1511.Google Scholar
Ylikarjula, J., Alaja, S., Laakso, J., Tesar, D.. Effects of patch number and dispersal patterns on population dynamics and synchrony. J. Theor. Biol., 207 (2000), 377-387.CrossRefGoogle ScholarPubMed
Zhu, H.Y., Yuan, W., Ou, C.H.. Justification for wavefront propagation in a tumour growth model with contact inhibition. Proc. R. Soc. Lond. A, 464 (2008), 1257-1273.CrossRefGoogle Scholar