On the form of smooth-front travelling waves in a reaction-diffusion equation with degenerate nonlinear diffusion

Mathematical Modelling of Natural Phenomena

Volumn 5, Issue 5, 2010, Pages 64-79

  Sherratt, J A ^a  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

nonlinear diffusion; sharp front; smooth front; travelling waves

Indexed keywords

ASYMPTOTIC APPROXIMATION; BACTERIAL COLONIES; BIOLOGICAL PHENOMENA; CELL BIOLOGY; NONLINEAR DIFFUSION; NUMERICAL RESULTS; REACTION DIFFUSION EQUATIONS; SHARP FRONT; SINGULAR PERTURBATION THEORY; SMOOTH FRONT; TRAVELLING WAVE FRONTS; TRAVELLING WAVES;

CYTOLOGY; DIFFUSION IN LIQUIDS; LINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PERTURBATION TECHNIQUES; WAVE TRANSMISSION;

NONLINEAR EQUATIONS;

EID: 79960786372     PISSN: 09735348     EISSN: 17606101     Source Type: Journal    
DOI: 10.1051/mmnp/20105505     Document Type: Article

References (54)

1. 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.
2. B.P. Ayati, G.F. Webb, A.R.A. Anderson. Computational methods and results for structured multiscale models of tumor invasion. Multiscale Modeling & Simulation, 5 (2006), 1-20. (Pubitemid 46388873)
3. R.D. Benguria, M.C. Depassier. Speed of fronts of the reaction-diffusion equation. Phys. Rev. Lett., 77 (1996), 1171-1173. (Pubitemid 126625642)
4. Z. Birö. Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type. Adv. Nonlinear Stud., 2 (2002), 357-371.
5. A.Q. Cai, K.A. Landman, B.D. Hughes. Multi-scale modelling of a wound healing cell migration assay. J. Theor. Biol., 245 (2007), 576-594. (Pubitemid 46365739)
6. I. Cohen, I. Golding, Y. Kozlovsky, E. Ben-Jacob, I.G. Ron. Continuous and discrete models of cooperation in complex bacterial colonies. Fractals, 7 (1999), 235-247. (Pubitemid 129619640)
7. P. Feng, Z. Zhou. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Comm. Pure Appl. Anal., 6 (2007), 1145-1165.
8. R.A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7 (1937), 353-369.
9. G. García-Ramos, F. Sánchez-Garduño, P.K. Maini. Dispersal can sharpen parapatric boundaries on a spatially varying environment. Ecology, 81 (2000), 749-760. (Pubitemid 30437998)
10. R.A. Gatenby, E.T. Gawlinski. A reaction-diffusion model of cancer invasion. Cancer Res., 56 (1996), 4740-4743.
11. B.H. Gilding, R. Kersner. A Fisher/KPP-type equation with density-dependent diffusion and convection: travelling-wave solutions. J. Phys. A: Math. Gen., 38 (2005), 3367-3379. (Pubitemid 41236951)
12. W.S.C. Gurney, R.M. Nisbet. The regulation of inhomogeneous population. J. Theor. Biol., 52 (1975), 441-457.
13. W.S.C. Gurney, R.M. Nisbet. A note on nonlinear population transport. J. Theor. Biol., 56 (1976), 249-251.
14. A. Hastings, K. Cuddington, K.F. Davies, C.J. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holland, J. Lambrinos, U. Malvadkar, B.A. Melbourne, K. Moore, C. Taylor, D. Thomson. The spatial spread of invasions: new developments in theory and evidence. Ecol. Lett., 8 (2005), 91-101. (Pubitemid 40172142)
15. J.J. Hellmann, J.E. Byers, B.G. Bierwagen, J.S. Dukes. Five potential consequences of climate change for invasive species. Conserv. Biol., 22 (2008), 534-543. (Pubitemid 351813451)
16. D. Hilhorst, R. Kersner, E. Logak, M. Mimura. Interface dynamics of the Fisher equation with degenerate diffusion. J. Differential Equations, 244 (2008), 2870-2889.
17. E.J. Hinch. Perturbation Methods. Cambridge University Press, 1991.
18. S. Kamin, P. Rosenau. Convergence to the travelling wave solution for a nonlinear reactiondiffusion equation. Rend. Mat. Acc. Lincei, 15 (2004a), 271-280. (Pubitemid 40854383)
19. S. Kamin, P. Rosenau. Emergence of waves in a nonlinear convection-reaction-diffusion equation. Adv. Nonlinear Stud., 4 (2004b), 251-272. (Pubitemid 39289310)
20. K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda, N. Shigesada. Modeling spatiotemporal patterns generated by Bacillus subtilis. J. Theor. Biol., 188 (1997), 177-185. (Pubitemid 27464187)
21. J. Kevorkian, J.D. Cole. Multiple Scale and Singular Perturbation Methods. Springer-Verlag, New York, 1996.
22. S.Y. Kim, R. Torres, H. Drummond. Simultaneous positive and negative density-dependent dispersal in a colonial bird species. Ecology, 90 (2009), 230-239.
23. F. Lutscher. Density-dependent dispersal in integrodifference equations. J. Math. Biol., 56 (2008), 499-524.
24. P.K. Maini, S. McElwain, D. Leavesley. A travelling wave model to interpret a wound healing migration assay for human peritoneal mesothelial cells. Tissue Eng., 10 (2004), 475-482. (Pubitemid 38580268)
25. L. Malaguti, C. Marcelli. Finite speed of propagation in monostable degenerate reactiondiffusion- convection equations. Advanced Nonlinear Studies, 5 (2005), 223-252.
26. L. Malaguti, C. Marcelli. Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations. J. Differential Equations, 195 (2003), 471-496. (Pubitemid 37495886)
27. M.B.A. Mansour. Traveling wave solutions of a nonlinear reaction-diffusion-chemotaxis model for bacterial pattern formation. Applied Mathematical Modelling, 32 (2008), 240-247. (Pubitemid 47506561)
28. E. Matthysen. Density-dependent dispersal in birds and mammals. Ecography, 28 (2005), 403-416. (Pubitemid 40816785)
29. G.S. Medvedev, K. Ono, P.J. Holmes. Travelling wave solutions of the degenerate Kolmogorov-Petrovski-Piskunov equation. Eur. J. Appl. Math., 14 (2003) 343-367. (Pubitemid 36938271)
30. W.I. Newman. Some exact solutions to a nonlinear diffusion problem in population genetics and combustion. J. Theor. Biol., 85 (1980), 325-334. (Pubitemid 10010011)
31. 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.
32. K.J. Painter, J.A. Sherratt. Modelling the movement of interacting cell populations. J. Theor. Biol., 225 (2003), 327-339. (Pubitemid 37353056)
33. B.-E. Saether, S. Engen, R. Lande. Finite metapopulation models with density-dependent migration and stochastic local dynamics. Proc. R. Soc. Lond. B, 266 (1999), 113-118. (Pubitemid 29048004)
34. F. Sánchez-Garduño, P.K. Maini. Existence and uniqueness of a sharp front travelling wave in degenerate nonlinear diffusion Fisher-KPP equations. J. Math. Biol., 33 (1994a), 163-192.
35. F. Sánchez-Garduño, P.K. Maini. An approximation to a sharp front type solution of a density dependent reaction-diffusion equation. Appl. Math. Lett., 7 (1994b), 47-51.
36. F. Sánchez-Garduño, P.K. Maini. Travelling wave phenomena in some degenerate reactiondiffusion equations. J. Differential Equations, 117 (1995), 281-319.
37. F. Sánchez-Garduño, P.K. Maini, E. Kappos. A review on travelling wave solutions of one-dimensional reaction diffusion equations with non-linear diffusion term. FORMA, 11 (1996a), 45-59.
38. F. Sánchez-Garduño, E. Kappos, P.K. Maini. A shooting argument approach to a sharp type solution for nonlinear degenerate Fisher-KPP equations. IMA J. Appl. Math. 57 (1996b), 211-221. (Pubitemid 126534632)
39. R.A. Satnoianu, P.K. Maini, F. Sánchez-Garduño, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete and Continuous Dynamical Systems B, 1 (2001), 339-362. (Pubitemid 33771899)
40. B.G. Sengers, C.P. Please, R.O.C. Oreffo. Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration. J. R. Soc. Interface, 4 (2007), 1107-1117. (Pubitemid 350303863)
41. J.A. Sherratt. 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.
42. J.A. Sherratt, J.D. Murray. Models of epidermal wound healing. Proc. R. Soc. Lond. B, 241 (1990), 29-36. (Pubitemid 20256945)
43. J.A. Sherratt, B.P. Marchant. Non-sharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion. Appl. Math. Lett. 9 (1996), 33-38. (Pubitemid 126360226)
44. J.A. Sherratt, M.A.J. Chaplain. A new mathematical model for avascular tumour growth. J. Math. Biol., 43 (2001), 291-312. (Pubitemid 33772356)
45. N. Shigesada, K. Kawasaki, E. Teramoto. Spatial segregation of interacting species. J. Math. Biol., 79 (1979), 83-99.
46. M.J. Simpson, K.A. Landman, K. Bhaganagarapu. Coalescence of interacting cell populations. J. Theor. Biol. 247 (2007), 525-543. (Pubitemid 46935211)
47. M.J. Simpson, K.A. Landman, B.D. Hughes, D.F. Newgreen. Looking inside an invasion wave of cells using continuum models: proliferation is the key. J. Theor. Biol., 243 (2006), 343-360. (Pubitemid 44647320)
48. J.G. Skellam. Random dispersal in theoretical populations. Biometrika, 38 (1951), 196-218.
49. M.J. Smith, J.A. Sherratt, X. Lambin. The effects of density-dependent dispersal on the spatiotemporal dynamics of cyclic populations. J. Theor. Biol., 254 (2008), 264-274.
50. A. Tremel, A. Cai, N. Tirtaatmadja, B.D. Hughes, G.W. Stevens, K.A. Landman, A.J. OConnor. Cell migration and proliferation during monolayer formation and wound healing. Chem. Eng. Sci., 64 (2009), 247-253.
51. R.R. Veit, M.A. Lewis. Integrodifference models for persistence in fragmented habitats. Bull. Math. Biol., 59 (1997), 107-137. (Pubitemid 27065392)
52. Y. Wu. Existence of stationary solutions with transition layers for a class of cross-diffusion systems. Proc. R. Soc. Ed., 132A (2002), 1493-1511. (Pubitemid 36170682)
53. J. Ylikarjula, S. Alaja, J. Laakso, D. Tesar. Effects of patch number and dispersal patterns on population dynamics and synchrony. J. Theor. Biol., 207 (2000), 377-387.
54. H.Y. Zhu, W. Yuan, C.H. Ou. Justification for wavefront propagation in a tumour growth model with contact inhibition. Proc. R. Soc. Lond. A, 464 (2008), 1257-1273. (Pubitemid 351390758)