Nonsharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion

Applied Mathematics Letters

Volumn 9, Issue 5, 1996, Pages 33-38

  Sherratt, J A ^a   Marchant, B P ^a,b  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Degenerate diffusion; Matched asymptotic analysis; Mathematical model; Nonlinear diffusion; Perturbation theory; Travelling waves

Indexed keywords

EID: 0040892330     PISSN: 08939659     EISSN: None     Source Type: Journal    
DOI: 10.1016/0893-9659(96)00069-9     Document Type: Article

References (20)

1. R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7, 353-369 (1937).
2. A. Kolmogorov, I. Petrovsky and N. Piscounov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Bull. Math. 1, 1-25 (1937).
3. J.D. Murray, Mathematical Biology, Springer-Verlag, Berlin, (1989).
4. J.G. Skellam, Random dispersal in theoretical populations, Biometrika 38, 196-218 (1951).
5. E.E. Holmes, M.A. Lewis, J.E. Banks and R.R. Veit, Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology 75, 17-29 (1994).
6. J.A. Sherratt and J.D. Murray, Models of epidermal wound healing, Proc. R. Soc. Lond. B 241, 29-36 (1990).
7. J.A. Sherratt and J.D. Murray, Epidermal wound healing: The clinical implications of a simple mathematical model, Cell Transplantation 1, 365-371 (1992).
8. E.A. Carl, Population control in arctic ground squirrels, Ecology 52, 395-413 (1971).
9. J.H. Myers and C.J. Krebs, Population cycles in rodents, Sci. Am. 6, 38-46 (1974).
10. W.S.C. Gurney and R.M. Nisbet, The regulation of inhomogeneous population, J. Theor. Biol. 52, 441-457 (1975).
11. W.S.C. Gurney and R.M. Nisbet, A note on nonlinear population transport, J. Theor. Biol. 56, 249-251 (1976).
12. N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Math. Biol. 79, 83-99 (1979).
13. A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks and herds, Adv. Biophys. 22, 1-94 (1986).
14. D.G. Aronson, Density dependent interaction systems, In Dynamics and Modelling of Reactive Systems, (Edited by W.H. Steward et al.), pp. 1161-176, Academic Press, New York, (1980).
15. W.I. Newman, Some exact solutions to a nonlinear diffusion problem in population genetics and combustion, J. Theor. Biol. 85, 325-334 (1980).
16. W.I. Newman, The long time behaviour of the solution to a nonlinear diffusion problem in population genetics and combustion, J. Theor. Biol. 104, 473-484 (1983).
17. F. Sánchez-Garduño and P.K. Maini, Existence and uniqueness of a sharp front travelling wave in degenerate nonlinear diffusion Fisher-KPP equations, J. Math. Biol. 33, 163-192 (1994).
18. F. Sánchez-Garduño and P.K. Maini, An approximation to a sharp front type solution of a density dependent reaction-diffusion equation, Appl. Math. Lett. 7 (1), 47-51 (1994).
19. F. Sánchez-Garduño and P.K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Diff. Eq. 117, 281-319 (1995).
20. F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. R. Soc. Edin. 80A, 213-234 (1978).