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Existence and uniqueness of a sharp travelling wave in degenerate non- linear diffusion Fisher-KPP equations. (English) Zbl 0822.92021

Summary: We use a dynamical systems approach to prove the existence of a unique critical value \(c^*\) of the speed \(c\) for which the degenerate density- dependent diffusion equation \(u_ t = [D(u)u_ x]_ x + g(u)\) has:
1. no travelling wave solutions for \(0 < c < c^*\),
2. a travelling wave solution \(u(x,t) = \phi (x - c^* t)\) of sharp type satisfying \(\phi (-\infty) = 1\), \(\phi(\tau) = 0\) \(\forall \tau \geq \tau^*\); \(\phi' (\tau^{*-}) = - c^*/D' (0),\) \(\phi' (\tau^{*+}) = 0\) and
3. a continuum of travelling wave solutions of monotone decreasing front type for each \(c > c^*\).
These fronts satisfy the boundary conditions \(\phi (-\infty) = 1\), \(\phi' (-\infty) = \phi(+\infty) = \phi' (+\infty) = 0\). We illustrate or analytical results with some numerical results.

MSC:

92D40 Ecology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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[1] Andronov, A. A., Leontovich, E. A., Gordon, I.I., Maier, A. G.: Theory of dynamical systems on a plane. Israel Program for Scientific Translations, Jerusalem 1972
[2] Arnold, V. I.: Ordinary differential equations. Cambridge: MIT Press 1980 · Zbl 0432.34001
[3] Aronson, D. G.: Density-dependent interaction-systems. In: Steward, W. H., Ray, W. H., Conley, C. C. (eds.) Dynamics and modelling of reactive systems. New York: Academic Press 1980
[4] Aronson, D. G.: The role of the diffusion in mathematical population biology: Skellam revisited. In: Fasano, A., Primicerio, M. (eds.) Lecture Notes in Biomathematics 57. Berlin Heidelberg New York: Springer 1985 · Zbl 0583.92016
[5] Arrowsmith, D. K., Place, C. M.: An introduction to dynamical system. Cambridge: Cambridge University Press 1990 · Zbl 0702.58002
[6] Britton, N. F.: Reaction-diffusion equations and their applications to biology. New York: Academic Press 1986 · Zbl 0602.92001
[7] Carl, E. A.: Population control in Arctic ground squirrels. Ecology 52, 395-413 (1971) · doi:10.2307/1937623
[8] Carr, J.: Applications of centre manifold theory. Berlin Heidelberg New York: Springer 1981 · Zbl 0464.58001
[9] de Pablo, A., Vázquez, J. L.: Travelling waves in finite propagation in a reaction-diffusion equation. J. Diff. Equ. 93, 19-61 (1991) · Zbl 0784.35045 · doi:10.1016/0022-0396(91)90021-Z
[10] Engler, H.: Relations between travelling wave solutions of quasilinear parabolic equations. Proc. Am. Math. Soc. 93, 297-302 (1985) · Zbl 0535.35042 · doi:10.1090/S0002-9939-1985-0770540-6
[11] Fife, P. C.: Mathematical aspects of reacting and diffusing systems. Berlin Heidelberg New York: Springer 1979 · Zbl 0403.92004
[12] Fisher, R. A.: The wave of advance of advantageous genes. Ann. Eugen 7, 353-369 (1937) · JFM 63.1111.04
[13] Grindrod, P., Sleeman B. D.: Weak travelling wave fronts for population models with density-dependent dispersion. Math. Meth. Appl. Sci. 9, 576-586 (1987) · Zbl 0653.92015 · doi:10.1002/mma.1670090136
[14] Gurney, W. S. C., Nisbet, R. M.: The regulation of inhomogeneous population. J. Theor. Biol. 441-457 (1975)
[15] Gurney, W. S. C., Nisbet, R. M.: A note on non-linear population transport. J. Theor. Biol. 56, 249-251 (1976) · doi:10.1016/S0022-5193(76)80056-2
[16] Gurtin, M. E., MacCamy, R. C.: On the diffusion of biological populations. Math. Biosci. 33 35-49 (1977) · Zbl 0362.92007 · doi:10.1016/0025-5564(77)90062-1
[17] Hadeler, K. P.: Travelling fronts and free boundary value problems. In: Albretch, J., Collatz, L., Hoffman, K. H. (eds.) Numerical Treatment of Free Boundary Value Problems. Basel: Birkhauser 1981 · Zbl 0482.35050
[18] Hadeler, K. P.: Free boundary problems in biological models. In: Fasano, A., Primicerio, M. (eds.) Free Boundary Problems: Theory and Applications, Vol. II. London: Pitman 1983 · Zbl 0525.92023
[19] Kolmogorov, A., Petrovsky, L., Piskounov, I. N.: Study of the diffusion equation with growth of the quantity of matter and its applications to a biological problem. (English translation containing the relevant results) In: Oliveira-Pinto, F., Conolly, B. W. (eds.) Applicable mathematics of non-physical phenomena. New York: Wiley 1982
[20] Malajovich, G.: Programa Tracador de Diagramas de Fase. Instituto de Matematica. Universidade Federal do Rio de Janeiro, Brazil 1988
[21] Montroll, E. W., West, B. J.: On an enriched collection of stochastic processes. In: Montroll, E. W., Lebowitz, J. L. (eds.) Fluctuation phenomena. Amsterdam: North-Holland 1979
[22] Murray, J. D.: Mathematical biology. Biomathematics Texts 19. Berlin Heidelberg New York: Springer 1989 · Zbl 0682.92001
[23] Myers, J. H., Krebs, Ch. J.: Population cycles in rodents. Sci. Am. 6, 38-46 (1974) · doi:10.1038/scientificamerican0674-38
[24] Newman, W. I.: Some exact solutions to a non-linear diffusion problem in population genetics and combustion. J. Theor. Biol. 85, 325-334 (1980) · doi:10.1016/0022-5193(80)90024-7
[25] Newman, W. I.: The long-time behaviour of the solution to a non-linear diffusion problem in population genetics and combustion. J. Theor. Biol, 104, 473-484 (1983) · doi:10.1016/0022-5193(83)90240-0
[26] Sánchez-Garduño, F.: Travelling waves in one-dimensional degenerate non-linear reaction-diffusion equations. D. Phil. thesis., University of Oxford (1993)
[27] Sánchez-Garduño, F., Maini, P. K.: Travelling wave phenomena in some degenerate reaction-diffusion equations. J. Diff. Equ. (in press) · Zbl 0821.35085
[28] Sánchez-Garduño, F., Maini, P. K.: Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations (submitted) · Zbl 0887.35073
[29] Shiguesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Math. Biol. 79, 83-99 (1979)
[30] Skellam, J. G.: Random dispersal in theoretical populations. Biometrika 38, 196-218 (1951) · Zbl 0043.14401
[31] Skellam, J. G.: The formulation and interpretation of mathematical models of diffusionary processes in population biology. In: Bartlett, M. S. et al. (eds.) The mathematical theory of the dynamics of biological population. New York: Academic Press 1973
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