Wave Patterns in One-Dimensional Nonlinear Degenerate Diffusion Equations
Several different types of wave patterns occur in physiology, chemistry and biology. In many cases such phenomena are modelled by reactive-diffusive parabolic systems (see, for example, Fisher 1937; Kolmogorov et al. 1937; Winfree 1988; Murray 1989; Swinney & Krinsky 1992). In many biological and physical situations, dispersal is modelled by a density-dependent diffusion coefficient, for example, the bacterium Rhizobium diffuses through the roots of some leguminosae plants according to a nonlinear diffusive law (Lara-Ochoa & Bustos 1990); nonlinear diffusion has been observed in the dispersion of some insects (Okubo 1980) and small rodents (Meyers & Krebs 1974).
Unable to display preview. Download preview PDF.
- Aronson, D. G. 1980. Density-dependent interaction-diffusion systems. In Dynamics and Modelling of Reactive Systems, Stewart, Warren E. (ed). Academic Press.Google Scholar
- Fife, P. 1979. Mathematical aspects of reaction diffusing systems. Vol. 28. Springer-Verlag. Lecture Notes in Biomathematics.Google Scholar
- Fisher, R. A. 1937. The wave of advance of advantageous genes. Ann. Eugenics., 7, 353–369.Google Scholar
- Kolmogorov, A., Petrovsy, I., & Piskounov, N. 1937. Study of the diffusion equation with growth of the quantity of matter and its applications to a biological problem. In Applicable Mathematics of Non-Physical Phenomena, Oliveira-Pinto, F., & Conolly, B. W. (eds). John Wiley and Sons, 1982 edn.Google Scholar
- Murray, J. D. 1989. Mathematical biology. New York: Springer-Verlag, Berlin.Google Scholar
- Okubo, A. 1980. Diffusion and ecological problems: mathematical models. In Biomathematics, volume 10. Springer-Verlag.Google Scholar
- Sánchez-Garduño, F., & Maini, P. K. 1992. Travelling wave phenomena in some degenerate reaction-diffusion equations. (Submitted for publication).Google Scholar
- Swinney, H. L., & Krinsky, V. I. 1992. Waves and Patterns in Chemical and Biological Media. Special Issue of Physica D.Google Scholar
- Winfree, A. 1988. When time breaks down. Princeton University Press.Google Scholar