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Nonlinear density-dependent diffusion for competing species interactions: Large-time asymptotic behaviour

Published online by Cambridge University Press:  20 January 2009

Anthony W. Leung
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In many biological diffusion-reaction studies, it was found that one should include the effect of density dependent rates, drift terms and spatially varying growth rates, in order to obtain more accurate results. (See e.g. [7],[10], [8] , [3]). On the other hand, many recent mathematical results on reaction-diffusion systems do not include such general setting. This article investigates the behaviour of competing-species reaction-diffusion model under this more general situation. Efforts are made to obtain results concerning coexistence, survival and extinction, by methods similar to that in [5], [6].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 27 , Issue 2 , June 1984 , pp. 131 - 144
Copyright
Copyright © Edinburgh Mathematical Society 1984

References

1.Brown, P., Decay to uniform states in ecological interactions, SI AM J. Appl. Math. 38 (1980), 22–37.CrossRefGoogle Scholar
2.Fife, P. and Tang, M., Comparison principles for reaction-diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation of disturbances, J. of Diff. Eq. 40 (1981), 168–185.CrossRefGoogle Scholar
3.Kareiva, P., Dispersal as a passive diffusion: an analysis of field experiments involving herbivorous insects, Manuscript (1981).Google Scholar
4.Ladyzenskaja, O., Solonnikov, V., and Ural'Ceva, N., Linear and quasilinear equations of parabolic type, Transl. of Math. Monograph AMS 23 (1968).CrossRefGoogle Scholar
5.Leung, A., Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl. 73 (1980), 204–218.CrossRefGoogle Scholar
6.Leung, A., Stabilities for equilibria of competing-species reaction-diffusion equations with homogeneous Dirichlet conditions, Funk. Ekv. (Ser. Interna.) 24 (1981), 201–210.Google Scholar
7.Levin, S., Models of population dispersal, differential equations and applications, Ecology, Epidemics and Population Problems, edited by Busenberg, S. and Cooke, K. (Academic Press, 1981).Google Scholar
8.Mimura, M., Nishiura, Y. and Yamaguti, M., Some diffusive prey and predator systems and their bifurcation problem, Annals of the N.Y. Academy of Sciences 316 (1979), 490–510.CrossRefGoogle Scholar
9.De Mottoni, P., Schiaffino, A., and Tesei, A., On stable space dependent stationary solutions of a competition system with diffusion, Reprint, (1981).Google Scholar
10.Okubo, A., Diffusion and Ecological Problems: Mathematical Models (Springer-Verlag, Berlin, 1980).Google Scholar
11.Rothe, F., A nonhomogeneous predator-prey system with patchiness, Preprint, (1981).Google Scholar
12.Zhou, L. and Pao, C., Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Anal. Th. Meth. and Appl. 6 (1982), 1163–1184.CrossRefGoogle Scholar