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Bifurcation of steady-state solutions in predator-prey and competition systems

Published online by Cambridge University Press:  14 November 2011

J. Blat  and
K. J. Brown
J. Blat
Affiliation:
Department of Mathematics, Heriot–Watt University, Riccarton, Currie, Edinburgh EH14 4AS
K. J. Brown
Affiliation:
Department of Mathematics, Heriot–Watt University, Riccarton, Currie, Edinburgh EH14 4AS
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Synopsis

We discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 21 - 34
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

1Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620–709.Google Scholar
2Brown, K. J.. Spatially inhomogeneous steady-state solutions for systems of equations describing interacting populations. J. Math. Anal. Appl. 95 (1983), 251–264.Google Scholar
3Cohen, D. S. and Laetsch, T. W.. Nonlinear boundary value problems suggested by chemical reactor theory. J. Differential Equations 7 (1970), 217–226.CrossRefGoogle Scholar
4Conway, E., Gardner, R. and Smoller, J.. Stability and bifurcation of steady-state solutions for predator-prey equations. Adv. in Appl. Math. 3 (1982), 288–334.Google Scholar
5Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Fund. Anal. 8 (1971), 321–340.CrossRefGoogle Scholar
6Henry, D.. Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).Google Scholar
7Leung, A.. Monotone schemes for semilinear elliptic systems related to ecology. Math. Methods Appl. Sci. 4 (1982), 272–285.Google Scholar
8Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
9Sattinger, D. H.. Topics in stability and bifurcation theory. Lecture Notes in Mathematics 309 (Berlin: Springer, 1973).Google Scholar
10Schiaffino, A. and Tesei, A.. Competition systems with Dirichlet boundary conditions. J. Math. Biol. 15 (1982), 93–105.Google Scholar
11Zhou, L. and Pao, C. V.. Asymptotic behaviour of a competition-diffusion system in population dynamics. Nonlinear Anal. 6 (1982), 1163–1183.Google Scholar