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Stability of non-monotone waves in a three-species reaction—diffusion model

Published online by Cambridge University Press:  14 November 2011

Patrick D. Miller
Patrick D. Miller
Affiliation:
Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA ([email protected])
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Extract

A stability theorem is proved for non-monotone waves in a reaction–diffusion system modelling three competing species, where few stability results currently exist for systems with more than two species. Asymptotic stability with respect to the nonlinear equations is established by showing that the spectrum of the linearized operator has no unstable eigenvalues and that the zero eigenvalue associated with translation invariance is simple. The result is obtained in a singular regime where strong pattern formation occurs and solutions to the linear equations can be separated into particular solutions related to the fast–slow structure of the underlying wave and its singular limit. In this system both the fast and slow waves can contribute an instability and the global characterization of these solutions must address certain difficulties not present in lower-dimensional systems. The topological index of Alexander, Gardner and Jones is used to count the eigenvalues of the linear operator.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 125 - 152
Copyright
Copyright © Royal Society of Edinburgh 1999

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