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Published January 1976 | Published
Journal Article Open

Bifurcation of Localized Disturbances in a Model Biochemical Reaction


Asymptotic solutions are presented to the nonlinear parabolic reaction-diffusion equations describing a model biochemical reaction proposed by I. Prigogine. There is a uniform steady state which, for certain values of the adjustable parameters, may be unstable. When the uniform solution is slightly unstable, the two-timing method is used to find the bifurcation of new solutions of small amplitude. These may be either nonuniform steady states or time-periodic solutions, depending on the ratio of the diffusion coefficients. When one of the parameters is allowed to depend on space and the basic state is unstable, it is found that the nonuniform steady state which is approached may show localized spatial oscillations. The localization arises out of the presence of turning points in the linearized stability equations.

Additional Information

© 1976 Society for Industrial and Applied Mathematics. Received by the editors August 5, 1974; Published online 12 July 2006. James A. Boa would like to thank Professor J. B. Keller for some illuminating discussions. This work was supported in part by the U.S. Army Research Office (Durham) under Contract DAHC-04-68-C-0006 and by the National Science Foundation under Grant GP 18471.

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