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A model biochemical reaction

Asymptotic solutions are presented to the non-linear 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 non-uniform steady states or time-periodic solutions, depending on the ratio of the diffusion coefficients. In the limit that one of the diffusion coefficients is infinite, multiple steady states of finite amplitude are found. When one of the parameters is allowed to depend on space and the basic state is unstable, it is found that the non-uniform 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. When diffusion is absent it is shown how kinematic concentration waves arise. Detailed calculations using singular perturbation techniques are made of the basic oscillation giving rise to these waves, which is a relaxation oscillation. It is found that the equations in its asymptotic approximation are not obtained from the full equations as the result of a limit process.

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A model biochemical reaction