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Limit Cycles in Reaction Systems with Second Order Autocatalysis



As a model for biochemical reactions the autocatalytic formation of the reactand X from a raw material Y is studied. It is shown that the system forms limit cycles; numerical examples are presented. As a second problem the above reaction is considered in the case of two boxes coupled by linear diffusion exchange of the raw material which is described by a term D(Y1 - Y2) (Yi - concentration in box i). In the case of weak coupling in both boxes beating between two frequencies are found. In the case of strong coupling damped oscillations and disappearance of Xi in one box are observed. The third problem is the stochastic theory of the given reactions. the corresponding master-equation is formulated and properties of the solution are discussed. Results of a computer simulation of the given birth and death process are represented.
It is well known that under appropriate conditions nonlinear physical, chemical or biological systems show various instabilities such as, for example, transitions leading to multistability, limit cycle oscillations or chaotic motion [1–3]. These instabilities correspond to bifurcations of the underlying dynamic equations. A realistic treatment requires one to take into account fluctuations, which appear as thermal fluctuations or are due to the discrete change of particle numbers in nonlinear chemical reactions. Another origin of noise may be the influence of the environment—which in general varies more or less randomly—on the thermodynamically open systems. Usually the description of complex systems, e.g. biochemical oscillations, is restricted to simplifying models which represent the observed behaviour only qualitatively. Then fluctuations may appear as a result of “hidden” reactions neglected in the model.
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