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Stochastic Investigation of a Limit Cycle System: Generalized Lotka-Selkov Reaction

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Abstract

For a special reaction scheme, (--> Y, Y --> X (autocatalytic), X --> ) which is a generalization of the well-known schemes of Lotka and Selkov, the transition to a limit cycle regime is investigated. Analogies to phase transitions of first and second kind are studied. Analytical approximations for the solutions of the master equations are given and especially the phase fluctuations in the case of limit cycles are discussed.
... Note that by including quadratic terms into F (X), the passive system can be converted into an active one [10,38,39]. ...
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First we discuss some early work of Ulrike Feudel on structure formation in nonlinear reactions including ions and the efficiency of the conversion of chemical into electrical energy. Then we give some survey about energy conversion from chemical to higher forms of energy like mechanical, electrical and ecological energy. We consider examples of energy conversion in several natural processes and in some devices like fuel cells. Further, as an example, we study analytically the dynamics and efficiency of a simple "active circuit" converting chemical into electrical energy and driving currents which is roughly modeling fuel cells. Finally we investigate an analogous ecological system of Lotka - Volterra type consisting of an "active species" consuming some passive "chemical food". We show analytically for both these models that the efficiency increases with the load, reaches values higher then 50 percent in a narrow regime of optimal load and goes beyond some maximal load abrupt to zero.
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