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Stability of Spatial Steady State Solutions for Hypercyles Replication System

Vladimir Posviansky, Alexander Bratus
The system of semilinear parabolic equations which described the mechanism and dynamics organizing of complicated macromolecules in the process of prebyology evolution is considered [Eigen, Schuster, 1979]. The aim of investigation is searching and analyzing stability of spatial non uniform steady state solutions (SNSS solutions) of the system. It is proved that if the diffusion coefficients are sufficiently small then there exist SNSS solutions of the system in the form of over fall’s waves or cycling wave with the m-th humps. This solution is not stable in usual sense. It is proved exclusion principal in case of a spatial dynamics of replication macromolecules by auto- catalyzing reaction. SNSS solutions of the system are stable in sense of mean integral values in case of replication dynamics macromolecules by hypercycles process. For open models of hypercycles replication reaction the analogous results was proved too. With help of the Galerkin method numerical solutions for various cases of realization SNSS solutions was obtained.
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