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Selection Theorem for Systems with Inheritance

Alexander N Gorban

Published 2004 in Mathematical Modelling of Natural Phenomena

ABSTRACT

The problem of flnite-dimensional asymptotics of inflnite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distribu- tions has generically flnite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and eco- nomics. This conservation of support has a biological interpretation: inheritance. The flnite-dimensional asymptotics demonstrates efiects of \natural" selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equa- tion with conservation of support becomes a flnite set of narrow peaks that become increas- ingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to flxed positions, and the path covered tends to inflnity as t ! 1. The drift equations for peak motion are obtained. Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). Models of self-synchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in inflnite-dimensional space is discussed, and the notion of \completely thin" sets is introduced.

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