Selection Theorem for Systems with Inheritance

Abstract

The problem of finite-dimensional asymptotics of infinite-dimensional dynamicsystems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of “natural” selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do nottend to fixed positions, and the path covered tends to infinity as t $\to$ ∞. The drift equationsfor peak motion are obtained. Various types of distribution stability are studied: internalstability (stability with respect to perturbations that do not extend the support), externalstability or uninvadability (stability with respect to strongly small perturbations that extendthe support), and stable realizability (stability with respect to small shifts and extensions ofthe density peaks). Models of self-synchronization of cell division are studied, as an exampleof selection in systems with additional symmetry. Appropriate construction of the notionof typicalness in infinite-dimensional space is discussed, and the notion of “completely thin”sets is introduced.