Selection theorem for systems with inheritance

TL;DR

This paper investigates finite-dimensional asymptotics in infinite-dimensional systems with inheritance, demonstrating how solutions evolve into narrow peaks that exhibit natural selection effects, with implications across biology, physics, chemistry, and economics.

Contribution

It introduces a new approach to analyze systems with inheritance, showing how their solutions tend to finite-dimensional attractors and deriving drift equations for peak dynamics.

Findings

Solutions form narrow peaks over time, indicating natural selection.

Finite-dimensional asymptotics are established for systems with inheritance.

Various stability types of distributions are analyzed.

Abstract

The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems 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 not tend to fixed positions, and the path covered tends to infinity as t goes to infinity. 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 infinite-dimensional space is discussed, and the notion of "completely thin" sets is introduced. Key words: Dynamics; Attractor; Evolution; Entropy; Natural selection