Asymptotics for a class of iterated random cubic operators

TL;DR

This paper studies the long-term behavior of a class of cubic stochastic operators modeling genetic type frequencies, showing that random compositions almost surely converge to equilibria with varying numbers of types present.

Contribution

It introduces a new analysis of the asymptotic dynamics of random cubic operators, revealing convergence to equilibria with different genetic compositions depending on the distribution.

Findings

Random compositions almost surely converge to equilibria.

Long-term dynamics depend on the distribution of the maps.

Equilibria can have one, two, or three genetic types.

Abstract

We consider a class of cubic stochastic operators that are motivated by models for evolution of frequencies of genetic types in populations. We take populations with three mutually exclusive genetic types. The long term dynamics of single maps, starting with a generic initial condition where in particular all genetic types occur with positive frequency, is asymptotic to equilibria where either only one genetic type survives, or where all three genetic types occur. We consider a family of independent and identically distributed maps from this class and study its long term dynamics, in particular its random point attractors. The long term dynamics of the random composition of maps is asymptotic, almost surely, to equilibria. In contrast to the deterministic system, for generic initial conditions these can be equilibria with one or two or three types present (depending only on the distribution).