Density behavior of spatial birth-and-death stochastic evolution of mutating genotypes under selection rates

TL;DR

This paper analyzes the long-term density behavior of a stochastic model for genotypes evolving through birth, death, and mutation, considering both homogeneous and heterogeneous spatial distributions.

Contribution

It introduces a model for genotypes with length-dependent birth, death, and mutation rates, and studies their asymptotic density behavior in space.

Findings

Asymptotic density behavior characterized for infinite genotype populations

Differences in behavior between space homogeneous and heterogeneous cases

Insights into mutation effects on genotype distribution

Abstract

We consider birth-and-death stochastic evolution of genotypes with different lengths. The genotypes might mutate that provides a stochastic changing of lengthes by a free diffusion law. The birth and death rates are length dependent which corresponds to a selection effect. We study an asymptotic behavior of a density for an infinite collection of genotypes. The cases of space homogeneous and space heterogeneous densities are considered.