On a model of evolution of subspecies

TL;DR

This paper extends a virus evolution model by incorporating a preferential attachment mechanism, resulting in power law behavior in the distribution of fitness, contrasting with the exponential decay in the original model.

Contribution

It introduces a modified model with preferential attachment, leading to different asymptotic behavior, and analyzes its impact on the evolution of fitness distributions.

Findings

Power law behavior in fitness distribution asymptotics

Modification of the original model to include preferential attachment

Contrast with exponential behavior in prior models

Abstract

Ben-Ari and Schinazi (2016) introduced a stochastic model to study `virus-like evolving population with high mutation rate'. This model is a birth and death model with an individual at birth being either a mutant with a random fitness parameter in $[0,1]$ or having one of the existing fitness parameters with uniform probability; whereas a death event removes the entire population of the least fit site. We change this to incorporate the notion of `survival of the fittest', by requiring that a non-mutant individual, at birth, has a fitness according to a preferential attachment mechanism, i.e., it has a fitness $f$ with a probability proportional to the size of the population of fitness $f$. Also death just removes one individual at the least fit site. This preferential attachment rule leads to a power law behaviour in the asymptotics, unlike the exponential behaviour obtained by Ben-Ari and Schinazi (2016).