A stochastic model of evolution

TL;DR

This paper introduces a stochastic evolution model where species with varying fitness undergo births and deaths, revealing a phase transition and a critical fitness threshold that determines species survival or extinction.

Contribution

It presents a novel stochastic model of evolution with a phase transition and analyzes the distribution of species fitness over time.

Findings

Existence of a sharp phase transition at a critical birth-death probability ratio.

Species with fitness above a critical value tend to uniformly distribute.

Species below the critical fitness level eventually go extinct.

Abstract

We propose a stochastic model for evolution. Births and deaths of species occur with constant probabilities. Each new species is associated with a fitness sampled from the uniform distribution on [0,1]. Every time there is a death event then the type that is killed is the one with the smallest fitness. We show that there is a sharp phase transition when the birth probability is larger than the death probability. The set of species with fitness higher than a certain critical value approach an uniform distribution. On the other hand all the species with fitness less than the critical disappear after a finite (random) time.