A stochastic model for the evolution of species with random fitness

TL;DR

This paper introduces a generalized stochastic model for species evolution, where species with random fitness levels are created and removed, revealing a critical fitness threshold that determines long-term survival or extinction.

Contribution

It extends previous models by allowing arbitrary fitness distributions and analyzes the resulting critical fitness threshold and survival dynamics.

Findings

Existence of a critical fitness $f_c$ separating extinction and survival regimes.

Uniform convergence of the distribution of surviving species.

New phenomena observed with non-uniform fitness distributions.

Abstract

We generalize the evolution model introduced by Guiol, Machado and Schinazi (2010). In our model at odd times a random number X of species is created. Each species is endowed with a random fitness with arbitrary distribution on $[0, 1]$. At even times a random number Y of species is removed, killing the species with lower fitness. We show that there is a critical fitness $f_c$ below which the number of species hits zero i.o. and above of which this number goes to infinity. We prove uniform convergence for the distribution of surviving species and describe the phenomena which could not be observed in previous works with uniformly distributed fitness.