On a link between a species survival time in an evolution model and the Bessel distributions

TL;DR

This paper analyzes a stochastic species evolution model where species survival times are linked to Bessel distributions, depending on their fitness relative to a critical threshold, revealing a phase transition in survival behavior.

Contribution

It establishes a connection between species survival times in an evolution model and Bessel distributions, providing explicit critical fitness thresholds.

Findings

Survival time distribution depends on fitness relative to a critical value.

Explicit computation of the critical fitness threshold.

Survival times exhibit different behaviors for f<f_c, f=f_c, and f>f_c.

Abstract

We consider a stochastic model for species evolution. A new species is born at rate lambda and a species dies at rate mu. A random number, sampled from a given distribution F, is associated with each new species at the time of birth. Every time there is a death event, the species that is killed is the one with the smallest fitness. We consider the (random) survival time of a species with a given fitness f. We show that the survival time distribution depends crucially on whether f<f_c, f=f_c or f>f_c where f_c is a critical fitness that is computed explicitly.