A differential equation for the asymptotic fitness distribution in the Bak--Sneppen model with five species

TL;DR

This paper derives a differential equation describing the steady-state fitness distribution in a five-species Bak--Sneppen model, providing insights into the asymptotic behavior of larger systems and their potential uniform distribution.

Contribution

It introduces a linear differential equation with hypergeometric coefficients to characterize the fitness distribution for five species in the Bak--Sneppen model.

Findings

Steady-state fitness distribution for five species is obtained via a fifth-order differential equation.

The approach may extend to larger systems to analyze asymptotic fitness distribution.

Potential to determine if fitness becomes uniformly distributed as the number of species grows.

Abstract

The Bak--Sneppen model is an abstract representation of a biological system that evolves according to the Darwinian principles of random mutation and selection. The species in the system are characterized by a numerical fitness value between zero and one. We show that in the case of five species the steady-state fitness distribution can be obtained as a solution to a linear differential equation of order five with hypergeometric coefficients. Similar representations for the asymptotic fitness distribution in larger systems may help pave the way towards a resolution of the question of whether or not, in the limit of infinitely many species, the fitness is asymptotically uniformly distributed on the interval [f,1] with f approximately equal to 2/3.