Bak-Sneppen Backwards

TL;DR

This paper investigates the backwards Markov chain of the Bak-Sneppen evolution model, deriving equations for its stationary distribution and applying them to specific multi-species models.

Contribution

It introduces the reversibility equations for the backwards chain and derives differential equations for stationary distributions in various Bak-Sneppen models.

Findings

Reversibility equations involve the stationary distribution.

Differential equations characterize stationary distributions.

Unified derivation of known distributions for specific models.

Abstract

We study the backwards Markov chain for the Bak-Sneppen model of biological evolution and derive its corresponding reversibility equations. We show that, in contrast to the forwards Markov chain, the dynamics of the backwards chain explicitly involve the stationary distribution of the model, and from this we derive a functional equation that the stationary distribution must satisfy. We use this functional equation to derive differential equations for the stationary distribution of Bak-Sneppen models in which all but one or all but two of the fitnesses are replaced at each step. This gives a unified way of deriving Schlemm's expressions for the stationary distributions of the isotropic four-species model, the isotropic five-species model, and the anisotropic three-species model.