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

- The Bak-Sneppen model represents evolving systems in punctuated equilibrium.
- We analyze in detail the case of five species.
- The steady-state fitness distribution satisfies an ODE with hypergeometric coefficients.

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 [fc, 1] with fc ≳ 2/3.