Bak-Sneppen backwards

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.

On a Local Version of the Bak–Sneppen Model

“…Indeed, since all sampled fitnesses are in [δ, 1 δ ], and all initial fitnesses for Ξ are > 1 δ , f − (Ξ t ) = 0, and similarly f + (Ξ t ) = 0. Therefore, on this event, f…”

“…, N − 1} and E = {{n, n + 1 mod N} : n ∈ V}. Also, in the original model, the fitnesses assigned were U[0, 1] rather than Exp (1), which, as is easy to see, has no effect on the dynamics. Furthermore, a deterministic transformation maps one model in a one-to-one fashion into the other (e.g.…”

“…A key element in the proof of the second statement is the notion of locking thresholds. It can be shown that if at time t = 0 the fitnesses are IID Exp (1), then at each time t ∈ Z + , the model has an associated (random) function L t : V → [0, ∞), such that conditioned on L t :…”

“…We note that the stationary distribution for the original Bak-Sneppen model was computed for N = 4 and N = 5 in [15] and [16], respectively, as well as in [1]. In the latter work, the authors obtained a formula for the stationary distribution for a variant of the Bak-Sneppen in which at each step all species are replaced except for one (coinciding with original Bak-Sneppen if N = 4) or two (coinciding with original Bak-Sneppen if N = 5).…”