Study of Generalizations of the Discrete Bak-Sneppen Model

Abstract: In 1993, Per Bak and Kim Sneppen proposed a model of co-evolution between species, where survival of a particular species affects the survival of its neighbouring species. In the discrete case of the model, each species, or an entry in a set with periodic boundary conditions, is an element x_i ∈ {0, 1}, in the set of size N, where x_i represents the fitness. An entry of the least fitness is chosen and replaced together with its two neighbours each with Bernoulli(p), p ∈ [0, 1] random variables. If the parameter p is larger than some p_cr, the whole set is eventually consumed by 1. In this paper, we study the generalizations of the discrete case of Bak- Sneppen model and evaluate p_cr both analytically and numerically. For that end, we first examine the case where in each iteration a vertex x_i and its both neighbours x_{i−1}, x_{i+1} are replaced by the same Bernoulli(p) variable. Then, we study the case where the type of the model - whether the entry is replaced alone or with its neighbours - is determined by a Bernoulli(r) variable. Finally, we find a non-trivial p_cr for a 2-dimentional set of entries.