The Bak-Sneppen model for species survival is a discrete-time Markov process defined as follows: A population of species with finite fixed size is arranged on a circle. Initially, each species is assigned a random fitness, independent and uniform on [0,1]. At each unit of time, the species with the least fitness and its two immediate neighbors are eliminated and are replaced by three new species, each with independent uniform [0,1] fitness. Numerical simulations indicate that as the size of the population grows to infinity, the stationary distribution converges to the product of uniform distributions on [b_c,1], for some b_c close to 0.667, suggesting the emergence of self-organized criticality. A proof has not been found yet. In the first part of the talk I will present some known results on the model. In the second part I will focus on a variation of the Bak-Sneppen model, recently proposed by Guiol, Marchado, and Schinazi, in which the number of species in the population is given by a birth and death process. For this model, the authors proved that - asymptotically in time - the distribution of the fitnesses is uniform on some interval [f_c,1]. I will present new sharper asymptotic results on the distribution of the fitnesses, essentially a central limit theorem and a law of iterated logarithm for the number of species above a certain fitness. Joint with Alexander Roitershtein and Anastasios Matzavinos (Iowa State).
