On a species survival model

Abstract

In this paper we provide some sharp asymptotic results for a stochastic model of species survival recently proposed by Guiol, Marchado, and Schinazi.1 Introduction and statement of results Recently, Guiol, Marchado, and Schinazi [7] proposed a new mathematical framework for modeling species survival which is closely related to the discrete Bak-Sneppen evolution model. In the original Bak-Sneppen model [3] a finite number of species are arranged in a circle, each species being characterized by its location and a parameter representing the fitness of the species and taking values between zero and one. The number of species and their location on the circle are fixed and remain unchanged throughout the evolution of the system. At discrete times n = 0, 1, . . . , the species with the lowest fitness and its two immediate neighbors update simultaneously their fitness values at random. The Bak-Sneppen evolution model is often referred to as an "ecosystem" because of the local interaction between different species. The distinguishing feature of the model is the emergence of self-organized criticality [1,2,6,8] regardless the simplicity and robustness of the underlying evolution mechanism. The Bak-Sneppen model has attracted significant attention over the past few decades, but it has also been proven to be difficult for analytical study. See for instance [6] for a relatively recent survey of the model.The asymptotic behavior of the Bak-Sneppen model, as the number of species gets arbitrarily large, was conjectured on the basis of computer simulations in [3]. It appears that the distribution of the fitness is asymptotically uniform on an interval (f crit , 1) for some critical parameter f crit , the value of which is close to 2/3 [1,8].Guiol, Marchado, and Schinazi [7] were able to prove a similar result for a related model with a stochastically growing number of species. Their analysis is based on a reduction to the study of a certain random walk, which allows them to build a proof using well-known results from the theory of random walks. The main result of [7] is *

On a link between a species survival time in an evolution model and the Bessel distributions

“…Observe that from Ben Ari et al[2] it would be also possible to get a Central Limit Theorem and a Law of the Iterate logarithm for R t .3Construction of the process. The construction uses ideas from Harris Graphical method for Markov Processes and basically take advantage from projections properties of a bi-dimensional Poisson process with rate 1.…”

A stochastic model for the evolution of species with random fitness

“…We note that for the GMS model and its generalizations, with µ ∼ U([0, 1]) (where U(I) is the uniform distribution on I), the fraction of surviving species in any I ⊆ [f c , 1] is proportional to µ(I). This is still true in our case when I ⊆ (f c , 1], but it does not hold for 1])/n → 0 (the exact rate of convergence for the GMS is studied in [3]), while again this needs not to be true if f c is an atom for µ. In this example we choose µ := αδ 1/2 + (1 − α)ν (where ν ∼ U([0, 1])); the case α = 0 is discussed in [6].…”

“…Observe that in Theorem 2.2 we used Z n as a normalizing factor for Z n (A) but there are two other natural choices: n (to compare with [2,9]) and N n (to compare with [3,6]).…”

“…Several papers have studied the GMS model: [3] gives a law of the iterated logarithm and a central limit theorem for number of species with supercritical fitness which go extinct (this number is negligible with respect to n); [5] studies the maximal fitness ever appeared in the subcritical case.…”

On a Local Version of the Bak–Sneppen Model

“…It is shown that this model obeys power law behavior for avalanche duration and size, behavior observed in numerical simulations for the original Bak-Sneppen model. Another variant of the Bak-Sneppen model was introduced in [9], studied in [5], and generalized in [14]. Here the number of species follows a path of reflected random walk transient to infinity.…”