Infinite Series of Exact Equations in the Bak-Sneppen Model of Biological Evolution

Recently Bak and Sneppen [1] introduced a particularly simple toy model of biological evolution (the BS model). It provides a coarse-grained description of the behavior of the ecosystem of interacting species driven by mutation and natural selection. The features of the real evolutionary process, which may be correctly reproduced by this model, include the intermittent behavior (punctuated equilibrium), the apparent scale invariance of large extinction events [2], and the power law probability distribution of the lifetimes of species. In the simplest variant of the BS model the ecosystem consisting of L species is characterized by L numbers fi arranged on a line. The number fi represents the effective barrier for a successful mutation of ith species. At every time step the smallest number in the system is located, and this species is selected for mutation. As a result of this mutation this number, as well as two of its nearest neighbors (representing the species that strongly interact with a mutated one), are replaced with new uncorrelated random numbers drawn from the uniform distribution between 0 and 1. The generalization of these rules to higher spatial dimensions is straightforward. The BS model may describe the evolution on the longest time scale, where, due to universality, the exact microscopic details are of no importance. In fact, there exists a whole class of models where the rules consist of selecting the site with the extremal (global maximal or minimal) value of some variable and then changing this variable and its nearest neighbors according to some stochastic rule. These models, referred to as extremal models, were extensively studied (for a recent review, see [3]). They were employed to describe a variety of physical phenomena such as fluid invasion in disordered porous media [4], low temperature creep [5], earthquake dynamics [6], etc. Among these models, the BS model occupies a special place similar to that of the Ising model in the equilibrium statistical mechanics, since many observations can be rigorously proven for the BS model and then applied to other extremal models based on numerical simulations and less rigorous arguments. The feature of interest in the BS model (as well as in other extremal models) is its ability to organize itself into a scale-free stationary state. The dynamics in this critical state is given in terms of bursts of activity or avalanches, which form a hierarchical structure [1,3] of subavalanches within bigger avalanches. In the biological context these avalanches represent big extinction events. In this work we introduce a “master” equation for the avalanche hierarchy. It describes the cascade process in which smaller avalanches merge together to form bigger ones as the critical parameter is changed. From this equation we derive an infinite series of exact equations, relating different moments of temporal duration S and spatial area R d of individual avalanches. The master equation connects undercritical and overcritical regions of parameters. Given the existence of the scaling, we rigorously prove that the exponents of the BS model are the same above and below the transition. From our results it follows that all terms of the Taylor series of the scaling function fsxd for the avalanche distribution are uniquely and explicitly determined by two critical exponents of the model. We expect that the usual restrictions on the shape of fsxd indirectly relate these two exponents and, therefore, reduce the number of independent critical exponents in the BS model to just one. As was described in [3], the avalanches in extremal