The Generalized Bak-Sneppen Model

A generalized Bak-Sneppen model (GBS model) of biological evolution with interaction strength p (0<p<1) is introduced. Simulations of the model on regular and Bethe lattice are given. According to the simulations we found that as long as there exists interaction between species, regardless of the strength of the interaction, the model can always self-organize to a critical state, and the critical threshold fc(p) decreases as p increases. The GBS model reduces to BS model as p=1. The staircase-like increasing of fmin(p,s) obeys an exact differential gap equation, which may provide a good criterion for judging the emergence of criticality and establishing exact scaling relations among the critical exponents of PMB avalanche. Forward and backward avalanche were studied, the forward avalanche critical exponent is an exact universal value2, and the backward avalanche critical exponent tp(p) and the PMB avalanche critical exponent t(p) satisfy tb(p)+t(p)=3. According to the simulations on one-to six-dimension we found that, when p was determined, fc(d) decreases as d increases, and the change is getting more and more slowly with the increasing of d According to the probability flow conservation, an exact master equation describing the hierarchical structure of avalanches is given. From the master equation, we derived an infinite set of exact equations that linked different (sk)(k=0,1,2,...)...