Avalanche Dynamics And Scaling Behaviors In The Process Of Evolution

This dissertation is concerned with scaling behaviors and avalanche 4 dynamics in the critical state of the evolution model. A new quantity, average fitness f(s), is proposed to prescribe the gen- eral feature of the ecosystem consisting of interacting species. f(s) is related to the species of the individual species of the whole system. The staircase-like increasing of f(s) obeys an exact differential gap equation, which can be solved analytically. Features of f(s) and its stability in the critical state suggest that f(s) may provide a good criterion for judging the emergence of criticality. The critical value of f(s) is related to the self-organized threshold f~ through an exact equation: fc = (1 + f~)/2. A new hierarchy of avalanches, LC avalanche, is observed through the average fitness. Based on the global level, LC avalanche embodies the nat- ural feature of the system. Two fundamental critical exponents, avalanche distribution -r and avalanche dimension D are vastly different from the counterparts for other avalanches. This manifests that LC avalanche is a different type of avalanche. The hierarchical structure of LC avalanche is described by an exact master equation. An infinite series of exact equations, relating different orders of (Sk), can be derived from the master equation. The master equation and the gap equation respectively give universal, exact values of two critical exponents, -y = 1 and p = 1. Exact scaling relations are established among the critical exponents of LC avalanches: T, D, y, p, a and i? Two exact equations, depicting the under- and over-critical LC iii it ~ DOCTORAL DISSERTAHON avalanche respectively, yield two infinite series of exact equations. ~Yu and ~3, two non-critical exponents, are determined by the configurations of the system. The exact explicit form of scaling function for LC avalanche size distri- bution is obtained from the master equation. Avalanche moments (Sk) is defined and calculated analytically. LC avalanche moments yield universal, exact values of critical exponents, -y~ = k(k = 0, 1, 2...), confirmed by sim- ulations. The asymptotic behavior of scaling function around the critical state yields the universal, exact value of the scaling exponent: /3~ = 1. Interaction strength, denoted by cq, is proposed to prescribe the degree of interaction among species in an ecosystem. ai is related to the inter- action factor, a, introduced in the generalized evolution model. a~ = 0 and ai = 1 corresponds to NIB model and Bak-Sneppen evolution model, respectively. 0