# Critical Exponent of Species-Size Distribution in Evolution

C. Adami

California Institute of Technology

http://www.krl.caltech.edu/~adami/
Ryoichi Seki

CalState Northridge and California Institute of Technology

Robel Yirdaw

CalState Northridge

### Abstract

We analyze the geometry of the species-size distribution in evolving and adapting populations of single-stranded self-replicating genomes: here programs in the Avida world. We find that a scale-free distribution (power law) emerges only in complex landscapes that achieve a separation of two fundamental time scales: the relaxation time (time for population to return to equilibrium after a perturbation) and the time between mutations that produce fitter genotypes. These scales are set by a number of parameters, most importantly by the type of landscape the population is adapting to and by the mutation rate. We show that the distribution for flat landscapes is exponential rather than of power-law type (geometric) and that this also occurs in the limit of large mutation rates. By extrapolating to zero mutation rates, we determine the critical exponent of the abundance distribution in the critical regime where the relaxation time is much smaller than the time between mutations producing fitter genotypes.