Statistics of certain models of evolution

In a recent paper, Newman [J. Theo. Bio. 189, 235 (1997)] surveys the literature on power law spectra in evolution, self-organized criticality and presents a model of his own to arrive at a conclusion that self-organized criticality is not necessary for evolution. Not only did he miss a key model $($Ecolab) that has a clear self-organized critical mechanism, but also Newman’s model exhibits the same mechanism that gives rise to power law behavior, as does Ecolab. Newman’s model is, in fact, a “mean field” approximation of a self-organized critical system. In this paper, I have also implemented Newman’s model using the Ecolab software, removing the restriction that the number of species must remain constant. It turns out that the requirement of constant species number is nontrivial, leading to a global coupling between species that is similar in effect to the species interactions seen in Ecolab. In fact, the model must self-organize to a state where the long time average of speciations balances that of the extinctions; otherwise, the system either collapses or explodes. In view of this, Newman’s model does not provide the hoped-for counterexample to the presence of self-organized criticality in evolution, but does provide a simple, almost analytic model that can be used to understand more intricate models such as Ecolab.