Statistics of Certain Models of Evolution

Abstract

In a recent paper, Newman surveys the literature on power law spectra in evolution, self-organised criticality and presents a model of his own to arrive at a conclusion that self-organised criticality is not necessary for evolution. Not only did he miss a key model (Ecolab) that has a clear self-organised critical mechanism, but also Newman's model exhibits the same mechanism that gives rise to power law behaviour as does Ecolab. Newman's model is, in fact, a ``mean field'' approximation of a self-organised critical system. In this paper, I have also implemented Newman's model using the Ecolab software, removing the restriction that the number of species remains constant. It turns out that the requirement of constant species number is non-trivial, 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-organise 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 counter example to the presence of self-organised criticality in evolution, but does provide a simple, almost analytic model that can used to understand more intricate models such as Ecolab.