Speciation and Extinction in a simple model of evolution

We introduce a simple model of macro-coevolution which allows the branching and termination of species lines and also incorporates external influences to the ecosystem. The strength of the external influences and the likelihood of speciation and extinction are defined from the fitness landscapes by two new parameters, $\delta g$ and $\delta s$. Results from numerical simulations show that the total number of species fluctuates about a natural system size $N_{\infty}$. We present a mean-field theory which predicts $N_{\infty}\propto(K-1)\delta s/\delta g^{2}$, where $K-1$ is the system connectivity and $\delta s$ is small. This result compares well with the numerical simulations. For large $\delta s$, we demonstrate why this expression changes to $N_{\infty}\propto(K-1)\delta s^{2}/\delta g^{2}$. We compare the model to the fossil record, and comment on the role of ecological niches in models of evolution.