stochastic individual based version of the model, so that demographic noise is explicitly included. The number of individuals in each species is an integer. At each time step, of duration $\Delta$, a Poisson number of immigrants of each species, with mean $\lambda \Delta$, is generated. A Poisson number of offspring of each species, with mean $N_i \Delta$, is generated as well. The $N_i$ veteran inhabitants are subject to death, with the number of individuals of species $i$ that expire drawn from a binomial distribution with parameters $N_i$ and probability $\Delta \sum_j (\delta_{i,j} + Cc_{i,j})N_j $. Clearly, $\Delta$ needs to be chosen to be sufficiently small that this probability does not exceed unity. The number of individuals of species $i$ after this process is then updated to reflect the new immigrants, offspring and deaths. On average, these changes are exactly those given by the deterministic model.

The stochastic model is specified by four parameters: $K$, $\lambda$, $C$ and $\sigma$. In what follows we will focus our attention on the phases in the $\sigma-C$ plane, keeping $K$ and the migration rate fixed. Once this behavior is understood it is straightforward to figure out at least the qualitative features of the dynamics for 