generic behaviors, which we shall call “phases". Very briefly, in the first, low $C$, phase all $Q$ species are present essentially all the time. In the second, some species are basically no longer present, supported only by the infrequent stochastic immigration of new individuals, while the rest have a more or less continuous existence, with occasional temporary extinctions. The third phase at yet higher $C$ is the most complicated, with the system jumping from one state to another. These states would be stable in the absence of immigration, but a successful colonization by some specific species drives the system to a new quasi-stable state. In the last phase, the system spends a preponderance of time in some stable state. In the following, we will attempt to explicate in more detail the various features of each of these phases.

![image](N20p05gallery1){width="85.00000%"}

One way to quantify the different behaviors is via the “inverse participation ratio", IPR, defined as $$\textrm{IPR} \equiv \frac{\left(\sum_{i=1}^Q \langle N_i  \rangle \right)^2 }{\sum_{i=1}^Q \langle N_i \rangle ^2}$$ The angle brackets refer to an average over time. The IPR varies from 1 to $Q$. In the case where only one species is present, it takes the value unity, and if all 