species have equal abundance, its value is $Q$. Thus, the IPR is a measure of how many different species are active in the system. It can differ enormously from the time average of the instantaneous number of species present. We show in Fig. \[figIPR\] the IPR as a function of $C$ for runs with the same parameters as in Fig. \[fig all\]. We see that the IPR is not monotonic in $C$. It initially decreases from $Q$, reaches a minimum and then starts to increase. It then achieves a local maximum and then starts to decrease again. This change in behavior of the IPR is clearly reflective of the different patterns captured in Fig. \[fig all\]. We shall elaborate on the behavior of the IPR as we investigate each phase.

![The inverse participation ration, $\textrm{IPR}$, as a function of $C$. $N=20$, $K=100$, $\sigma=0.5$, $\lambda=0.01$.[]{data-label="figIPR"}](invprN20){width="45.00000%"}

Phase I: The heterogeneous coexistence phase
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We start by considering the leftmost region of Fig. 1, the area where $C$ is relatively small. Increasing $C$ from zero, at a fixed value of $\sigma$, corresponds to an increase of interspecific competition; for example, different species of birds that leave happily together, each having a different diet 