only does the number of species in the deterministic solution decrease, but a significant fraction of these have very small abundances. The upshot is that the number of species in the stochastic simulation with significant abundance decreases significantly. This is reflected in the sharp fall of the IPR in this range of $C$.

![The time averaged abundance (open circles) vs. species number for an extended run of the second panel of Fig. \[fig all\]. $C=0.25$. $N=20$, $K=100$, $\sigma=0.5$, $\lambda=0.01$. Also shown is the deterministic time-independent solution (filled squares). The lines are shown just to guide the eye.[]{data-label="figdetvsmean"}](MeanvsDetN20p5p25){width="45.00000%"}

Phase III: The “Chaotic" Phase
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At some point, $C_{2}$, the partial exclusion picture breaks down. This is clear from an examination of the fourth and fifth panels of Fig. \[fig all\]. There is no longer a fixed set of “resident" species which are always present in large numbers. Instead, there is a constant turnover in the set of species present. This is reflected in the rise seen in the IPR in Fig. \[figIPR\] for $C\gtrsim 0.45$. Another way to see the same point is to examine what we will term the “occupancy" of each species, the fraction of time that it is 