The dotted black line to the right of the solid black line is the same line for $K=200$, $\lambda=0.01$. The rightmost, blue, line marks the region where the system spends more than $20\%$ of its time in a particular state, for $K=100$, $\lambda=0.01$. The May line was determined by averaging over 100 different realizations of the interaction matrix. The other lines result from averages over 10 or 20 matrices.[]{data-label="phase"}](N20phase){width="45.00000%"}

Increasing $C$ in this manner in the partial coexistence phase, one obtains a nested hierarchy of solutions, each with less diversity, which are immune to invasion by any of the extinct species. As mentioned, this hierarchy is completely independent of the carrying capacity $K$. This prescription eventually breaks down. At some point, there is a “resurrection" of one of the eliminated species. This happens due to the fact that the species in question was strongly suppressed by another species. As $C$ increases, this suppressor species is itself reduced in abundance, and so the suppressed species is able to stage a comeback. Thus there is a value of $C$ at which this suppressed species is able to invade. At this point, it needs to be added back to the reduced system, since 