are not supported anymore on the island. Weak competitors (species that suffer from strong competition against others that do not suffer as much) are selected out.

This is the deterministic picture without migration. With migration, “exclusion" does not correspond to exactly zero density. Instead, above the May transition the deterministic density of these species is $\cal{O}(\lambda)$. Since we assume that $\lambda$ is small, demographic noise induces frequent extinction of these transient species. As opposed to the other, “semi-resident" species (the “semi-" prefix taking account of the possibility of a short-lived absence due to demographic fluctuations), the growth rate of a transient species immigrant is negative, rendering its average persistence times small, independent of $K$.

Technically speaking, the May transition takes place at a critical value of the competition parameter $C_1$, where the smallest $N_i$ reaches zero. Above the transition one can define a reduced system, eliminating the row and column of $B$ corresponding to the eliminated species. This reduced system does admit a solution where all the (remaining) species have positive abundance. This is clear, since at the exact value of $C$ at which the next species vanishes, the remaining $N_i$’s constitute a solution of the reduced problem. In addition, 