for the length $Q$ column vector consisting of all one’s, and $B$ is the matrix $$B_{i,j}=\delta_{i,j} + C c_{i,j}.$$

Figure \[fig2\] illustrates the process. For $C$ small enough that coexistence fixed point solution is physical, such that all $N_i$ are positive, there is a unique stable solution and all the $Q$ mainland species will be present on the island. Their abundances decrease as a function of $C$, but no species goes extinct in the deterministic theory. As long as this remains true, for reasonably large $K$, one can more or less neglect the noise, given the stability of the fixed point solution. The species that have lower abundance are those who suffer more from the competition, and even if demographic noise drives a few of them to extinction, the effect on the rest of the network is minor and the system will restore itself by immigration. This situation is illustrated by the first panel of Fig. \[fig all\], the case $C=0.05$, where there are no extinctions seen in the time-frame shown. Examined over a much longer period, there are indeed a few occasional extinctions of all the various species. The most extinction prone species, for example, was seen to go 