of worms, may start to interact with each other if the supply of worms is decreased and different species begin to consume the same resource. In such a case species-specific niches are “squeezed" towards each other, increasing the niche overlap.

Once the species start to compete, the heterogeneity of the $c_{i,j}$’s (as reflected in the parameter $\sigma$) implies that some species are impacted by the competition more than the others. As a result, the abundance $N_i$ of the $i$th species is no longer $K$ (as on the Wilson-MacArthur line $C=0$); instead, all abundances are reduced by a species-dependent amount and one gets a distribution of species abundance values. Still, as long as $C$ is not too large, the deterministic dynamics of Eq. (\[GLV\]) supports a single attractive fixed point that corresponds to the case where all the $Q$ species of the mainland are represented on the island. In this case, we can approximate the system by the $\lambda=0$ system, since the small external flux does not qualitatively change the steady-state, which in any case has all species present. Eq. \[GLV\] with $\lambda
= 0$ allows for an explicit stationary state solution, $$N_i = K B^{-1} \mathbb{1}
 \label{steady}$$ where $\mathbb{1}$ stands 