species, but does not swamp the intrinsic competitive dynamics on the island. The interaction matrix takes the form $d_{i,j} = C c_{i,j}$, where C sets the overall strength of the interaction and the $c_{i,j}$-s are normalized such that $$\sum_{i \ne j} c_{i,j} = Q(Q-1)$$ In other words, the average magnitude of a $c_{i,j}$ is unity. We consider here the case where the $c_{i,j}$ are chosen randomly from a distribution with unit mean and variance $\sigma^2$; the $c_{i,j}$ values are kept fixed throughout the process. The current work considers the case of purely competitive community without symbiosis or food web features. Accordingly, we do consider here the case of interaction matrices with a modular or nested structure. As we shall show be, this structureless matrix provides a natural generalization of the Hubbell neutral theory [@Hubbell2001unifiedNeutral]. For our simulations, the $c_{i,j}$ were chosen from a Gamma distribution with probability distribution function $$P(c) = \frac{c^{1/\sigma^2-1}\sigma^{-2/\sigma^2} e^{c/\sigma^2}}{\Gamma(1/\sigma^2)}$$ The final form of our GCLV model is then $$\label{GLV1}
\dot{N}_i = \lambda +  N_i - \frac{N_i}{K} \left( N_i + C
\sum_{j\ne i} c_{i,j} N_j \right).$$

For small immigration rate $\lambda$, extinctions and recolonizations play a crucial role in the dynamics. We treat this by constructing a 