with a certain probability per unit time, and we denote the average rate at which individuals of the $i$th species reach the island as $\lambda_i$.

Denoting by $N_i$ the abundance of the $i$th species on the island, the deterministic part of the dynamics satisfies $$\label{GLV}
\dot{N}_i = \lambda_i + \alpha_i N_i - \frac{N_i}{K_i} \left( N_i +
\sum_{j\ne i} d_{i,j} N_j \right).$$ Here $\alpha_i$ is the growth rate of the $i$-th species on the island, and $K_i$ sets the carrying capacity of the $i$-th species *in the absence of competition with other species*. Interspecific interactions are expressed by the elements of the matrix $d_{i,j}$. Here we study a purely competitive system where all the $d_{i,j} \ge 0$.

As our goal is to identify the different phases of this model, not to fit it to a specific empirical system, we make a few simplifications. First, we assume (as in [@bell2000distribution], for example) that all species have the same flux of immigrants from the mainland, $\lambda_i = \lambda$, the same linear growth rate that we scale to one ($\alpha_i=1$) and the same carrying capacity $K$. We are interested here specifically in small values of $\lambda$, so that immigration primarily serves to “rescue" extinct 