structure. Our primary focus is on an individual based stochastic version of the model which incorporates demographic noise. We will see that the model, despite its simplicity, is very rich and exhibits a wide range of different behaviors. Our goal here is to exhibit this panoply of “phases" and understand their origins. A parallel study of the much more tractable deterministic version of the model will be a key tool in unraveling the dynamics.

In particular, we shall show below that the most celebrated models of island biogeography - the Wilson-MacArthur theory of island biogeography [@macarthur1967theory; @losos2009theory] and Hubbell’s neutral theory of biodiversity [@Hubbell2001unifiedNeutral; @TREE2011] are two special cases of this model. Following a few recent publications that emphasized some specific aspects of the dynamics [@fisher2013niche; @pigolotti2013species], we would like to show how the phase structure of the model is governed to a large extent by these two special limits. Finally, we will consider the relevance of this model to the empirical findings.

The stochastic GCLV model
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In this section we introduce our model and set notation. Let us consider a regional pool of $Q$ species on the mainland. Each individual of these species may immigrate to the island 