The theory is very simple. The world is divided up into many patches, all of which are potentially habitable. Populations on inhabited patches go extinct with a density-independent probability, e. Occupied patches all contain the same population density, and produce migrants (propagules) at a rate m per patch. Empty patches are colonized at a rate proportional to the total density of propagules and the availability of empty patches that are suitable for colonization. The response variable is the proportion of patches that are occupied, p. The dynamics of p, therefore, are just gains minus losses, so
At equilibrium dp/dt = 0, and so
giving the equilibrium proportion of occupied patches p* as
This draws attention to a critical result: there is a threshold migration rate (m = e) below which the metapopulation cannot persist, and the proportion of occupied patches will drift inexorably to zero. Above this threshold, the metapopulation persists in dynamic equilibrium with patches continually going extinct (the mean lifetime of a patch is 1/e) and other patches becoming colonized by immigrant propagules.
The simulation produces a moving cartoon of the occupied (light ...