  We consider the approximation of the equilibrium of a metapopulation model,
in which a finite number of patches are randomly distributed over a bounded
subset $\Omega$ of Euclidean space. The approximation is good when a large
number of patches contribute to the colonization pressure on any given
unoccupied patch, and when the quality of the patches varies little over the
length scale determined by the colonization radius. If this is the case, the
equilibrium probability of a patch at $z$ being occupied is shown to be close
to $q_1(z)$, the equilibrium occupation probability in Levins's model, at any
point $z \in \Omega$ not too close to the boundary, if the local colonization
pressure and extinction rates appropriate to $z$ are assumed. The approximation
is justified by giving explicit upper and lower bounds for the occupation
probabilities, expressed in terms of the model parameters. Since the patches
are distributed randomly, the occupation probabilities are also random, and we
complement our bounds with explicit bounds on the probability that they are
satisfied at all patches simultaneously.
