  We analyse metapopulation dynamics in terms of an individual-based,
stochastic model of a finite metapopulation. We suggest a new approach, using
the number of patches in the population as a large parameter. This approach
does not require that the number of individuals per patch is large, neither is
it necessary to assume a time-scale separation between local population
dynamics and migration. Our approach makes it possible to accurately describe
the dynamics of metapopulations consisting of many small patches. We focus on
metapopulations on the brink of extinction. We estimate the time to extinction
and describe the most likely path to extinction. We find that the logarithm of
the time to extinction is proportional to the product of two vectors, a vector
characterising the distribution of patch population sizes in the quasi-steady
state, and a vector -- related to Fisher's reproduction vector -- that
quantifies the sensitivity of the quasi-steady state distribution to
demographic fluctuations. We compare our analytical results to stochastic
simulations of the model, and discuss the range of validity of the analytical
expressions. By identifying fast and slow degrees of freedom in the
metapopulation dynamics, we show that the dynamics of large metapopulations
close to extinction is approximately described by a deterministic equation
originally proposed by Levins (1969). We were able to compute the rates in
Levins' equation in terms of the parameters of our stochastic, individual-based
model. It turns out, however, that the interpretation of the dynamical variable
depends strongly on the intrinsic growth rate and carrying capacity of the
patches. Only when the growth rate and the carrying capacity are large does the
slow variable correspond to the number of patches, as envisaged by Levins. Last
but not least, we discuss how our findings relate to other, widely used
metapopulation models.
