## Separation of Time Scales and Convergence to

the Coalescent
in Structured Populations

*Magnus Nordborg and Stephen M. Krone*
**Abstract**

Many biological phenomena can be thought of, and modeled as generalized population structure,
with individuals belonging to different states. Certain groups of states in such models are
often connected by ``migration'' that occurs on a time scale that is much faster than the
coalescent time scale. We show that, when viewed on the coalescent time scale, the structure
associated with such groups of states collapses, and is replaced by a kind of averaging over
the states. When the entire structure collapses, the standard coalescent is retrieved.
The effect of the population structure on the coalescent is then captured by a simple scaling
factor, that is related to various notions of ``effective population size.'' It is also
possible for parts of the structure to collapse, leading to a simpler structured coalescent.