Mutant Fixation for a Stochastic Evolutionary Model in Fragmented Populations

Population fragmentation is a common feature of many biological systems.

Understanding mutant fixation in such systems is challenging because the underlying stochastic dynamics are high-dimensional.

In this work, we develop a general mathematical framework for analyzing stochastic evolution in fragmented populations connected by rare migration.

The framework is sufficiently general to accommodate heterogeneous deme sizes, deme-dependent birth and death processes, and migration on arbitrary strongly connected directed networks with asymmetric migration rates.

We show that, in the limit where migration occurs on a much slower timescale than within-deme dynamics, the full stochastic process can be reduced to a lower-dimensional Markov chain whose states correspond to configurations of fully mutant and fully wild-type demes.

The reduction theorem establishes that fixation probabilities and absorption times of the original process are asymptotically determined by the corresponding quantities of a reduced chain.

As an application, we derive explicit formulas for mutant fixation probabilities and fixation times in fragmented populations initiated by the introduction of a single mutant.

The results provide a general and tractable approach for studying evolutionary dynamics in complex fragmented populations.