Mutant Fixation for a Stochastic Evolutionary Model in Fragmented Populations
Abstract
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.
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BibTeX
@misc{fu2026mutant,
title = {{Mutant Fixation for a Stochastic Evolutionary Model in Fragmented Populations}},
author = {Yi Fu and Natalia L. Komarova},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.23946},
eprint = {2606.23946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23946},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Mutant Fixation for a Stochastic Evolutionary Model in Fragmented Populations
AU - Yi Fu
AU - Natalia L. Komarova
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2606.23946
UR - https://arxiv.org/abs/2606.23946
AB - 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.
ER -