Cannings models, population size changes and multiple-merger coalescents.

Area of Science:

• Population Genetics
• Mathematical Biology
• Stochastic Processes

Background:

• Kingman's n-coalescent is a standard model for population genealogy, but it does not capture complex population structures.
• Multiple-merger coalescents, such as [Formula: see text]-n-coalescents, offer a more flexible framework for modeling diverse population histories.
• [Formula: see text]-n-coalescents arise as limit processes of discrete population models (Cannings models) with rescaled time and population size.

Purpose of the Study:

• To generalize the construction of time-changed [Formula: see text]-n-coalescents.
• To demonstrate how population size fluctuations in discrete models translate to time changes in continuous coalescent limits.
• To extend the understanding of coalescent theory to populations with more complex demographic histories.

Main Methods:

• Analysis of limit processes of discrete Cannings models with variable population sizes.
• Construction of time-changed [Formula: see text]-n-coalescents through stochastic calculus and probability theory.
• Mathematical derivation of coalescent dynamics under arbitrary time changes.

Main Results:

• A general method for constructing time-changed [Formula: see text]-n-coalescents is presented.
• This construction applies to a broader class of [Formula: see text]-n-coalescents than previously shown.
• The results explicitly link arbitrary time changes in discrete population models to the continuous coalescent limit.

Conclusions:

• Time-changed [Formula: see text]-n-coalescents provide a powerful and general tool for modeling population genealogies.
• This work extends the applicability of coalescent theory to populations with complex and fluctuating demographic histories.
• The findings facilitate more realistic simulations and analyses of genetic diversity in diverse populations.