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On multi-type Cannings models and multi-type exchangeable coalescents
1Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.
Theoretical Population Biology
|February 17, 2024
Summary
This study analyzes multi-type population models with migration and mutation. As population sizes grow, ancestral processes converge to a multi-type coalescent, revealing insights into population genetics.
Area of Science:
- Population Genetics
- Mathematical Biology
- Stochastic Processes
Background:
- Cannings population models are foundational for studying genetic drift and population structure.
- Understanding multi-type models is crucial for analyzing populations with distinct subpopulations or types.
Purpose of the Study:
- To analyze a multi-type neutral Cannings population model with migration.
- To investigate a related multi-type Cannings model incorporating mutation and varying subpopulation sizes.
- To examine the asymptotic behavior of these models as population sizes increase.
Main Methods:
- Analysis of a multi-type neutral Cannings model with fixed subpopulation sizes and migration.
- Application of coalescent theory, extending single-type models.
- Decomposition of transition probabilities into reproduction and migration components.
- Forward and backward time analysis of a model with mutation and stochastic subpopulation sizes.
Main Results:
- Convergence of the time-scaled ancestral process to a multi-type coalescent under specific conditions.
- Demonstration of exchangeability and consistency properties in the multi-type coalescent.
- Identification of multi-type limiting branching processes for large population sizes in the forward-time analysis.
- Characterization of the backward structure for the model with mutation.
Conclusions:
- The study provides a rigorous mathematical framework for understanding multi-type population dynamics.
- The findings contribute to the theoretical underpinnings of population genetics, particularly for structured populations.
- Asymptotic results offer insights into the behavior of large, complex populations over evolutionary time.
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