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Two-type branching processes with immigration, and the structured coalescents

Part of: Markov processes

Published online by Cambridge University Press:  29 September 2025

María Emilia Caballero ,
Adrián González Casanova Open the ORCID record for Adrián González Casanova [Opens in a new window]  and
José Luis Pérez Open the ORCID record for José Luis Pérez [Opens in a new window]
María Emilia Caballero*
Affiliation:
Universidad Nacional Autónoma de México
Adrián González Casanova*
Affiliation:
Arizona State University
José Luis Pérez*
Affiliation:
Centro de Investigación en Matemáticas
*
*Postal address: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Área de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, Coyoacán 04510, Cuidad de México, México. Email: marie@matem.unam.mx
**Postal address: School of Mathematical and Statistical Sciences, Wexler Hall, 901 Palm Walk Room 216, Tempe, AZ 85281. Email: agonz591@asu.edu
***Postal address: Departamento de Probabilidad y Estadística, Centro de Investigación en Matemáticas, A.C. Calle Jalisco S/N C.P. 36240, Guanajuato, Mexico. Email: jluis.garmendia@cimat.mx
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Abstract

We consider a population consisting of two types of individuals, each of which can produce offspring on two different islands (in particular, the islands can be interpreted as active or dormant individuals). We model the evolution of the population of each type using a two-type Feller diffusion with immigration and study the frequency of one type on each island, when the total population size on each island is forced to be constant at a dense set of times. This leads to the solution of a stochastic differential equation, which we call the asymmetric two-island frequency process. We derive properties of this process and obtain a large population limit as the total size of each island tends to infinity. Additionally, we compute the fluctuations of the process around its deterministic limit. We establish conditions under which the asymmetric two-island frequency process has a moment dual. The dual is a continuous-time two-dimensional Markov chain that can be interpreted in terms of mutation, branching, pairwise branching, coalescence, and a novel mixed selection–migration term.

Keywords

Feller diffusion with immigrationasymmetric two-island frequency processesmoment duality

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D15: Problems related to evolution 92D25: Population dynamics (general)

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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