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Branching processes with interactions: subcritical cooperative regime

Part of: Markov processes Special processes

Published online by Cambridge University Press:  17 March 2021

Adrián González Casanova ,
Juan Carlos Pardo  and
José Luis Pérez
Adrián González Casanova*
Affiliation:
Universidad Nacional Autónoma de México
Juan Carlos Pardo*
Affiliation:
Centro de Investigación en Matemáticas
José Luis Pérez*
Affiliation:
Centro de Investigación en Matemáticas
*
*Postal address: Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, 04510, México, D.F. Email address: adriangcs@matem.unam.mx
**Postal address: Centro de Investigación en Matemáticas, A.C., Calle Jalisco s/n, Col. Valenciana CP 36023 Guanajuato, Gto., México, Apartado Postal 402, CP 36000.
**Postal address: Centro de Investigación en Matemáticas, A.C., Calle Jalisco s/n, Col. Valenciana CP 36023 Guanajuato, Gto., México, Apartado Postal 402, CP 36000.
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Abstract

In this paper, we introduce a family of processes with values on the nonnegative integers that describes the dynamics of populations where individuals are allowed to have different types of interactions. The types of interactions that we consider include pairwise interactions, such as competition, annihilation, and cooperation; and interactions among several individuals that can be viewed as catastrophes. We call such families of processes branching processes with interactions. Our aim is to study their long-term behaviour under a specific regime of the pairwise interaction parameters that we introduce as the subcritical cooperative regime. Under such a regime, we prove that a process in this class comes down from infinity and has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype, and whose parameters have a classical interpretation in terms of population genetics. The moment dual is an important tool for characterizing the stationary distribution of branching processes with interactions whenever such a distribution exists; it is also an interesting object in its own right.

Keywords

Branching coalescing processesstabilitymoment dualitypopulation genetics

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 251 - 278
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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