Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Changing the branching mechanism of a continuous state branching process using immigration

Romain Abraham and Jean-François Delmas

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Abstract

We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type, this construction is a natural way to model neutral mutation. It also provides in some sense a dual construction of the particular pruning at nodes of continuous state branching process introduced by the authors in a previous paper. For a critical or sub-critical quadratic branching mechanism, it is possible to explicitly compute some quantities of interest. For example, we compute the Laplace transform of the size of the initial population conditionally on the non-extinction of the whole population with immigration. We also derive the probability of simultaneous extinction of the initial population and the whole population with immigration.

Résumé

Nous considérons une population initiale dont la taille évolue selon un processus de branchement continu. Nous ajoutons ensuite à ce processus une population migrante (qui évolue selon le même mécanisme de branchement que la population initiale), avec un taux d’immigration propotionnel à la taille de la population totale. Nous montrons que ce processus de branchement continu avec immgration proportionnelle à sa taille est encore un processus de branchement continu. En voyant cette immigration comme l’apparition d’un nouveau type, cette construction est un moyen naturel de modéliser des mutations, neutres vis à vis de l’évolution. Elle peut être également vue comme la construction duale de l’élagage aux noeuds de l’arbre généalogique associé à la population totale, introduit par les auteurs dans un article précédent. Lorsque le mécanisme de branchement est quadratique et critique ou sous-critique, il est possible de calculer explicitement certaines quantités intéressantes. Par example, nous calculons la transformée de Laplace de la taille de la population initiale conditionnellement à la non-extinction de la population totale. Nous en déduisons également la probabilité d’extinction simultanée de la population initiale et de la population totale.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 1 (2009), 226-238.

Dates
First available in Project Euclid: 12 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1234469979

Digital Object Identifier
doi:10.1214/07-AIHP165

Mathematical Reviews number (MathSciNet)
MR2500236

Zentralblatt MATH identifier
1171.60374

Subjects
Primary: 60G55: Point processes 60J25: Continuous-time Markov processes on general state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx]

Keywords
Continuous state branching processes Immigration process Multitype populations

Citation

Abraham, Romain; Delmas, Jean-François. Changing the branching mechanism of a continuous state branching process using immigration. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 226--238. doi:10.1214/07-AIHP165. https://projecteuclid.org/euclid.aihp/1234469979


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