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

Abraham, Romain; Delmas, Jean-François

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 226-238.

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.

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.

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