Height and the total mass of the forest of genealogical trees of a large population with general competition

¹ Department of Mathematics, Hanoi National University, 334 Nguyen Trai Str., Hanoï, Viêt Nam.
levi121286@gmail.com
² Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France.
etienne.pardoux@univ-amu.fr

Received: 30 July 2013
Revised: 30 October 2013

Abstract

Consider a continuous time branching process, which is integer or real valued (in the latter case it is called a continuous state branching process, and we restrict ourselves to the class of Feller branching diffusions) which models the time evolution of a population, to which we superimpose an interaction between the branches (which destroys the branching property). In the case of a large population, the interaction is of the type of a competition, which increases the individual death rate. We give precise conditions on the competition term, in order to decide whether the extinction time (which is also the height of the forest of genealogical trees) remains or not bounded as the ancestral population size tends to infinity, and similarly for the total mass of that forest of genealogical trees.