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.
DOI : 10.1051/ps/2014019
Mots clés : Population with competition, extinction time, total mass of genealogical tree
@article{PS_2015__19__172_0,
author = {Vi, Le and Pardoux, Etienne},
title = {Height and the total mass of the forest of genealogical trees of a large population with general competition},
journal = {ESAIM: Probability and Statistics},
pages = {172--193},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2014019},
mrnumber = {3386369},
zbl = {1333.60188},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2014019/}
}
TY - JOUR AU - Vi, Le AU - Pardoux, Etienne TI - Height and the total mass of the forest of genealogical trees of a large population with general competition JO - ESAIM: Probability and Statistics PY - 2015 SP - 172 EP - 193 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014019/ DO - 10.1051/ps/2014019 LA - en ID - PS_2015__19__172_0 ER -
%0 Journal Article %A Vi, Le %A Pardoux, Etienne %T Height and the total mass of the forest of genealogical trees of a large population with general competition %J ESAIM: Probability and Statistics %D 2015 %P 172-193 %V 19 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2014019/ %R 10.1051/ps/2014019 %G en %F PS_2015__19__172_0
Vi, Le; Pardoux, Etienne. Height and the total mass of the forest of genealogical trees of a large population with general competition. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 172-193. doi : 10.1051/ps/2014019. http://www.numdam.org/articles/10.1051/ps/2014019/
, The continuum random tree I. Ann. Probab. 19 (1991) 1–28. | MR | Zbl
W.J. Anderson, Continuous-Time Markov Chains. Springer, New York, MR1118840 (1991). | MR | Zbl
, and , The effect of competition on the height and length of the forest of genealogical trees of a large population, in Malliavin Calculus and Related Topics, a Festschrift in Honor of David Nualart. Springer Proc. Math. Statis. 34 (2012) 445–467. | MR | Zbl
M. Ba and E. Pardoux, Branching processes with interaction and generalized Ray Knight theorem. To appear in Ann. Inst. Henri Poincaré Probab. Statist. (2015). | MR
, Almost sure comparison of birth and death processes with application to M/M/s queueing systems. Queueing Systems 1 (1986) 103–127. | MR | Zbl
, , , , and , Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37 (2009) 1926–1969. | MR | Zbl
and , Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2012) 813–857. | MR | Zbl
O. Kallenberg, Foundations of Modern Probability. Springer, New York (1997). | MR | Zbl
S. Karlin and H.M. Taylor, A First Course in Stochastic Processes, 2nd ed. Academic Press, New york (1975). | MR | Zbl
, and , Trees under attack: a Ray Knight representation of Feller’s branching diffusion with logistic growth. Probab. Theory Rel. Fields 155 (2013) 583–619. | MR | Zbl
E. Pardoux, Markov processes and applications. Wiley Series in Probability and Statistics. John Wiley and Sons, Ltd., Chichester, Dunod, Paris (2008). | MR | Zbl
and , From Brownian motion with a local time drift to Feller’s branching diffusion with logistic growth. Elec. Comm. Probab. 16 (2011) 720–731. | MR | Zbl
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer, New York (1999). | MR | Zbl
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