## On the effect of competition on the height and length of the genealogical tree of a population

#### Etienne PardouxUniversité d'Aix-Marseille I (Université de Provence)

We consider a population generating a forest of genealogical trees in continuous time,
with m roots (the number of ancestors). In order to model competition within the population,
we superimpose to the traditional Galton-Watson dynamics (births at constant rate b, deaths at constant rate d)
a death rate which is c times to the size of the population alive at time t to some power r>1 (r=1 is a case without competition).

If we take the number of ancestors at time 0 to be equal to [xN], weight each individual by the factor 1/N,
choose adequately b, d and c as functions of N, then the population process converges as N goes to infinity
to a Feller SDE with a negative polynomial drift. The genealogy in the continuous limit is described by a real
tree (in the sense of Aldous).

In both the discrete and the continuous case, we study the height and the length of the genealogical tree, as an
(increasing) function of the initial population. We show that the expectation of the height of the tree remains
bounded as the size of the initial population tends to infinity iff r>1, while the expectation of the length of the tree
remains bounded as the size of the initial population tends to infinity iff r>2.

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