The Effect of Competition on the Height and Length of the Forest of Genealogical Trees of a Large Population

Abstract

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 μ, deaths at constant rate λ) a death rate which is γ times the size of the population alive at time t raised to some power α > 0 (α = 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 and choose adequately μ, λ and γ 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 (Ann Probab 19:1–28, 1991)]. 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 α > 1, while the expectation of the length of the tree remains bounded as the size of the initial population tends to infinity iff α > 2.

AMS Subject Classification 2000:(Primary) 60J80, 60F17 (Secondary) 92D25.

Received 11/27/2011; Accepted 4/15/2012; Final 6/1/2012