Coalescence in the recent past in rapidly growing populations

Abstract

In a rapidly growing population one expects that two individuals chosen at random from the n th generation are unlikely to be closely related if n is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton-Watson branching process with an offspring distribution {p j } such that p 0 =0 and ψ(x)=∑ j p j I {j≥x} is asymptotic to x -α L(x) as x→∞ where L(·) is slowly varying at ∞ and 0<α<1 (and hence the mean m=∑jp j =∞) it is shown that if X n is the generation number of the coalescence of the lines of descents backwards in time of two randomly chosen individuals from the n th generation then n-X n converges in distribution to a proper distribution supported by ℕ={1,2,3,⋯}. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We show that if the offspring mean m satisfies 1<m≡∑jp j <∞ and p 0 =0 then the coalescence time X n does converge to a proper distribution as n→∞, i.e., coalescence does take place in the remote past.