The Size of Elephant Herds

Abstract

Populations of large mammals such as elephants tend to group themselves into herds. As these herds wander about, it may happen that two of them meet and amalgamate, or that a large one may split into two smaller ones, (see John Hillaby, “Elephants as a pest control problem”, New Scientist, 12 (1961), 736-8). Suppose that a given area contains n animals, and let pk (t) denotes the probability that at time t they are grouped into k herds. Conditional on this, let x1(t), …, xk(t) denote the sizes of the herds arranged say, in non-increasing order. The chance that two herds will meet and amalgamate will increase with the number of herds, and it may, for instance be taken to be proportional to the excess of this number over unity. Thus the probability of an amalgamation in the interval (t, t + δt), given that there are k herds at time t, is