The dynamics of group formation

A general continuous model is presented for animal group size distribution. Attention is restricted to a fixed size population divided into groups of various dynamic sizes, but the approach extends easily to populations of variable size. The basic idea is to relate group size distribution to two functions, the (density-dependent) rates of fusion and fission. These functions can be estimated from data and can ultimately be related to the behavior of individuals and the dynamics of groups. For various functional forms, the stationary distributions of group sizes are sought. In several prototype cases, the stationary distribution has a peak value, the "most frequent group size," which emerges endogenously from the dynamics. The authors determine when such a peak emerges and more generally show the existence and uniqueness of the stationary distribution. Stability of stationary solutions is discussed. Progress is shown, but a general treatment remains refractory.