School Size Statistics of Fish

A system of a fixed population size is considered in which fish schools break up and unite with other schools. The size distribution of schools is investigated on the basis of a balance equation, which corresponds to the mean-field theory of Smoluchowski-equation model of the coagulation–fragmentation process. The rates of fission and fusion are determined from a simple dynamic viewpoint of schooling. The size distribution, in effect, follows a power law up to a cutoff size, which can be fitted to data. The power index and the cutoff size depend on the population size. It is also elucidated how statistical properties of the system are regulated by the total population size. As the population size increases the number of schools increases, and asymptotically approaches a fixed value. If the population size is large, the mean school size depends linearly upon the population size. The standard deviation of the school-size distribution is proportional to the mean school size which is checked with data.