Mathematical model for the size distribution of fish schools

Abstract

A mechanistic approach to the size distribution of fish schools foraging in patchy environment is presented. Because many schools nomadize in a limited area, interaction between schools occurs so that two of them meet and join, or so that a school splits into smaller ones. This problem is assumed to be described by a Markov process model, where it is asked about the probability that a fish engages in schools of social size N. The stationary distribution of school size which is realized with an overwhelming probability is given by the use of the H-theorem subject to two constraints: the probability distribution is normalized and the population-mean size of school is kept fixed constant. In order to determine the joining or splitting rates of schools, the schooling behavior is mechanistically investigated by examining the function on the foraging for patchily distributed food resources. The most frequently observed size of schools is likely to be regulated by food supply. Faced with a decreasing food supply, the most probable size becomes larger. Moreover, the existence of the optimal school size maximizing the long-term average rate of per capita net energy intake is elucidated. This paper deals with the connection between individual-level performance and patterns at the population level, offering the method to couple the ecologically scaled schooling behavior (i.e., school size distribution) with the environmental condition (i.e., food supply). Data on the size distribution of schools are fitted to the theoretical result fairly well.