A STOCHASTIC MODEL FOR THE SIZE OF FISH SCHOOLS 



James Jay Anderson^ 



ABSTRACT 



A model is presented that reproduces the frequency distribution offish school diameters observed 

 acoustically in waters off southern California. The rate of change of school diameter is described by an 

 equation thai nicludes an entrance rate offish into a school, which is mdependent of the numberof fish 

 in the school, and an exit rate, which is proportional to the number. The number in a school is assumed 

 to be proportional to the square of the school diameter implying the average shape is disklike. 

 Fluctuations in school size from unknown factors are approximated by stochastic rate terms for the 

 entrance and exit rates and the diameter-number relation. This gives a stochastic dynamic equation for 

 the rate of change of diameter and the probability distribution of the diameter is analyzed with a 

 Fokker-Planck probability equation. A sensitivity analysis indicates two basic distribution types occur. 

 Large exit rates and small stochastic fluctuations produce a narrow range of small diameters, while 

 large entrance rates and large fluctuations produce a wide range of large diameters. Qualitative 

 inferences from the model indicate schools of large fish should have a wide range of large diameters 

 while small fish should have a narrow range of small diameters. Also fishing activity could decrease 

 entrance rates and increase exit rates, and the combination would shift the probability distribution to a 

 narrow range of small diameters. 



When fish are mutually attracted they form 

 schools, either in random orientation or in highly 

 organized structures (Shaw 1978). The shape of 

 the schools are diverse and changeable, and typi- 

 cally range from ribbonlike to spherical with the 

 latter being uncommon (Radakov 1973). The at- 

 traction is mostly keyed visually and in northern 

 anchovy, Engraulis mordax, schools off southern 

 California the shape is disklike during the day and 

 generally more diffuse and elongated at night 

 (Squire 1978). Often within a school fish arrange 

 in a lattice structure with the density of fish per 

 unit volume related to fish length (Breder 1976; 

 Serebrov 1976). 



Models for the interactions of fish in a school 

 have been postulated by a number of authors (see 

 Breder 1976 for review and Okubo et al. 1977), but 

 the processes controlling the size of a school in 

 terms of the number offish in a school or its physi- 

 cal dimensions have not been considered in 

 mathematical terms. Considering what factors 

 may be important in controlling size it is apparent 

 the problem is complex and could include species 

 behavior, light, predator-prey interactions, turbu- 

 lence, life cycle stages, the stock population, and 

 the size of the schools themselves. 



'Contribution No. 563 College of Fisheries, University of 

 Washington, Seattle, Wash. 



^Fisheries Research Institute, Universitv of Washington. 

 Seattle, WA 98195. . --v^ 



Manuscript accepted September 1980. 

 FISHERY BULLETIN: VOL. 79, NO. 2, 198L 



v3l^' 



An interesting set of observations on the size- 

 frequency distribution of unidentified schools off 

 southern California, shows a well-defined peak 

 frequency at a diameter of about 15 m (Smith 

 1970). Towards larger and smaller diameters the 

 frequency distribution decreases in an exponen- 

 tiallike manner (Figures 1, 2). 



This simple, and relatively stable, distribution 

 is particularly interesting considering the possi- 

 ble complexity of the schooling process. The obser- 

 vations might be produced in one of two funda- 



1 20 



1 -  



3 

 6 



4 



2 







+ « * ^ * 



-t 1 1 1 1 1 1 1 ^H H 



2 4 f-". ^1 



X 



S 1 



Figure l. — Frequency distribution offish schools (F) vs. school 

 diameter iX) for unidentified schools observed with sonar by 

 Smith (1970) from San Francisco, Calif., to Cabo San Lazaro, 

 Baja California, in May 1969 (Oi and June 1969 ( + >. F in num- 

 bers of schools and X in meters. 



315 



