FISHERY BULLETIN: VOL. 79, NO. 2 



I y y -►• 



4 06 -■ 



; n y 



1 - • 



+ + + + + + + + + 

 1 1 (- — \ H 1 H 1 — ^H± — ^ 



fi 2 fi 4 fi fi fi 8 1 



X 



Figure 2. — Frequency distribution offish schools (F) vs. school 

 diameter (X) for composite of Smith's (1970) sonar surveys off 

 California and Mexico in January, February, April, May, June, 

 and Julv 1969. 



mental ways. Either the frequency distribution is 

 coupled to the school size or it is coupled to other 

 factors such as the species composition of the ob- 

 servations. In the first case, the factors relating to 

 environmental conditions, species, and stock 

 numbers may also be acting on the school size but 

 could be hidden in the averaging, inherent in the 

 frequency mode of data display. In the second case, 

 size may be insignificant and the distribution 

 might reflect other factors. For example, if each 

 species has a preferred school size, then a particu- 

 lar frequency distribution of species could produce 

 an apparent frequency size correlation. 



In this paper I explore the situation where the 

 schooling dynamics are dependent on the school 

 size and other factors, like species composition, 

 have effects randomly distributed over the range 

 of school sizes. 



OBSERVATIONS 



Smith (1970) observed fish schools over a 

 200,000 nmi^ area between San Francisco, Calif, 

 and Cabo San Lazaro, Baja California. The targets 

 were recorded using sonar with a transducer fixed 

 at a 90° relative bearing giving a beam perpen- 

 dicular to the path of the ship. A 30 kHz frequency 

 was used with a 10° conic beam (at -3 dB), at 

 ranges from 200 to 450 m from the ship. Observa- 

 tions were made during daylight hours only and a 

 complete survey of the area could be made in <2 

 mo. The dimensions of the schools were measured 

 at right angles and parallel to the ship and the 

 number of schools in 5 m intervals of diameter was 



estimated, correcting for bias from schools that 

 only fell partially within the 200-450 m observa- 

 tion window of the sonar. 



Frequency distributions for surveys in May and 

 June 1969 were composed of 525 and 650 observa- 

 tions, respectively, in the diameter range 0-99 m 

 (Figure 1). A composite frequency distribution of 

 surveys in January, February, April, May, June, 

 and July 1969 contained 2,549 observations of 

 schools between and 99 m diameter (Figure 2). 



Smith (1970) estimated to a first order that 

 about 30% of the observed schools were adult 

 northern anchovy. Other common schooling fish in 

 the area, the CalCOFI area, include northern an- 

 chovy juveniles; jack mackerel, Trachurus sym- 

 metricus , ju-veni\es; Pacific bonito, Sarda chilien- 

 sis; Pacific mackerel. Scomber Japonicus; and 

 Pacific sardine, Sardinops sagax. 



THE MODEL 



To describe the frequency distribution of school 

 diameters first consider an equation for the rate of 

 change of the number offish in a school. Define 

 this rate in terms of a deterministic equation, 

 which is related to school size, plus a stochastic 

 equation which is taken to approximate the re- 

 maining unknown fluctuating behavior of the 

 rate. The deterministic and stochastic equations 

 are combined to give a stochastic dynamic equa- 

 tion for the rate of change offish in a school as 



dN 

 dt 



a + 3{t)-fiN + y(t)N. 



(1) 



The number offish in a school is N and its change 

 with time is determined as the difference between 

 the rate fish enter the school, which is independent 

 of N, and the rate fish exit from the school, which 

 is proportional to N. The deterministic entrance 

 and exit rates are a and fiN where a and f3 are 

 constant and represent averages over an ensemble 

 of schools. The stochastic entrance and exit rates 

 are S(^) and y(t)N where 8(t) and y(t) represent 

 white noise fluctuations which vary rapidly com- 

 pared with variations in A^, and are not affected by 

 past conditions. The mean values of the stochastic 

 terms are zero and so the stochastic parts repre- 

 sent fluctuations about the deterministic rate 

 terms. 



To express the frequency distribution in terms of 

 school diameters we note Squire's (1978) observa- 

 tions in which the average shape of northern an- 



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