FISHERY BULLETIN: VOL. 79, NO. 2 



distribution as is evident by the similarity of the 

 curves for relative values cr, = 0.1 and 1 (Figure 8). 



DISCUSSION 



The frequency distribution of fish school diam- 

 eters observed acoustically off the coast of south- 

 ern California by Smith (1970) has a well defined 

 most frequent, or peak, diameter. The distribution 

 is also skewed towards small diameters and is 

 relatively stable from one month's observations to 

 the next. These data likely represent a range of 

 fish sizes and species with northern anchovy prob- 

 ably being the dominant group. 



The frequency distribution can be modeled by a 

 probability equation that is based on a dynamic 

 equation that contains deterministic and stochas- 

 tic rates for the entrance and exit of fish from a 

 school. The entrance rates are taken to be inde- 

 pendent of the number offish in a school while the 

 exit rates are taken to be proportional to the 

 number. The number of fish in a school is trans- 

 formed to a diameter by assuming the average 

 school shape is disklike so the number is propor- 

 tional to the horizontal area, as expressed by the 

 square of the diameter. 



The effects of environmental conditions, fish 

 sizes, species, predator-prey interactions, and 

 stock size are assumed to be contained in the 

 stochastic parameters of the dynamic equation. 

 This assumption requires these basically un- 

 known factors have white noise character. 



In effect the deterministic behavior of the rate of 

 change of diameter is represented by a dynamical 

 equation, dxidt = fix), where fix) is the deter- 

 ministic rate and is a function of jc. The remaining 

 unknown fluctuating behavior, due to other fac- 

 tors, is approximated by eix)jit) where jit) is 

 white noise fluctuation and eix) gives the x depen- 

 dence of the stochastic rate. Combining the rates 

 we obtain a dynamical stochastic equation for x 

 and the probability analysis of the process can be 

 carried out using a Fokker-Planck equation, 

 which is a diffusion equation for probability in x 

 and t. The solution of the Fokker-Planck equation 

 gives the probability curve PiX). 



Fitting the curve PiX) to Smith's (1970) obser- 

 vations yields the equation constants or fitting 

 parameters a, b, and c. These fitting parameters 

 are ratios of the dynamic parameters of the 

 dynamical stochastic equation for x. 



The sensitivity analysis of PiX) to relative 

 changes in the dynamic parameters indicates two 



basic probability distributions can be produced: 1) 

 a narrow probability distribution, favoring a nar- 

 row range of small diameters with the occurrence 

 of large schools unlikely and 2) a wide distribution 

 in which a wide range of larger diameters have low 

 but essentially equal probabilities, and small di- 

 ameters are unlikely. Wide distributions are fa- 

 vored by large entrance rates and a large amount 

 of randomness to the schooling process. Narrow 

 distributions are favored by large exit rates and 

 low randomness in the schooling process. Addi- 

 tionally, wide distributions are favored for schools 

 with a low fish density per cubic meter, and narrow 

 distributions of diameters are favored with high 

 density schools. The density of fish in schools is 

 related to the fish length so the analysis infers that 

 large fish should have a wide probability distribu- 

 tion of large diameters and small fish should de- 

 velop a narrow probability distribution of small 

 diameters. 



From a commercial fishing viewpoint factors 

 that affect school sizes are important and so, 

 briefly, we consider a possible qualitative response 

 of school diameter to fishing activity. If we envi- 

 sion the fishing process as an event that divides a 

 school and removes one of the fractions, then we 

 expect fishing should at least affect the deter- 

 ministic parameters of the model. The dividing of 

 the school, by fishing, decreases the mean time 

 interval between school divisions and this, in turn, 

 would increase the exit rate coefficient /3. The fact 

 that part of the stock is removed by fishing may 

 increase the time interval between school encoun- 

 ters and thus decrease the entrance rate a. A con- 



06 T 



100 



Figure 9.— Probability distribution (P) vs. school diameter (X) 

 in meters. Curve A, distribution corresponding to Smith's ( 1970) 

 observations. Curve B, distribution postulated for fishing activ- 

 ity that increases /3 and decreases « by 507f . 



322 



