FISHERY BULLETIN: VOL 79. NO. 2 



the ratio of their "incremental standard devia- 

 tions." These are not the typical standard devia- 

 tions, but if the correlation functions q, and qj are 

 similar the ratio of the incremental standard de- 

 viations should approach the ratio of the typical 

 standard deviations of the variables, i.e., 

 r^SDii{t))/SD ijit)) as qiit'Vqjit')^!. 



Now the stochastic dynamical equation for fish 

 school diameter is wTitten 



dX 

 dt 



a 



/3X Jit) 



2pX 



IpX 



+ X 



(10) 



o 



M2 (X) = — (r/pX + Xy . ( 13) 



4 



From the similarity of frequency distributions for 

 different months (Figures 1, 2) we assume a 

 steady-state probability distribution and from 

 Goel and Richter-Dyn (1974) this is expressed as 



P{X) = 



M2{X) 



exp 



{{ 



2 ds 



ll4) 



The probability characteristics of X can be ana- 

 lyzed according to the Fokker-Planck equation 

 ( Goel and Richter-Dyn 1974). This is also known as 

 the forward diffusion equation for probability in X 

 and t. The probability density for the system hav- 

 ing a diameter X at time t when it had diameter Y 

 at time zero is denoted P{X/Y,t^ and for Equation 

 (10) the Fokker-Planck equation is 



mx,t) 



+  



a 



1 

 2 



2 a 



3X 



a 



8 dX\pX 



IpX 



+ x' 



-/3X 



P{X, t) 



where C is a constant determined by the condition 

 that the total probability equals one 



1 = r PiX)dX 







(15) 



Using Equations (12) and (13) in Equation (14) the 

 steady-state probability distribution, or probabil- 

 ity density, for school diameter is 



PiX) 



kXe 



-2{a + bc)Kc+X^) 



(^+^2)(l-25) 



where 



(16) 





(11) 



The term a^ is the incremental variance of the 

 diameter shrinkage rate and is in fact equivalent 

 to a^j in Equation (7). The term has a dimension of 

 t ~^ and is also the diffusion coefficient of probabil- 

 ity P(X,t) in X space. In this manner it quantifies 

 the level of randomness in the schooling process. 



For the Fokker-Planck equation the growth rate 

 of the mean value of X is 



Mi(X) 



a 



/3X 



2pX 



a 



+ — (r/pX+X)(l-r/pX2) 

 2 



and the growth rate of the variance of X is 

 318 



(12) 



a = Oi/po^ , b = i3/CT^, c = r/p. 



k = AC/o' 



(17) 



The dimensions of the above constants with / for 

 length, t for time, and n for number offish are as 

 follows. The parameters a and c have dimensions 

 of /^, b is dimensionless, and k has dimensions of 

 ^1+46 rpj^g dimension of a is /^ ^ /^ and o-^ have 

 t-\ p has rt/-^ r has n, and C has r^l^ + *^ . 



FITTING THE MODEL TO DATA 



Equation (16) can be fit to Smith's (1970) data 

 through a number of methods all of which adjust 

 the free parameters a,b, and c to obtain a best fit 

 according to visual or statistical criterion. To ob- 

 tain a first order estimate of the free parameters 

 we will use a simple algorithm in which the proba- 

 bility curve is made to go through the observed 



