ANDERSON: A STOCHASTIC MODEL OK FISH SCHOOL SIZE 



probability distribution at the most common, or 

 peak, diameter and one other diameter in the dis- 

 tribution. This fixes two of the free parameters and 

 with trial values of the remaining free parameter 

 the model is fit to observations. 



We begin by converting the probability equation 

 to the frequency equation 



F{X) 



P{K) 



(18) 



where F(X) is the number of fish observed at di- 

 ameter X and /j'"' is a constant that is a function of 

 the number of schools observed. At the most com- 

 mon diameter Xo, the frequency is maximum so 

 dF(Xo)ldX = and from Equations (16) and (18) 

 we obtain 



1 1 



a = Xl {b+-)--c^/Xl 

 4 4 



(19) 



With Equation (19) in FiX) the log of F(Z) yields 

 equations for b and k* . These can be solved 

 explicitly with observations F(Zo), F(Zi), Xo, Xi, 

 and the free parameter c where Xi is a diameter 

 greater than the peak diameter Xo. The equations 

 are 



A{X,)-A{X^) + C{X,)-C{Xo) 

 B(Xi)-B(Xo) 



k = exp -A{Xo) + bB{Xo) - C(Xo) 



where A(X.) = - (cVX^ -X^)/(c+X5) 



+ ln(X./(c+X]) 



B(X.) = 2(c+X2)/(c+x5) 



+ 21n(c+X^) 

 C(X,) = -ln(F(X.)) 



with i = and 1. 



(20) 



(21) 



For the composite of observations depicted in 

 Figure 2 we takeXo = 14 m,Xi = 40 m,F(Xo) = 

 491, andF(X,) = 115. A good fit (Figure 3) is ob- 

 tained withe = 60 m^ and the remaining constants 

 from Equations (19), (20), and (21) area = 133 m^fo 

 = 0.452, and ^* = 4716244. 



I y T 



4 -■ I \ 



;00 



ii y y  • 



1 -■ / 



A,+ 



.. J 



+, 



"t..> 



I 



-.. + 

 +■"■-.. 



-f-. 



H 1 H 



20 



H 1 1 H 



'--^^^=^ 



4 I? 



X 



S 1 



Figure .3. — Fit of probability equation of school diameter to 

 frequency distribution of Figure 2. Probability Equation (16) is 

 equated to frequency according to Equation (18). 



SENSITIVITY ANALYSIS 



The sensitivity of P(X) to variations in the 

 model parameters has been investigated for P(X) 

 in the configuration of the observed distribution 

 (Figure 3). 



We note that the curve P(X) is defined by three 

 free parameters a, 6, and c which are in effect 

 fitting parameters. These are ratios of the coeffi- 

 cients of the stochastic dynamic Equation (10). The 

 coefficients are the dynamic parameters of the sys- 

 tem and are a, fi, a, cr,, and p. The relationship 

 between the fitting parameters and the dynamic 

 parameters is given by Equation (17) where cr, is 

 related to r by Equation (9). Because the dynamic 

 parameters are only known in ratios in this model 

 we can only investigate their effect on PiX) in 

 terms of relative changes. The relative value y', of 

 a dynamic parameter y, can be defined 



y/yo 



(22) 



where yo is the value of the dynamic parameter 

 corresponding to the fit to the observed frequency 

 distribution. 



To investigate the equation sensitivity, each 

 dynamic parameter is varied while the others are 

 held constant at their yo values. For each set of 



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