Parrack: Estimating stock abundance from size data 



305 



The model suggested by Chapman (1961) and Rich- 

 ards (1959) may be used to include growth. Letting A, 

 m, b, and k be parameters, s the size, and t the time 

 from birth, the general model 



1 



St = (Ai-™ - b e-"^ t)i^ 



is the "logistic" function of Verhulst if m = 2, the Brody 

 (monomolecular, von Bertalanffy) model if m = 0, and 

 it approaches the Gompertz function as m approaches 

 unity. Using the rationale of Fabens (1965) where Si 

 is the size at time tj and S2 is the size at time t2 , the 

 above growth model leads to 



1 

 S2 = (A1-™ - (Ai-'"-Sii-"') e-Mt2-ti))T^. (1) 



This satisfies assumption (3), without reference to the 

 actual age of individuals, by expressing size as a con- 

 tinuous function of time, but if growth is intermittent 

 or has changed, a specialized model is most appropriate. 

 From (1), or a more suitable model, let 



I ' = the size of a fish on date yx that was size s on 



date yt, 

 u' = the size of a fish on date yx that was size s -t- 1 



on date yt, 

 a' = the size of a fish on date c^ that was size s on 



date yt, and 

 b' = the size of a fish on date Cj, that was size s -t- 1 



on date yt, 



where V , u', a', and b' fall in size-classes /, u, a, and 

 b. Including size in the abundance equation gives 



Nt,s = 



j Nx.wdwe^fyT-y.) -I- X f Ck.wdw 



(2) 



;z(<^k-yt). 



If size-classes are suitably narrow, the frequency of 

 size within size-classes tends to be proportional to size. 

 The frequency of size within a class is therefore approx- 

 imated by a trapezoid (i.e., trapezoidal integral approx- 

 imation). The number of fish within the size class is 



s+l 



F, = r f^dw = V2(s + l-s)(f, + fs^i) 



s 



= V2(f3-fs,l), 



where s is a size class, fg is the frequency at size s, 



and Fs is the number within size-class s. Let the 

 largest fish fall in class S: 



fg = VeFs because fg+i = 0. 

 Rewriting gives fs = 2 Fg. 



Proceeding to smaller sizes, 



Fs-i = V2(fs_i + fs) =V2(fs_i + 2Fs) 



sofs_i = 2(Fs_i-Fs). 



Fs-2 = V2(fs_2 + fs-i) =V2(fs_2 + 2Fs_i + 2Fs) 



sofs_2 = 2(Fs_2-Fs-i + Fs). 



F, = V2(f3 + fs.l) 



= V2(f, + 2F,,i-2F3,2+---±2Fs). 



Rearrangement gives the general expression for the 

 frequency at size-class bounds: 



fs = 2(Fs-Fs,i + F,,2-Fs.3 + • • • ± Fs). 



The frequency of any size, s, within class s is also 

 required: 



fs = f s + ^^^ (S'-S) = f s + (S'-S) (fs.i-fs). 

 S-l-l-S 



The approximate integrals for equation (2) are thus: 



\i\x = I: C 



Nt,w dw = 



V2(u'-0(l/ + (^'-0(lui-'7;) + (u'-0('7/+i -»];)■ 



or 



if u > ^: I Ntw dw = 

 I 



y2(l+l-r){r]i + il' -l)ir]ui-r]l) + m*l) +•■• 



. . .V2(u'-U)(2)1u-H(u'-U)(»1u.^i-I7u)) 



u-l 



+ I Nx.i, 



