FISHERY BULLETIN: VOL. 84, NO. 4 



where r 2 = t - t r . 



The biomass per recruit in fish whose growth can 

 be described by Equation (7) is, according to the 

 model of Beverton and Holt (1957; see also Ricker 

 1975, p. 253): 



| = W m (A, + A 2+ A 3 + A A ) 



(21) 



where A l = 



A 9 = 



A 9 = 



and 



A A = 



1 - e~ Zr 3 



-3 e~ Kr * (1 - e ^ z+K ^) 

 Z + K 



3 e -2gr 4 (! _ e -(Z + 2K)r^ 



Z + 2K 



_ e -3Kr A fa _ e -(Z+3K)r 3 } 



Z + 3K 



where r 3 = £ max - t r 



r A = t, - U. 



This model assumes, as does Equation (20), a stable 

 age distribution. 



Combining Equations (21) and (20) leads to the 

 model for estimating Q/B, which has the form: 



APPLICATION EXAMPLE AND 



SENSITIVITY ANALYSIS OF 



THE MODEL 



In the following application examples, the newly- 

 derived model (Equation (22)) is used to compare the 

 food consumption of a tropical carnivore (Epine- 

 phelus guttatus) with that of a tropical herbivore 

 (Holacanthus bermudensis). A list of the parameter 

 values used is given on Table 2. 



The solutions of Equation (22), inclusive of the in- 

 tegration of its numerator, were obtained by means 

 of a short BASIC microcomputer program available 

 from me. Note that the integration, which according 

 to Equation (22) should be performed for the inter- 

 val between two ages (t r and £ max ), can be per- 

 formed for the intervals between two sizes (W r , 

 W max ), the age corresponding to these sizes being 

 estimated from the inverse of Equation (7) ; i.e., 



t = t - ((UK) (log, (1 - WIWJ™)). (23) 



The results, i.e., the values of Q/B, expressed as 

 a percentage on a daily basis are 0.76 for E. gut- 

 tatus and 2.50 for H. bermudensis. 



A sensitivity analysis of Equation (22) was per- 

 formed, following the outline in Majkowski (1982). 

 The results are given in Figure 3, which shows that 

 of the six parameters of Equation (22), (3 is the one 

 which has the strongest impact on the estimates of 



Q 



B 



-max 



Qir f q - exp(-/Cr 1 )) 2 • exy(-(Kr 1+ Zr 2 )) 

 6K J 1 - (1 - exv(-Kr{))W " at 



(Ax + A 2 + A 3 + A 4 ) 



Equation (22) has only 6 parameters (K, t , t r , 

 t max , Z, and P); of these, K and t are estimated 

 from growth data, while t r and t max can be set more 

 or less arbitrarily (see text below and Figure 3). 

 Total mortality (Z), which is here the equivalent of 

 a production/biomass ratio (see Allen 1971) can be 

 estimated easily, e.g., from length-frequency data 

 and growth parameters (see Pauly 1982, 1984a: 

 chapter 5) and is an input required anyway by the 

 ECOPATH program (Polovina 1984). Thus only p 

 and a "hidden" value of W^ applicable to both food 

 experiment and growth data are needed in addition 

 to the easily obtainable parameters required by this 

 model. 



(22) 



Q/B, while t r has the least, the relationships be- 

 tween the importance of these parameters being 

 best summarized by 



p > K> Z » t max > to > t r 



(24) 



These results suggest that, when using this model, 

 most attention should be given to an accurate esti- 

 mation of p (see below). It should be also noted that 

 P and K have opposite effects on the estimation of 

 Q/B (see Figure 3). Thus, a biased (e.g., high) 

 estimate of W x will be associated with too low 

 values of p and K which partially compensate each 

 other. 



832 



