LENARZ ET AL.: YIELD PER RECRUIT OF ATLANTIC YELLOWFIN TUNA 



4. Size-specific rates of fishing mortality — 

 COHORT. 



5. Yield-per-recruit isopleths for multigear 

 fisheries with size-specific F — MGEAR. 



6. Optimum size at recruitment under differ- 

 ent levels of effort by two gears — OPSIZE. 



ANALYSIS 



As previously mentioned in the introduction, 

 we use two approaches in analyzing the data, 

 the knife-edged recruitment approach and the 

 size-specific F approach. 



Knife-Edged Recruitment Approach 

 Introduction 



Two commonly used models for computing 

 yield per recruit and determining the size at 

 recruitment which maximizes yield per recruit 

 are those of Beverton and Holt (1957) and 

 Kicker (1958). We employed both models for 

 knife-edged approximation analyses — the sim- 

 plified Beverton and Holt model, making use of 

 a wide range of parameter estimates or extra- 

 polations from fisheries for similar species, and 

 the Ricker model, making use of the best param- 

 eter estimates and giving a more detailed an- 

 alysis of yield per recruit. We used the Ricker 

 model instead of the Beverton and Holt model 

 for calculating yield-per-recruit isopleths be- 

 cause the Ricker model allows the use of expo- 

 nents in the length-weight relationship with 

 values other than 3. It is important to stress 

 that the material in the simplified Beverton 

 and Holt model involves fewer assumptions 

 than the material in subsequent sections. This 

 is important because as our approach becomes 

 more complex the data requirements become 

 more rigorous. It can be argued that we have 

 sufficient data for this simplified approach. In 

 the more complex approaches this assertion be- 

 comes more tenuous; because we use more as- 

 sumptions in the more complex approaches we 

 do not necessarily obtain more information, 

 even though it may appear that way. However, 

 it should be noted that the assumption of a 

 constant rate of mortality over the fishable life 

 span contained in the simplified approach may 

 be important, and we believe that it is not ful- 

 filled. These analyses are followed by sections 

 discussing the problems of determining the 



proper parameters which represent the cur- 

 rent situation of the fishery. 



Simplified Beverton and Holt Model 



The Beverton and Holt yield-per-recruit 

 model may be simplified such that relative yield 

 per recruit, Y\ is a function of three ratios: 



C = i,:iL^ 



Q = MIK 



E = FI(F + M) 



Y'= YI(RW^) 



and where // is the size (length) at recruit- 

 ment, W^ , L^ , and K are parameters of the 

 von Bertalanffy growth equation, Y is yield in 

 weight, and R is recruitment. Y' is tabulated in 

 Beverton and Holt (1966), but more extensive 

 calculations were performed with program 

 YPER.i" Beverton and Holt (1959) concluded 

 that, within reason, there exists a common 

 ratio between M and K within related species 

 groups. Therefore, a range of estimates for the 

 various parameters is utilized along with other 

 information obtained by examining parameter 

 estimates for M and K for yellowfin tuna from 

 areas other than the Atlantic. 



The range of values for the various parameters 

 is as follows: K = 0.28 to 0.53 and L^ = 175.2 

 to 223.0 cm from LeGuen and Sakagawa (1973), 

 Z = 0.91 to 1.82 from Lenarz and Sakagawa 

 (1972, see footnote 5), and M = 0.6 to 1.0. From 

 these ranges of e.stimates, a maximum range for 

 E is 0.0 to 0.67 and for Q is 1.13 to 3.57. Using 

 our most reasonable parameter estimates of K 

 = 0.42, M = 0.8, and Z = 1.4, however, a rea- 

 sonable range for E and Q was established by 

 allowing either the numerator or denominator 

 of the ratio to be one of our most reasonable 

 estimates — the reasonable ranges are E = 0.12 

 to 0.56 and Q = 1.42 to 2.86. With K = 0.42, 

 M = 0.8, and Z = 1.4, our most reasonable es- 

 timates of £■ and Q are 0.43 and 1.9. respectively. 

 Table 1 contains optimal values of size (cm) 

 at recruitment, /*/, for the maximum range 

 of estimates of E and Q (deleting the impossible 

 E = 0.0) for the range and most reasonable es- 

 timates of L^. The dashed lines enclose the 



,1 f.^n.A"'"^ ^^'"^^ °f '•• Table lib of Beverton and 

 Holt (1966) was slightly higher than computed by YPER- 

 this may be due to differing methods of rounding 



41 



