TEGNER ET AL.: BIOLOGY AND MANAGEMENT OF RED ABALONES 



N,,.=N,^,,_S,_,. (13) 



The number of individuals in any cohort t and 

 length interval J can now be determined if a further 

 assumption is made. It is assumed that the length 

 distribution of all recruited individuals within a 

 cohort t is a truncated normal distribution, whose 

 mean is L, as described by Equation (2). In reality, 

 the recruited segment of an older cohort will com- 

 prise several groups of individuals, each group hav- 

 ing been exposed to fishing for a different length 

 of time. Thus the length distribution of recruits in 

 a cohort is unlikely to be normal. However, it is not 

 possible to specify their distribution without speci- 

 fying how varying annual length increments are 

 distributed among individuals in a cohort. This is 

 complex and requires further assumptions, so in- 

 stead we have made the simplifying assumption 

 described. We believe that the results are not sensi- 

 tive to this assumption. 



The number of individuals in each cohort t and 

 each length interval j is given by 



N,,j = NV,^^ j<R (14) 



N,_^ = Q,,jNR,,J{l - QPR,) j^R 



and total annual egg production is 



E 



i. J 



= 11 iv,,7;. 



(=1 ; = m ■' ■" 



(15) 



Initial trials showed that this model is not unduly 

 sensitive to the choice of a and b in Equation (3). 

 The model was run with the same growth param- 

 eters used in the yield-per-recruit analysis; A = 25; 

 a and b in Equation (3) were set at 8.0 and 0.025 

 based on inspection of Figure 3; c and d in Equa- 

 tion (6) from the combined analysis in Table 2; and 

 hfWas set to 50. It is assumed that male and female 

 growth rates are the same. This may not necessar- 

 ily be the case (Doi et al. 1977; Shepherd and Hearn 

 1983), but we have no information on sexual dif- 

 ferences in growth of red abalones. F. M, and the 

 minimum legal size h^ were all varied in the same 

 way as in the yield-per-recruit analysis. 



Egg-per-recruit analyses were also made for pink 

 abalones, using the same growth data from females 

 used in the yield-per-recruit analysis. The fecundity 

 relation was obtained from Tutschulte (1976). It was 

 assumed that the number of eggs is proportional to 

 weight (2,078 eggs per gram whole weight in mature 

 females); and that weight is proportional to length: 



weight (g) = 2.66 x 10"'^ length (mm)='-2^ 



Lamda was set at 25; in the absence of data, a and 

 6 in Equation (3) were set to the same values as for 

 red abalones; M was set at 0.2; and minimum size 

 and F were varied as for yield-per-recruit analysis. 



Simultaneous Analysis 



Results from the yield- and egg-per-recruit anal- 

 yses were combined in a third analysis to examine 

 the relative performance of each combination of 

 minimum size and fishing mortality rate (Breen 

 1986). Results from each of the previous two analyses 

 are rescaled and then simply multiplied together: 



SV = (y/y max) EPV 



(16) 



where SV is the "strategy value" of a particular 

 combination of minimum size and fishing mortality 

 rate, and takes values from to 100; Y is the yield- 

 per-recruit at that combination; ymax is the max- 

 imum yield-per-recruit; and EPV is transformed 

 value of egg-per-recruit, given by 



EPV = 0.0 



E < 20.0 (17) 



EPV = (3.333 E) - 66.6667 20.0 <E < 50.0 



EPV = 100.0 E > 50.0 



where E is the egg production at that combination, 

 expressed as a percentage of the egg production in 

 the unfished situation. The relation of Equation (17) 

 is designed so that when egg production is less than 

 20% of the virgin egg production, SV is zero; when 

 E is greater than 50% of virgin egg production, SV 

 is equal to the scaled value of yield-per-recruit. The 

 values 20% and 50% are chosen as the values below 

 which most managers would consider egg produc- 

 tion to be dangerously low and above which egg pro- 

 duction is probably adequate to maintain the stock, 

 respectively. 



This analysis was also carried out for pink 

 abalones. 



RESULTS 



Field Studies 



Population Structure 



The size-frequency distributions of red abalones 

 recovered in the destructively sampled quadrats 



321 



