TEGNER ET AL.: BIOLOGY AND MANAGEMENT OF RED ABALONES 



Table 2 shows the regression constants from the 

 predictive regression of the natural logarithm of 

 fecundity on the natural logarithm of length for each 

 data set, and the range of lengths encompassed by 

 each data set. Because the methods were widely dif- 

 ferent among the various sets and because the ab- 

 solute numbers of eggs were not considered impor- 

 tant, the four sets were combined in the following 

 way. Each data set was scaled by dividing the ob- 

 served number of eggs by 1 million times the num- 

 ber of eggs predicted for a length of 125 mm, from 

 Table 2. The length 125 mm was chosen because it 

 was most nearly common to all four data sets. The 

 constant 1 million was used to minimize distortion 

 when taking natural logarithms of the scaled data. 

 Then the predictive regression of the natural log- 

 arithm of fecundity on the natural logarithm of 

 length was calculated using data from all four sets. 

 The constants obtained are also shown in Table 2. 



The validity of this procedure was tested by using 

 the egg-per-recruit model described below and com- 

 paring results from each of the fecundity regressions 

 in Table 2. Results obtained usingM = 0.15, F = 1.0 

 and 2.0, and minimum legal sizes of 179 and 191 mm 

 are also shown in Table 2. The low variation between 

 estimates using different fecundity relations, and 

 the general similarity of the isopleths, allowed us 



to conclude that egg-per-recruit analysis was not 

 sensitive to our treatment of the fecundity data. 



Egg-Per-Recruit Analysis 



In egg-per-recruit modelling, one wishes to com- 

 pare the egg production of a fished population with 

 that of the equilibrium virgin population. The model 

 described here is a simple, deterministic, age-struc- 

 tured model that allows individual variability in 

 length around mean length-at-age. Thus cohorts can 

 be partially recruited to the fishery. 



The unfished population is considered first. The 

 number of females of a particular size within a 

 cohort of an unfished population is represented as 

 NV, j, where t indexes cohort age and j indexes 

 length. Each length hj lies within one of the J 

 intervals 



{hj - wl2, hj + 



wl2); j = I, J 



where hj = h^ + (j - l)w is the midpoint of thejth 

 interval and each interval has width w. UNq is the 

 abundance of females at age zero, the female abun- 

 dance of each cohort in the unfished population is 



NV,, = No exp(-Mt) 



(1) 



Table 2.— Fecundity analysis. 



Predictive linear regression constants for equations relating fecundity to lengtfi in the four data 

 sets described in tfie text. Tfie equations are 



In (fecundity) = a + b \n (length) 

 where length is in mm. The final set of constants describes the equation for the combined data 

 - see text for the method used. 



Data set 



Range of lengths 



Results of egg-per-recruit modelling using the regression estimates above. Natural mortality rate 

 was set to M = 0.15. Fishing mortality rate F and minimum legal size (MLS) were vaned as shown. 

 Results are expressed as percentages of egg production in the virgin population. 



1.0 



F = 2.0 



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