684 



Fishery Bulletin 89(4), 1991 



Figure 2 



Growth rings in a cross section of a Haliotis iris shell. 



Mortality estimation 



Age-length keys (Ricker 1975) were constructed from 

 the ring-count data. The keys were used to estimate 

 the catch curve from the length-frequency sample for 

 each region; total mortality rate was then estimated 

 from the regression of the natural log of estimated 

 number-at-age on 'age' (Robson and Chapman 1961). 

 The first 'age' included in the procedure was chosen 

 to represent the first 'age' class with a mean length 

 greater than minimum legal size. 



Total mortality rates were also estimated from the 

 length-frequency samples using the method of Four- 

 nier and Breen (1983), using a strategy similar to that 

 of Tegner et al. (1989). To obtain the fishing mortality 

 rate F, we subtracted an assumed value of M = 0.10, 

 based on the work of Sainsbury (1982), from the total 

 mortality rate Z. Natural mortality rate is difficult to 

 estimate, especially in fished populations (Shepherd and 

 Breen 1991); Sainsbury used several methods to 

 estimate M in a population protected from fishing. The 

 method of Fournier and Breen (1983) can estimate 

 growth and mortality rates simultaneously but is bet- 

 ter applied by constraining the growth estimates, to 

 be consistent with independent estimates. We applied 

 this method to length-frequencies from the northern 

 Marlborough Sounds and D'Urville I., using the growth 

 estimates obtained from mark-recapture data in those 

 regions. This method could not be applied to the other 



regions because we had no independent estimates of 

 growth. 



In using the Fournier and Breen (1983) method, the 

 assumed number of age classes N was determined by 

 examining results using different values of N. We chose 

 the highest value for which the procedure estimated 

 a substantial proportion of the population to be con- 

 tained within each age-class. At unrealistically high 

 values of N, the procedure estimates that some age- 

 classes are "empty." The parameter NFULL (the 

 index of the first cohort whose abundance is used in 

 estimating mortality) was chosen with reference to the 

 mean cohort length and the minimum legal size. 

 Estimated standard deviations of lengths-at-age 

 around mean length-at-age were constrained to values 

 near 4.1, based on examination of length-frequencies 

 in which cohort modes were clear (Schiel, unpubl. data). 

 L M was unconstrained except for D'Urville I., where 

 a minimum constraint was required to obtain estimates 

 consistent with the mark-recapture data. Population 

 proportions were left unconstrained unless the pro- 

 cedure made unrealistically high estimates for single 

 age-classes. Variance of mean lengths from the von 

 Bertalanffy growth curve and variance of the estimated 

 population proportions from an exponential decay 

 curve were both constrained to a maximum of 1.0. Sen- 

 sitivity of the procedure to changes in N, NFULL, K, 

 and the standard deviations of length-at-age was ex- 

 amined by varying these parameters (see Tegner et al. 

 1989). 



Modeling 



Yield-per-recruit (YPR) modeling was done on a 

 spreadsheet incorporating the YPR model of Ricker 

 (1975). We report results from two regions: the nor- 

 thern Marlborough Sounds and D'Urville I. Length- 

 weight relations were calculated from field sampling 

 results (Table 1). Egg production modeling was done 

 for these regions using a simple spreadsheet model of 

 the form described by Sluczanowski (1984, 1986). The 

 additional information required for this modeling is the 

 length-fecundity relation. Relevant data are available 

 from Poore (1973), Sainsbury (1982), and Wilson (1987). 

 Based on the size range of paua in each of these 

 sources, we used Wilson's data to derive 



Eggs = (9.32 x 10- 12 ) x shell length (mm) 8 - 408 . 



The natural mortality rate M was assumed to be 0.10. 

 We assessed the sensitivity of the results to this 

 assumption by varying M. For the northern Marlbo- 

 rough Sounds, we also varied M and the size-at-first- 

 capture and developed the response surface of YPR 



