Worthington et al.: Alternative size limits for Haliotis rubra in New South Wales, Australia 



553 



(a > 0, b < 0), logistic (a > 0, b = -1), and Gompertz 

 (a > 0, b = 0) models (Schnute, 1981). Francis (in 

 press) reparameterized this model to express ob- 

 served increments in length during tagging (AY") as 

 a function of the observed length at tagging (Y t ) and 

 time at liberty (At). The model can be repara- 

 meterized further, so that growth can be described 

 in terms of average annual growth for individuals of 

 a given size (see Francis, 1988a). The general model 

 where both a * and 6*0, was 



AY 



-Y l+ {Y t b 



+ c(l-e 



t 



where 



At 



and 



y 2 +g2- 



The model was fitted to the observed increments by 

 using maximum likelihood (Francis, 1988b). Param- 

 eters estimated during this process include the mean 

 annual growth rates (g 1 andg 2 ) at two sizes (y x and 

 y 2 ) and the parameter b. These parameters together 

 combine to define the parameter a, and hence the 

 shape of the fitted growth curve. 



Several other parameters were also considered, 

 and were included in the model if they significantly 

 improved the fit (see Francis, 1988b). Two param- 

 eters describing seasonal variation in growth were 

 examined by replacing At above with At + i(j) t - (f> t ) 

 where for any time it t ) 



_ u(sm(2n[t, -w])) 



and t, and t r are the times at tagging and recapture. 

 The parameters u and w then describe the ampli- 

 tude and phase of seasonality in growth, respectively. 

 A parameter describing variation in growth among 

 individuals was also examined, where the mean (u ) 

 and standard deviation (o ) of the expected incre- 

 ment in length were related by 



v <" £ 



where v is the coefficient of variation in growth. A 

 parameter describing contamination by outliers was 



also examined, but it never added significantly to 

 the model (see Francis 1988b). 



The error model used considered errors due to 

 variation in growth and measurement by 



AY 



obs 



AYe D 



where the observed increment in length (AY o6s ) was 

 a function of that predicted by the growth model (AY) 

 and errors due to variation in growth (e g ) and mea- 

 surement ie m ), assuming e g ~ Nil, o g ) and £^ ~ Niu m , 

 <J m ). As such, fi m represents measurement bias (i.e. 

 consistent difference in measurement between mark- 

 ing and recapture) and o m represents random error 

 in measurement. Since better estimates of the growth 

 parameters can be achieved when both \i m and o m 

 are known (Francis, in press), they were estimated 

 by repeatedly measuring the same set of shells in=50) 

 after the tagging program had finished. Measure- 

 ment bias was disregarded (^ m =0) because the same 

 people used identical techniques to measure abalone 

 at tagging and recapture. Random errors in measure- 

 ment among replicate readings and among readers 

 were similar in size, and as a consequence o m was 

 set to 0.65 for all analyses. Following fitting of the 

 model, plots of residuals against length at tagging 

 and time at liberty were investigated for any sys- 

 tematic lack of fit. 



The parameters^ andy 9 should be chosen to span 

 the lengths at tagging and to enable reliable esti- 

 mates ofgj and g 2 (Francis, 1988b). Because of dif- 

 ferences in the length of abalone tagged among the 

 sites, appropriate sizes for y x and y 2 also differed 

 among sites. To facilitate comparison, y 1 and y 2 for 

 each site were chosen from three standard lengths: 

 65, 90, and 115 mm. At sites where few small aba- 

 lone were tagged, annual growth rates were esti- 

 mated at 90 and 115 mm. Alternatively, at sites where 

 few large abalone were tagged, annual growth rates 

 were estimated at 65 and 90 mm. Once fitting of the 

 model was completed, the annual growth rate at the 

 third standard length was also calculated. This 

 growth rate is defined by the parameters chosen 

 during fitting of the model. Comparisons among es- 

 timated parameters were made by using £-tests. 



To facilitate the comparison of our estimates of 

 growth with those previously published, we also fit- 

 ted the traditional von Bertalanffy growth model. 

 This was done within the framework described above 

 (see Francis, 1988b). To further aid the comparison 

 of growth rates among studies, we converted the tra- 

 ditional von Bertalanffy parameters estimated from 

 tagging data (k and L m ) to parameters describing 

 estimates of annual growth at the standard sizes used 

 in this study. This was also done for previously pub- 



