Haddon et al.: Using an inverse-logistic model to describe growth increments of Haliotis rubra 
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a probabilistic interpretation of the parameters, the 
von Bertalanffy and Gompertz curves predict negative 
growth increments at initial lengths greater than L x , 
and the inverse-logistic predicts ever decreasing growth 
increments as the initial length at tagging increases. 
Thus, a realistic representation of the final size distribu- 
tion of larger abalone is provided by the inverse-logistic 
model without the complexity of a probabilistic inter- 
pretation of the parameters. In Tasmania, the growth 
of small abalone, at least above 10 mm, appears to be 
linear-like and to increase in relatively constant growth 
increments through time (Prince et al., 1988; Gurney 
et al., 2005). Although the von Bertalanffy and Gomp- 
ertz equations can approximate linear-like growth over 
these small sizes, linear growth limits their capacity to 
describe accurately the growth of larger animals at the 
same time. The von Bertalanffy curve predicts a linear 
relationship between growth increment and initial shell 
length. The Gompertz equation, on the other hand, pre- 
dicts that small abalone would have very small growth 
increments that initially increase with initial length 
and then decline again. Neither of these alternatives is 
consistent with observations in Tasmania of linear-like 
early growth. 
Growth pattern 
The tagging data on growth increments of blacklip aba- 
lone from various sites around the south of Tasmania 
were able to be grouped according to similarity of growth 
pattern. All regions exhibited a similar pattern of mean 
growth increments that were well described by a sym- 
metric inverse-logistic curve. 
Negative growth increments observed in the tagging 
data were not taken to be evidence of negative growth, 
but were rather taken to be a reflection of measurement 
or recording errors, a possible chipping of shell edges 
during collection, or an increased chance of shell ero- 
sion in disturbed animals (or a combination of these 
possibilities). Because of this, when simulating growth, 
negative increments were not included. 
The tagging data were, in some cases, truncated ei- 
ther in the smaller or the larger sizes. The fishing mor- 
tality rate on legal-size abalone is high and numbers of 
animals much larger than the minimum legal length 
are significantly reduced. In addition, the cryptic nature 
of undersize abalone means that obtaining representa- 
tive data across the whole size range can be difficult. It 
is also possible that the tagging process could influence 
the subsequent rate of growth. Intuitively, if there were 
an impact, it would probably be a negative bias on the 
growth increments that would increase the variation ob- 
served (by extending growth into smaller increments). 
Despite the limitations of the data, the proposed sys- 
tem of two linked inverse-logistic curves proved capable 
of fitting and simulating data from three sites in south- 
ern Tasmania. The inverse-logistic model was fully 
capable of producing transition matrices with predicted 
values across the full range of size classes required 
(60 mm to 210 mm). The truncation of the available 
data by high levels of fishing pressure did have effects, 
however. Surprisingly, the more complex 6-parameter 
model did not always provide the most workable de- 
scription of growth because the fit with six parameters 
could over-emphasize missing data; that is, the absence 
of data could influence the fitted curve, especially when 
the flexibility of the 6-parameter curve was used. It can 
be argued that the simpler 4-parameter model provides 
a more useful description of growth because it is less 
likely to be influenced by peculiarities or limitations of 
the available data. For example, at the legal minimum 
size limit (136 mm shell length at the time of data 
collection), abalone from the Bruny Island region were 
growing an average of 8 mm per annum (with a range 
from 2 mm to 15 mm). However, fishing mortality rates 
at Bruny Island were very high and few legal-size ani- 
mals remained for long at this site with the result that 
the tagging growth increment data were sparse above 
140 mm. The 6-parameter model describes the specific 
pattern of growth in the data from Bruny Island, trun- 
cating any growth beyond the maximum size available, 
whereas the 4-parameter model extrapolates the growth 
pattern beyond the maximum size available in the data 
and, in this case, provides a much more plausible solu- 
tion. For stock assessment purposes, the 4-parameter 
model would be more useful in practice. The symmetry 
of the inverse-logistic curve enables the 4-parameter 
model to project the growth dynamics into size classes 
for which there are few or no samples. 
Seasonal growth 
The independent samples from Actaeon Island and 
Sterile Island, which are close together geographically, 
generated very similar estimates of the timing of the 
seasonal changes in growth rates. These samples were 
so similar that the mean curves remained close until the 
abalone reached about 80 mm shell length. The abalone 
at Actaeon Island, however, continued growing rapidly 
for longer than the animals at Sterile Island; therefore 
these curves diverged. An implication of this difference 
in productivity is that instead of taking about 8 years to 
reach the legal minimum length, as at Actaeon Island, 
it takes 9 years at Sterile Island. 
The blacklip abalone at the Middle Ground and Ga- 
gens Point sites were selected by local abalone divers as 
having notoriously slow growth. Compared to Actaeon 
Island, these two sites in the Actaeon Island region did 
indeed have relatively low productivity compared to the 
seasonal sample from Actaeon Island (only about 308 
kg relative to 484 kg); and this occurred despite the 
MaxAL being about 23 mm for the Middle Ground and 
Gagens Point sites but only 21 mm at Actaeon Island. 
Productivity was strongly and positively correlated with 
the L n ^ 0 and the L'g 5 parameter values, which relate to 
how long the linear-like growth phase continues. 
The different sites in the Actaeon region are all rela- 
tively close together geographically and yet variation in 
the parameter estimates and consequent productivity 
among sites were high. This result is consistent with 
