Haddon et al.: Using an inverse-logistic model to describe growth increments of Haliotis rubra 
59 
Table 1 
Data by sites and regional groupings of sites (Fig. 1) in southern Tasmania where blacklip abalone ( Haliotis rubra) were collected 
to compare growth rates. Time step relates to whether the analysis was for annual or seasonal growth, count is the number of 
tags recovered, and Min-Max Init. L (in mm) are the minimum and maximum initial length of blacklip abalone at tagging for 
each site. 
Time step 
Longitude 
(East) 
Latitude 
(South) 
Regional 
grouping 
Site name 
Count 
n 
Min-Max Init. 
L 
Annual 
145.492 
-42.969 
Southwest 
Black Island 
116 
57-171 
Annual 
145.667 
-43.075 
Southwest 
Giblin River 
84 
83-173 
Annual 
145.781 
-43.226 
Southwest 
Hobbs Island 
57 
57-181 
Annual 
146.900 
-43.566 
Actaeon 
Gagens Point 
154 
50-142 
Annual 
146.972 
-43.549 
Actaeon 
Middle Ground 
353 
47-146 
Annual 
147.381 
-43.366 
Bruny Island 
Fluted Cape 
135 
83-154 
Annual 
147.385 
-43.111 
Bruny Island 
One Tree Point 
162 
52-153 
Seasonal 
146.996 
-43.534 
Actaeon 
Actaeon Island 
390 
61-176 
Seasonal 
146.990 
-43.550 
Actaeon 
Sterile Island 
373 
48-146 
Gorfine, 1998; Troynikov et al., 1998; Bardos, 2005). 
Like the von Bertalanffy model, the Gompertz equation 
is deterministic in predicting an asymptotic maximum 
length. The probabilistic forms of these two models pre- 
dict a more plausible range of final maximum lengths 
and some configurations of Francis’s (1995) model can 
also exhibit a spread of final sizes. However, like the 
von Bertalanffy and Gompertz models, Francis’s (1995) 
model fails to exhibit the linear-like early growth of 
small abalone that has been observed in Tasmania. 
An important characteristic of growth models is an 
ability to accurately model growth across a broad size 
range. Nine of the studies cited in Day and Fleming 
(1992) indicate that early growth in abalone is effec- 
tively linear. This linearity contrasts strongly with both 
the von Bertalanffy curve (which predicts faster early 
growth) and the Gompertz growth curve (which predicts 
slower early growth). Neither the von Bertalanffy nor 
the Gompertz growth models are consistent with ob- 
servations of effectively linear growth in small blacklip 
abalone in Tasmania, Australia (Prince et al., 1988; 
Gurney et al., 2005). Such early linear-like growth 
would imply constant growth increments in small ani- 
mals and would require a different structural model to 
represent such growth dynamics. 
Given the wide variation in maximum sizes found in 
natural abalone populations, an alternative approach to 
using deterministic models (with a known or even prob- 
abilistic asymptotic length) would be to model growth 
as indeterminate. Indeterminate growth would imply 
no specific upper limit and animals would be expected 
to continue growing, even if very slowly, until they die. 
This indeterminacy would have the disadvantage in that 
there would be no simple analytical solution for length- 
at-age, but nevertheless this strategy could provide for 
intuitively simple empirical descriptions of growth that 
avoid the complexities of fitting probabilistic models as 
proposed by Sainsbury (1982a) and Bardos (2005). 
Here we present a new empirical description of black- 
lip abalone growth using an inverse-logistic model for 
both the mean growth increment and the predicted 
variation about the mean increment for a given shell 
length. In contrast to both the von Bertalanffy and 
Gompertz growth curves, the growth description given 
here allows for both linear growth of small and juve- 
nile abalone as well as the option of either determinate 
growth (with a maximum shell length) or indeterminate 
growth (with a spread of maximum lengths). 
Materials and methods 
Examination of growth patterns 
To provide an initial empirical indication of growth 
patterns, the size of abalone at tagging were grouped 
into 10-mm classes and the mean growth increment in 
each class was then plotted on top of the raw data from 
the southwest area of Tasmania (a combination of three 
sites, Fig. 1; Table 1). 
Tagging methods and locations 
Two sets of data were used in the description of the 
inverse-logistic model. Firstly, to examine annual growth 
increments, we used tagging data from seven sites 
around the south of Tasmania (Fig. 1). Data from those 
sites were limited to tagged-and-recaptured abalone 
and the data were collected approximately one year 
apart (between 0.96 and 1.05 years apart). Abalone 
from some groups of sites were found to exhibit very 
similar growth patterns and the data from these sites 
were combined to generate three larger regions (namely, 
southwest, Actaeon, and Bruny Island) (Fig. 1; Table 1). 
Secondly, tagging data from two sites were used to 
examine seasonal growth (Table 1), and for this analysis, 
