64 
Fishery Bulletin 106(1 ) 
von Bertalanffy 
Gompertz 
Inverse-logistic 
25 50 75 100 125 150 
Initial size Lf (mm) 
175 
Figure 3 
Visual comparison of the von Bertalanffy, Gompertz, and inverse-logistic 
curves fitted to the annual data from the southwest region of Tasmania. 
The expected growth increments for smaller blacklip abalone ( Haliotis 
rubra) for each curve are also illustrated as extensions of the lines (to 
the left) and these demonstrate major differences between the curves. 
The horizontal line at zero represents the point of no growth. 
growth transition matrices describing the 
expected growth for different parts of the 
year. 
Comparison of productivity between areas 
With the inverse-logistic description of 
growth the parameter combinations do not 
provide an intuitively obvious indication of 
the productivity of different areas. A large 
MaxAL does not necessarily mean an area 
has high productivity if the large growth 
increments occur only for relatively small 
animals. An index of relative productivity 
can be obtained by applying the growth 
transition matrix derived for an area to a 
standard initial vector of numbers at size. 
The areas considered here were compared 
by generating an annual transition matrix 
for each area with 31 five-mm size classes 
from 60 mm up to 210 mm. This transi- 
tion matrix was multiplied with an initial 
numbers-at-size vector containing 1000 
individuals in the smallest size class. This 
multiplication was then repeated itera- 
tively for 10 years of growth (i.e., without 
mortality). The final numbers at size were 
converted to mass by using the standard 
Weight = a Length b where, in southern 
Tasmania, a = 5.669.E-05 and b = 3.1792, to provide a 
comparable index of relative productivity between areas 
in kilograms. 
Results 
Initial summary of growth patterns 
The general growth pattern apparent in the southwest 
Tasmania (Fig. 2) was also found at other sites around 
Tasmania, although its full expression was sometimes 
obscured because the range of available data was lim- 
ited or truncated by intense size-selective fishing on 
the larger abalone or because of difficulty in finding 
cryptic smaller abalone. The growth pattern begins 
with a relatively constant growth increment (implying 
linear-like growth) in the smaller-size abalone. This 
early linear-like growth is followed by a steady decline 
in growth increment, possibly approaching some mini- 
mum annual increment in what would be an asymptotic 
fashion. Not only do the growth increments follow this 
decreasing pattern, but a similar pattern is exhibited 
by the variability of the observed growth increments 
around the mean trend, although the decrease in varia- 
tion only occurs at larger sizes (Fig. 2). This pattern of 
growth differed markedly from the expectation of both 
the von Bertalanffy and the Gompertz growth models, 
even when these models were implemented with prob- 
ability density functions instead of constant parameters 
(Sainsbury, 1982a; Troynikov et ah, 1998; Bardos, 2005). 
Instead of these standard curves, the growth pattern 
observed indicated that some kind of inverse-logistic 
curve might describe the observations well across the 
range of available data. If the minimum predicted mean 
growth increment was greater than zero, it would indi- 
cate indeterminate growth in which the dynamics would 
permit growth to continue (possibly very slowly) until 
each animal died. An alternative way of looking at 
indeterminate growth is to note that there may be an 
upper size limit, at which the growth increment becomes 
zero, but it is so high that individuals never reach it 
before death. 
Comparison of the inverse-logistic, von Bertalanffy, 
and Gompertz models 
For the southwest region, the three different growth 
curves all predicted or described the expected growth of 
blacklip abalone reasonably well . The overlap between 
the von Bertalanffy and Gompertz curves was especially 
close over this range (Fig. 3). However, at smaller and 
larger sizes all three curves diverged significantly. The 
von Bertalanffy and Gompertz curves both predicted 
negative growth increments beyond the L x (although 
using a probabilistic version of these curves could pre- 
vent this problem). The major difference between the 
three curves was therefore found in what was predicted 
for smaller abalone. The von Bertalanffy curve predicted 
linearly decreasing growth increments (Fig. 3) as initial 
size increased. The Gompertz curve predicted initial 
exponential growth, starting from very small increments 
