Haddon et al.: Using an inverse-logistic model to describe growth increments of Haliotis rubra 
61 
MaxAL x 
AL = 
AZ + Csin^Tr^-p))- 
Csin(27r(z r -p) 
Ln(l9) 
(**-*&) 
+ £t 
(1) 
l + e 
i L % 5 - L 5 n o) 
The fact that L/L19) is used instead of -Ln(19) implies 
that the logistic is inverse and that the L gs parameters 
relates to the 95% point (Ln(15) would equate to the 
75% point). The inverse-logistic description of variation 
is general; however, if the expected length increments 
always remain greater than zero then the standard 
deviation of the residuals o L can be defined as the 
simpler 
where MaxAL = 
At = 
L t = 
J m — 
^ 50 - 
T m - 
•^95 “ 
c = 
t R and t T = 
P = 
the hypothetical asymptotic maximum 
growth increment at some initial size 
of abalone that sets the exponential 
term to zero; 
the interval between tagging and 
recapture (as a fraction of a year); 
the size when first tagged; 
the initial length at which the mid- 
way point between the MaxAL and 
lowest growth increment is reached; 
the initial length at which 95% of the 
difference between the smallest and 
maximum increment is reached; 
the amplitude of the seasonality effect 
for AL; 
the dates of recapture and tagging, 
respectively (as fractions of a year, 
e.g., June 30 th = 0.5; t R = t T + At); 
and 
the date of maximum growth rate (as 
a fraction of a year). 
The error term £ L t is additive and normal, and is assumed 
to have a mean of zero and standard deviation o L that 
can be defined either as a function of initial length, L t , 
or as a function of the predicted length increment A L t . 
If the expected length increments ever attain zero, or go 
negative, then the standard deviation of the residuals o L 
can be defined in terms of the initial length L t : 
Maxo L x 
°L t =- 
AZ + OrSm^/r^-p))- 
C^sin \2n[t T -p) 
L t -L* 
Ln( 19)— 50_ 
l + e 95 50 
where Maxo L = the hypothetical asymptotic maximum 
standard deviation of the residual 
values at some initial size of abalone 
that sets the exponential term to 
zero; 
L| 0 and L| 5 = the parameters describing the inverse- 
logistic for how the variability of resid- 
uals reduces with increasing L t ; and 
C a = the amplitude of the seasonality effect 
for the o, term. 
L t 
°L t 
= a 
(3) 
where a and jl are parameters of a power relationship 
with the expected length increment A L t and the season- 
ality is achieved from Equation 1. 
When seasonality is ignored (when estimating annual 
growth increments), the C and C a parameters are set 
to zero leaving the simple At so that any slight devia- 
tions from a At of one year are assumed to alter the 
predicted growth in a linear fashion. Thus, 0.95 of a 
year permits 95% of the growth increment of that year. 
With the use of At alone, there is the assumption that a 
simple linear scaling of growth increment with respect 
to time elapsed will provide sufficient adjustment for 
small deviations of At from one year. 
Using a normal distribution to describe the residuals, 
we found that there was an excellent match of this dis- 
tribution to available data. However, if some probability 
density function other than the normal distribution 
provided a better fit for some other species or popula- 
tion, then the equivalent measure of spread about the 
expectation would need to be implemented. 
Where the tagging interval is greater than one year, 
the expected growth increment is estimated in two 
steps. First, the expected growth increment and stan- 
dard deviation that would be expected during a year of 
growth are estimated, and then the growth increment 
for the fraction of the year remaining from the date 
of tagging to the date of recapture (after subtracting 
one year) is estimated by using the initial size plus 
the estimated yearly growth increment as the starting 
length for the second installment of growth. Thus, AL 
is first estimated with Equation 1 with the C and C a 
parameters set to zero, and At set to 1.0, and then the 
fraction of a year remaining from the date of tagging 
to the date of recapture (after subtracting one year) is 
used in the full version of Equation 1 and the L t is set 
to the original L t plus the AL predicted from one year 
of growth. The two sequential AL estimates are added 
together to obtain the total predicted growth incre- 
ment. Using Equation 2 to define the variation about 
the curve, we applied a similar sequential process to 
the estimation of the standard deviation of the respec- 
tive residual errors. In this case, the expectation was 
that the variability would reduce with increasing size 
so the C a parameter was expected to be negative rather 
than positive as was expected for the C parameter. 
The use of Equation 3 requires that it be applied to 
