Finer et al.: Bias in estimates of growth when selectivity in models includes effects of gear and availability of fish 
77 
e a 
p(a)= Y7°^’ (D 
where Z a = instantaneous total mortality rate 
at age a. 
We assume that the lengths in each age 
class are normally distributed around their 
mean length and the mean lengths at age 
are determined by the von Bertalanffy 
(1938) growth model (VBGM). The popula¬ 
tion proportions-at-age and proportions-at- 
length can be calculated by using 
p al = (P(l/a)p(a), (2) 
where conditional probability P(l/a) of being 
in a discrete length l given age a follows a 
normal distribution around the predicted 
length-at-age (L a ) based on the VBGM: 
L a = L inf (1 - e‘ K,a_a o ) + e, (3) 
where, L in f = the asymptotic length; 
K = the growth coefficient; 
£ = the error; and 
a 0 - the hypothetical age (expressed 
in years) when average length 
is zero. 
The error (e) is the variation in length at age 
and is assumed to be normal with a mean of 
0 and a standard deviation of CV a L a 
The observed proportions at age and 
length from the fleet is given by: 
Qa,i=d a s lPal , (4) 
where p a \ = the population proportion at age 
and length; and 
d a s\ - the combined effects of age- 
based probability that a fish 
is in the area where the fleet 
occurs (d a , availability) and the 
length-base probability that an 
encountered fish will be caught 
(S], gear selectivity). 
The observed length-based selectivity that 
includes the length-based gear selectivity 
that is adjusted for the approximation of 
age-based availability is given by 
V 1 — 
£a/>a,l 
where q a \ - the observed; and 
p a i = the population proportions-at- 
age and -length. 
The selectivity-adjusted proportions-at-age 
and proportions-at-length are given by 
m., --, (6) 
' Vj 
where q a \ = the observed proportion-at-age 
and proportion-at-length; and 
0123456789 10 
Age (year) 
Length (cm) 
c 
~ 0.10 
/ 
I 
/ 
f \ 
1 
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 
Length (cm) 
Figure t 
Results from a deterministic population dynamic model of a hy¬ 
pothetical population of a small, migratory pelagic fish in which 
selectivity includes effects of gear and availability of fish and was 
used in this study to examine bias in growth estimates. (A) Plot of 
true length-at-age (solid line), length-at-age observed by the fish¬ 
ery without accounting for selectivity effects (dotted line), and esti¬ 
mated length-at-age after accounting for the observed length-based 
selectivity that included both gear and an approximation of the age- 
based availability (dashed line). (B) Plot of the true length-based 
gear selectivity (solid line) and the observed length-based selectiv¬ 
ity that included the addition of an approximation of age-based 
availability (dotted line). (C) Plot of the true length distribution of 
age-3 fish (gray bars), length distribution observed by the fishery 
(dotted line), and estimated length distribution after accounting for 
the observed length-based selectivity that included an approxima¬ 
tion of age-based availability (dashed line). 
