He et ai.: Interactions of age-dependent mortality and selectivity functions in age-based stock assessment models 
215 
where R Y = expected recruitment at year y and is mod- 
eled by the Beverton-Holt stock recruit 
relationship: 
SO y 
a + /3SO v 
where R = recruitment at year y; 
SO = spawning output at year y; and 
a and j3 are recruitment parameters. 
(3) 
Annual catch by fishery at sex x, and age a is given by 
C =N 
w *,y,a M +F 
1 — e 
( 8 ) 
Landing by fishery f at year y, y/ , is given by 
^v = XI C w W x ^, (9) 
The relationship can be reparameterized by using a 
steepness parameter (h): 
where W x = weight of fish at sex x an age a, which is 
assumed to be constant for all years. 
and 
Rn 
4 (1 -h) 
4 h 
P = 
5 h - 1 
4 hR 0 
(4) 
(5) 
where B 0 is virgin spawning output {B 0 =SO 0 ), and 
R 0 is defined previously. 
Growth, weight and spawning output 
Growth and length-weight relationship are given by 
= L: (l_ e -Y-?)) 
(10) 
W „ = r, L T r ; , 
x,a 1 a 7 
(11) 
where L xa = length at sex x and age a; 
L“, K x , and t° x = growth parameters for sex x\ and 
Tjand r x = length-weight parameters. 
The “steepness” is the expected fraction of jR 0 at 0.2 
B 0 and is set to range from 0.2 to 1.0. When h = 0.2, 
recruits are a linear function of spawning output (j3=0, 
and if, =— so ). When h = 1, recruits are constant and in- 
a i 
dependent of spawning output (a= 0, and R , = )■ In the 
simulation, R y is replaced by actual annual recruitment 
CRi, see Eq. 15) which includes annual recruitment 
deviation (Ry)- 
Selectivity, fishing mortality, and catch 
Selectivity is same for both sexes. Logistic selectivity is 
given by the equation 
^ _j_ ^-ln(19Xa-t7 2 )/t7i ’ 
where /] 1 , and i] 2 are selectivity parameters. 
Double normal selectivity is a special function for 
selectivity used by the SS3 program with two normal 
functions jointed by smooth functions. Details of this 
special function are given in Methot (2009a). There are 
six parameters in this selectivity function: (i] 1 ) ascend- 
ing inflection age; (r/ 2 ) width of plateau expressed as 
logistic between maximum selectivity and maximum 
age; (rj 3 ) logarithm of ascending width; (q 4 ) logarithm 
of descending width; (rj 5 ) selectivity at age 0 expressed 
as logistic between 0 and 1; and ( q 6 ) selectivity at maxi- 
mum age expressed as logistic between 0 and 1. 
Fishing mortality is given by 
F x ,y,a = FF y S a , (7) 
where FF = full fishing mortality for year y; and 
S a = selectivity at age at. 
Annual biomass B v is given by 
= (12) 
Annual spawning output is given by 
SO = Y PN, G , (13) 
y Zw a l,y,a a ’ 
a 
where P a = proportion of mature females at age a; and 
G a = fecundity for female at age a. 
Abundance index 
The abundance index (I) for year y and survey i has 
the following relationship: 
^ = < 14) 
x a 
where = catchability coefficient for survey i\ 
N yxa = population abundance; and 
S xa = selectivity for sex x and age a. 
When the abundance index (/ ys ) is outputted to the 
assessment model, a new index (/' ) is created by add- 
ing sampling error to I (see Eq. 16). 
Recruit variability and sampling errors: 
Estimated annual recruitment (R'), annual survey indi- 
ces (I'), and annual landings (\\i') are all subject to 
log-normal errors with zero means and their respective 
standard deviations: 
R y = R y e R * (15) 
