14 
Fishery Bulletin 106(1 ) 
Length at age (l a ) was modeled with the von Berta- 
lanffy equation (von Bertalanffy, 1938), Z a =L 00 (l-e _if(a_4 o ) ), 
in which L x is the asymptotic length, K is the growth 
coefficient, and t 0 is the theoretical age at which length 
is zero (Z 0 = 0 assumed arbitrarily). Length at age was 
converted to weight at age ( w a ) by the allometric rela- 
tionship 
W a = V l l a 2 ’ (2) 
where u 2 and v 2 are constants under the assumption of 
isometric growth. This relationship was also used to 
model fecundity at age ( e a , eggs per mature female), 
e a = £ iC 2 > (3) 
where f 7 and S 2 are constants. Fecundity often scales 
nearly linearly with weight, such that £ 2 = u 2 = 3. 
Transition from female to male was modeled as a 
logistic function of age, 
1 
with p a the proportion male at age, j3 p the slope of 
sexual transition, and a p the age at 50:50 sex ratio. 
The same function was used to model female maturity 
at age ( g a ), with parameter j3 g = the slope, and a g = 
the age at 50% maturity. All males were considered to 
be mature on the basis of low numbers of transitional 
fish observed in the field and the apparent ability to 
complete sex transition between spawning seasons 
(Collins et al., 1987). 
Total egg production ( E ) was determined by the prod- 
uct of mature females and eggs per female, summed 
across ages, 
E = ^ N a^-Pa^a e a- < 5 > 
k , which can range from 0.2 to 1.0 (Fig. 1). A high 
value of k corresponds to a stock that can maintain its 
fertilization rate when males are scarce. In terms of 
life histories, one might expect group spawners to have 
higher k than pair spawners. The number of fertilized 
eggs (i/d under fishing rate F was computed as the 
product of fertilization rate and total egg production 
(t p-f(x F )E). 
Recruitment was computed from fertilized eggs (R(ijj)) 
with the Beverton-Holt spawner-recruit model, 
Because fertilization may become limited by sperm avail- 
ability, fertilization rate (/) was modeled as a function 
of sex ratio, 
f(x F ) = 
4k x F 
(l-Kr) + (5x-l)x F 
( 6 ) 
In Equation 6, x F is the ratio of the proportion of males 
in the population (in numbers) under fishing rate F to 
the proportion males at the unfished level, a measure 
of male depletion (x^E [0,1]). The fertilization rate 
function f is a form of the Beverton-Holt recruitment 
model scaled to one for x F = 1. It has similar shape to 
the fertilization function of Heppell et al. (2006) and 
has the following desirable properties. In the absence 
of males, / takes its minimum value of 0.0, and at the 
unfished sex ratio, f takes its maximum value, which 
is set arbitrarily to 1.0. In between these extrema, 
fertilization rate depends on the steepness parameter 
R( (//) = 
4hR 0 y/ 
R 0 (p 0 ( 1 - h ) + ( 5/z - 1 )y/ 
( 7 ) 
In this parameterization (Mace and Doonan, 1988), cp 0 
is the unfished level of fertilized eggs per recruit, R 0 is 
unfished recruitment, and h is steepness (analogous to 
k in the fertilization function). 
Parameter values 
Based on life-history theory and empirical study of pro- 
togynous fish, values of several parameters were related 
to natural mortality rate in order to avoid untenable 
parameter combinations and to maintain generality of 
results. Gardner et al. (2005) reported relationships 
between growth rate and natural mortality (K=0.64M), 
age at 50% maturity and natural mortality (e/_,=0.96/ 
M ), and size at 50:50 sex ratio and asymptotic length 
(L 50 = 0.77L oo ). The results from Gardner et al. (2005) 
