Fitzhugh et al.: Size- and age-dependence in batch spawning 
415 
Table 1 
Model equations used in our per-recruit analysis of spawning potential ratio and reproductive value. 
Equation 
Description 
AT 1= 1 
N a e~ ls ‘ F+M> 
Equilibrium number of females (N ) per recruit at age a — a function of selectivity at age (s a ), 
fishing mortality rate ( F ), and natural mortality rate ( M ) 
1 
Selectivity at age— a logistic function characterized by a slope parameter (p s ) and age of 50% 
selectivity (A s ) 
1 
Maturity at age (m a ) — a logistic function characterized by a slope parameter (p m ) and age of 50% 
maturity (A m ) 
w a = [ 
L„(l-e" & )] r 
Weight at age (W a )(von Bertalanffy, 1938) — a function of asymptotic length (L m ), somatic growth 
parameter ( K ), and an exponent (r) relating length to weight 
K = c i 
1 
Annual number of batches spawned at age (6 a ) per mature female — a function of 3 parameters 
(p-j, p 2 , A b ), scaled by a constant c { so that the area under the curve is the same for each spawning 
pattern t (Fig. 1) 
-jp,(a-A b )+p 2 (a-A b ) 2 ) 
1 + e ' ' 
fl = b i W a 
Annual fecundity at age (f a ) per mature female of spawning pattern i. Batch fecundity was assumed 
to be proportional to body weight 
Table 2 
j Model parameters used in the per-recruit analysis of spawning potential ratio and reproductive value. Means of values reported 
for Gulf of Mexico reef fishes are taken from Table 1 of Farmer et aid, excluding sand perch ( Diplectrum formosum) because of 
an apparent error and deepwater tilefishes and groupers because they inhabit waters typically colder than the waters of tropical 
and subtropical species. 
Parameter 
Value 
Description 
Source 
A 
max 
22 
Maximum age; ages modeled are 1, 2,..., A max 
Farmer et aid 
M 
M = 4.22/A max = 0.19 
Natural mortality rate 
Hewitt and Hoenig, 2005 
F 
Range [0.0, 0.61 
Fishing mortality rate 
Independent variable 
L m 
1.0 
Asymptotic length 
Assumed 
K 
0.19 
Somatic growth parameter 
Farmer et aid 
T 
3.0 
Exponent relating body length to weight 
Assumed 
3.16 
Age of 50% maturity 
Farmer et aid 
a s 
A-s = Anax ' /4 = 5.5 
Age of 50% selectivity 
Assumed 
Pm’ Ps 
1.0 
Slope of logistic maturity or selectivity, respectively 
Assumed 
Pi, P 2 > A b for i = 1 
03, 30, 0.0 
Spawning frequency constant with age 
Control variables 
Pi, P 2 , A b for i = 2 
0.5, 0.0, A max /2 
Spawning frequency increases with age 
Control variables 
Pi, P 2 ,A b fori = 3 
-0.5, 0.0, A max /2 
Spawning frequency decreases with age 
Control variables 
Pi, Pz, A b for 1 = 4 
0.5, -0.05, A max /4 
Spawning frequency dome-shaped with age 
Control variables 
1 Farmer, N. A., R. P. Malinowski, and M. F. McGovern. 2010. Species groupings for management of the Gulf of Mexico reef fish fishery. NOAA, 
NMFS, SERO-LAPP-2010-03 Rep., 32 p. 
where f a = annual fecundity at age a (the product of 
batch fecundity and number of batches per 
year); 
m a - maturity at age; and 
N a = relative abundance at age, discounted 
through time by natural mortality and fish- 
ing rates (Table 1). 
The denominator (0 O ) of this ratio was similarly com- 
puted, but with N a discounted only by natural mortality. 
Thus, Tp measured effects of fishing on the expected 
reproductive output per recruit relative to the output 
under no fishing. It decreased with increased fishing rate. 
Spawning potential ratio was suggested conceptu- 
ally by Goodyear (1977), and its use remains widely 
