Ault et al.: A multispecies assessment of coral reef fish stocks 
399 
K and - parameters of the von Bertalanffy 
equation. 
Thus, the only unknown variable in Equation 3 is the 
total mortality rate in year t, Z(t), which can be esti- 
mated fairly easily with an iterative algorithm called 
LBAR (Ault et al., 1996). Finally, by assuming that M, 
the annual instantaneous rate of natural mortality, is 
known and constant for the interval At, we can esti- 
mate the fishing mortality rate as F(t)= Z it)-M. 
Reef fish exploited-populafion simulation 
model 
To achieve a better understanding of the dynamics 
of multispecies tropical coral reef fish stocks, their 
response to exploitation, and the accuracy and pre- 
cision of statistical estimates from the sampling sur- 
veys, we developed an object-oriented computer simu- 
lation model for exploited reef fish populations in the 
Florida Keys fishery (REEFS [reef fish equilibrium 
exploitation fishery simulator (Ault 3 )]). The funda- 
mental population-dynamic processes of growth, 
mortality, and recruitment are relatively similar for 
fishes of the temperate, boreal, and tropical seas; 
however, some distinct differences in rates exist for 
tropical marine fishes as reflected by quasicontinuous 
growth, protracted spawning and recruitment, and 
competition-based population dynamics (Ault and 
Fox, 1990; Sparre and Venema, 1992; DeMartini, 
1993). To represent the continuous time dynamics of a 
tropical coral-reef fish population in the numerical 
model, following Ault and Olson (1996), we formalized 
the conservation law for population abundance as 
dN(a,t ) dN(a,t ) 
dN(a,t) = — — : — da + — dt = 
da dt 
- Zia,t)N(a,t)dt . 
(4) 
This partial differential equation expresses popula- 
tion age structure in terms of the average number of 
fish by age over time. The term dN/da is the contri- 
bution to the change in N(a,t) resulting from indi- 
viduals getting older. Because the variable a is tied 
stepwise to chronological age, for each time step t, a 
gets one unit older, so that da/dt = 1. This condition 
holds for t > 0 and a > 0. Equation 4 requires two 
conditions on N(a,t ): one initial condition for N(0,t); 
and a boundary condition in age 0 tied to reproduc- 
tion for N(0,t). Integration of Equation 4 with a 
growth function allows efficient estimation of popu- 
lation biomass and average size in the stock over 
3 Ault, J. S. 1998. Tropical coral reef fishery resource decision 
dynamics. Unpubl. manuscript. 
time. In the numerical model REEFS, we modified 
Equation 4 to a stochastic age-independent length- 
based population dynamics model to simulate effi- 
ciently the average or ensemble number at a given 
length for the entire population age structure as 
N(L) = J*R(t - a)Sia)Q(a)P(L\a}da , (5) 
t T 
where R( z-a) = recruitment lagged back to cohort 
birth date; 
S(a) = survivorship to age a; 
Q(a) - sex ratio at age a; and 
P(L | a) = the conditional probability of a fish 
being length L given that it is age a. 
The ensemble average length L at age a is repre- 
sented by the von Bertalanffy growth function. The 
conditional probability distribution for length and age 
was assumed to be bivariate normal. 
The reported maximum age of fish in the stock t f 
(equal to a generation), usually obtained from age 
and growth studies by using either scales or otoliths, 
allows application of a convenient and consistent 
method to normalize the annual instantaneous natu- 
ral mortality rate M to life span. First, we assume 
that Sit,), the fraction of the initial cohort numbers 
surviving from recruitment t to the maximum age, 
can be expressed as 
Nit,) 
N(t r ) 
) = e 
( 6 ) 
Then, assuming an unexploited equilibrium, by set- 
ting the probability of survivorship of recruits to the 
maximum age to be 5% (i.e. S(t A )=0.05), and letting t 
be equal to 0, we rearranged Equation 6 in order to 
provide an estimate of the natural mortality rate 
(7) 
Mortality and growth estimation in tropical fishery 
populations are normally approached from a size-based 
perspective because of difficulties in ageing fish. Aver- 
age size can be converted to mean age by making two 
assumptions: 1) that age a maps directly into, or is a 
function of, size L(a)\ and 2) that mean length-at-age 
from the von Bertalanffy equation can be inverted as 
-In 
L „ - Lia) 
K 
+ t n 
(8) 
