Lin et al.: Sensitivity of models to bias and imprecision in life history parameters 
303 
often used growth model, but different estimation ap- 
proaches (e.g., length-frequency methods versus read- 
ings of rings in calcified structures; see Lin and Tzeng, 
2009, versus Lin and Tzeng, 2010), different regions of 
study (Helser and Lai, 2004; Keller et ah, 2012), and 
aging errors from annuli readings (Lai and Gunderson, 
1987; Cope and Punt, 2007) may lead to biases in the 
von Bertalanffy growth coefficient (K). 
In another example of the potential for uncertainty, 
biases in the asymptotic length (L x ) may result from 
the regional variation in growth potential in different 
habitats (Beverton and Holt, 1957), from a latitudinal 
trend (Helser and Lai, 2004; Keller et ah, 2012), or 
from sampling schemes unrepresentative of the popu- 
lation size structure, because of its high dependency of 
maximum size in the sample (Formacion et ah, 1991; 
Froese and Binohlan, 2000). In addition, the uncertain- 
ty (or multiplicative error) in the growth curve (£qr) is 
related to the imprecision in the lengths-at-age scat- 
ter at a given age. The current fishing mortality rate 
(F cur ) can be estimated from various methods (e.g., Se- 
ber, 1982; Quinn and Deriso, 1999; King, 2007) with 
both the mean and variance of high uncertainty (Chen 
and Wilson, 2002). 
It is essential to incorporate and quantify these un- 
certainties in the assessment of population dynamics 
(Hilborn and Walters, 1992; Peterman, 2004). Further, 
parameter uncertainties can be categorized into bias 
uncertainty and precision uncertainty. The influence 
of parameter bias on YPR and SPR models has been 
investigated since the 1950s (e.g., Beverton and Holt, 
1957; Goodyear, 1993; Mace, 1994), but the effects of 
precision uncertainty have seldom been addressed. The 
effects of parameter imprecision with a limited number 
of comparisons (i.e., high or low variation) that were 
inadequate to reveal a full picture of the sensitivity to 
bias and imprecision in the parameters have been ex- 
amined in a few studies (e.g., Restrepo and Fox, 1988; 
Chen and Wilson, 2002; Chang et al., 2009). Systemat- 
ic examinations of different degrees of parameter bias 
and imprecision with a wider range and finer resolu- 
tion could provide detailed information about their ef- 
fects on model outputs (e.g., Goodyear, 1993), but few 
such examinations exist for per-recruit models. 
It is difficult to elucidate the sensitivity of per- 
recruit models to parameter bias and imprecision 
through analysis of the formulae because the effects 
of the parameters on Frrp s are nonlinear with compli- 
cated forms (Schnute and Richard, 1998). Alternatively, 
a sensitivity analysis through the use of a Monte Carlo 
simulation can be used to examine their effects on the 
estimation of FgRp s , on fishery status, and on manage- 
ment implications (Jiao et al., 2005). Therefore, the ob- 
jective of this study was to evaluate the effects of dif- 
ferent degrees of bias, represented by the differences in 
parameter means, and of imprecision, represented by 
the differences in parameter standard deviations (SDs), 
in the YPR and SPR model results, specifically in the 
FrrPs and the resultant composite risks of overfishing. 
Materials and methods 
To better understand the sensitivity of outputs from 
the YPR and SPR models, we changed the mean or SD 
of selected parameters one at a time over large ranges 
with fine increments (from 5% to 95% with increments 
of 5% for cases of under-specification and from 150% 
to 1000% with increments of 50% for cases of over- 
specification). The following parameters were selected: 
1) M\ 2) von Bertalanffy growth coefficients of K and 
Loo; and 3) F cur . In addition, the multiplicative error in 
the growth curve (£qr) and length-weight relationship 
(£rw) were selected as well. In this study, we used in- 
formation on the mean and SD values of these parame- 
ters from previous studies conducted during 1998-2006 
on the life history and fishery of the Japanese eel in 
the Kao-Ping River in Taiwan (Lin and Tzeng 2009; 
Lin et al. 2010a, 2010b) to examine the sensitivity of 
model results to effects of parameter bias and impreci- 
sion with a Monte Carlo simulation. 
We used wide ranges for the parameters for 3 rea- 
sons: 1) the scale of boundaries used; a factor of 10 was 
applied in a classical study (figure 4 in Goodyear, 1993); 
2) a wide range is more general because it can include 
all likely ranges; and 3) in our previous study (Lin and 
Sun, 2013), we found that the variation in estimates 
of M from several different indirect methods can vary 
from 0.10/year to 2.13/year. Also, the difference in K 
can be large, especially when different approaches are 
involved. For the same region, for example, in Kao-Ping 
River, K has varied from 0.12/year to 0.38/year, depend- 
ing on whether methods based on otoliths or length 
frequencies were applied (Lin and Tzeng, 2009, 2010). 
YPR and SPR models 
The YPR and SPR models were calculated according to 
the formulae in Quinn and Deriso (1999): 
YPR — e ~ MUc - tr) Fit)Nit)Wit)dt and (1) 
SPR = f' mM N(t)W(t)S(t)dt, (2) 
where Fit ) 
Nit) 
Wit ) 
Sit) 
t r 
t c 
/max 
the fishing mortality at age t; 
population size in numbers at age /; 
the weight at age t; 
the maturation proportion at age t\ 
the age at recruitment; 
the age at first capture; and 
the maximum age (the formulae for Fit), 
Nit), Wit), and Sit) are shown in Table 1). 
Because, in Kao-Ping River, females were 4 times more 
abundant than males (Han and Tzeng, 2006) and be- 
cause females contribute more directly to spawning 
biomass, the parameters of female Japanese eels were 
used in this study; they were also used in Lin et al. 
(2010a). Estimates of means and corresponding SDs of 
the parameters were derived from our previous works 
(Lin and Tzeng, 2009; Lin et al. ,2010b). 
