Fisheries Research 41 (1999) 133±153

A method to determine size-speci®c natural mortality applied to
westcoast steenbras (Lithognathus aureti) in Namibia
J.E. Beyera,*, C.H. Kirchnerb, J.A. HoltzhausenbaDanish Institute for Fisheries Research, Charlottenlund Castle, DK-2920, Charlottenlund, Denmarkb
Ministry of Fisheries and Marine Resources, Box 912, Swakopmund, Namibia
Received 12 February 1998; accepted 17 November 1998

Abstract
In traditional single-species ®sh stock assessment natural mortality is considered constant, independent of age or size. It isargued that predation mortality is generally related to prey size, that it decreases with increasing size and that such sizespeci®c signals can best be detected in stocks that are close to a virgin state. Size-speci®c natural mortality rates of westcoaststeenbras (Lithognathus aureti) were determined by using length frequencies of rod-caught ®sh from a lightly exploited andclosed population at Meob Bay, Namibia. It was assumed that natural mortality is inversely proportional to (body) length andapproaching a constant minimum rate (M1) as the ®sh grow bigger. Simple and new length-based catch curve methods weredeveloped using the traditional simpli®cation of neglecting variations in length at age. The von Bertalanffy growth parameters
for annual mean growth as well as the coef®cient of variation of length at age were estimated from analyzing age-length data.
A simple deterministic simulation model was developed to examine the robustness of the methods and the impact ofvariability in individual growth. The model assumes that ®sh grow with a constant coef®cient of variation in length at age. The
simple method works within 10% precision criteria in most real cases. It is shown that overestimating mean length at old age
1  counteracts the effects of overlapping lengths for consecutive age groups. This fact can be used to estimate theLfundamental mortality-to-growth ratio (M1/K) without any prior knowledge on growth. The application of M1 for steenbras
to obtain size-speci®c natural mortality rate for silver kob (Argyrosomus inodorus), as an input parameter for virtual
population analysis, is also proposed. # 1999 Elsevier Science B.V. All rights reserved.
Keywords: Natural mortality; Size speci®c rates; Length-based methods; Individual growth variation; Simulation; Lithognathus aureti

1. Introduction
Although the instantaneous rate of natural mortality(M) is a key factor in ®sh population dynamics, this isdif®cult to estimate accurately, particularly if little
information is available on the biology of the ®sh.
*Corresponding author. Tel.: +45 33 963445;
fax: +45 33 963333; e-mail: jeb@dfu.min.dk

Natural mortality accounts for decreasing ®sh stock
abundance potentially due to numerous other causes
apart from ®shing, including e.g. predation, cannibalism, disease, spawning stress, starvation and senescence.
As direct measurements are usually impossible toobtain, measurable quantities that are assumed proportional to M are often used for the estimation ofnatural mortality. A number of methods to determine

0165-7836/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 7 8 3 6 ( 9 9 ) 0 0 0 1 1 - 9

134

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

an instantaneous rate of size-independent natural mortality have been suggested in the literature, most using
life-history parameters (e.g. Beverton and Holt, 1959;
Rikhter and Efanov, 1976; Pauly, 1980; Gunderson
and Dygert, 1988; Jensen, 1996; Lorenzen, 1996). In areview on the estimation of size-independent natural
mortality rate in ®sh stocks, Vetter (1988) concluded
that M is far from constant for many ®sh stocks andthis within-stock variability should not be ignored as
the standard ®shery models can be sensitive to thevariations in M. Changes in natural mortality rate withbody size of ®sh within single ®sh stocks have been
discussed by e.g. Ricker (1969), Ware (1975) andPeterson and Wroblewski (1984). Beyer (1989);
Example 10, simply considered mass balance in the
predation-food-consumption process and deduced that®sh growing according to the von Bertalanffy growthequation (VBGE) was consistent with predation mortality being inversely proportional to length, L, i.e.
M L  M1

L1
;
L

(1)

where L1 as usual, denotes the mean length ofin®nitely old ®sh and M(L1)M1 similarly denotes
the mortality (predation) of in®nitely old ®sh ± anotation we retain in the present study.
Particularly in slow growing long-lived species
the usual assumption in single-species ®sh stock
assessment that natural mortality remains constantfor all age (size) groups appears to be unreasonable.
The simple assumption that natural mortality rateis inversely proportional to body length was adopted
as an attractive and promising hypothesis. This study
is an approach to quantify size-speci®c natural mortality for westcoast steenbras (Lithognathus aureti)
and further apply it to silver kob (Argyrosomus
inodorus).
In situations of zero ®shing mortality, new linearized length-based catch curves are derived from ®rst
principles to estimate M/K in the case of size-independent natural mortality and M1/K in the case of size
dependent mortality according to Eq. (1), where K as
usual denotes the curvature parameter in VBGE.
These simple methods require knowledge of L1.
The VBGE parameters for mature steenbras are
obtained from age-length data and further discussed
in the light of mark-recapture data. It is of note that our
mortality analysis is developed from the fundamental

basis of traditional ®sh population dynamics, i.e. a
year-class is considered to comprise identical ®sh.
This is generally a fallacy because it is well-known
that ®sh of the same age may differ considerably insize. It is therefore not clear how the length-based
methods perform as the coef®cient of variation inlength at age (CV) increases thereby creating a considerable amount of overlap in the size-range ofconsecutive age groups. This and related issues ofrobustness and bias are examined using a deterministic
simulation model assuming that the individual ®sh of acohort grow according to VBGE with the same K andwith the same speci®c growth rate (in length), i.e. theconstant CV-growth model of Beyer and Lassen(1994). The model is also used to demonstrate theimportant role of the mortality-to-growth ratio forlength-based population dynamics.
2. Materials and methods
This study was undertaken in the Meob Bay area
(248310 S, 148360 E) (Namibia, south-western Africa),
situated in the Namib Naukluft Park which falls under
the jurisdiction of the Ministry of Environment andTourism. The Namibian Sea Fisheries Act (no. 29 of
1992) prohibits recreational ®shing in restricted areas
and only ®shing for scienti®c purposes is allowed. The
Meob Bay area was therefore a unique study area as no®shing effort and hence zero ®shing mortality (F)(except for some biological sampling) occur. In theyears 1995±1997, regular surveys were undertaken at
Meob Bay by the Line®sh section as part of the
Namibian Fish Tagging Project (NAFTAP). Of theapproximately 16 400 steenbras that were marked andreleased at Meob Bay during the period 1992±1997,
146 were recaptured within this area and only eight
ranging from 35.5 to 70 cm fork length (FL) wererecaptured south of Walvis Bay in the Sandwich area
(238080 S, 148260 E) (which lies approximately 160 km
from Meob Bay). As only 5.5% of westcoast steenbras
were recaptured outside the Meob Bay area, thepopulation under investigation was treated as being
closed and unexploited. Note that this 5.5% is mostcertainly an overestimate as the ®shing pressure in the
Sandwich area is much higher than in the Meob Bay
area because this is open to angling for the generalpublic.

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153135Fig. 1. Length frequency distributions of L. aureti sampled at Meob bay, Namibia, with rod and reel from the shore throughout 1995
(n954124), 1996 (n964183) and 1997 (n977867). (Shown lengths refer to lower class limits; 2.5 cm classes).

Westcoast steenbras were caught from the shorewith rod and reel and measured to the nearest 0.5 cm
FL. Length frequency distributions in 2.5 cm classes
are shown in Fig. 1. Preliminary experimentation with
hook-size selectivity was also carried out (Fig. 2).
2.1. Determination of growth parameters forwestcoast steenbras
Estimated values for the von Bertanlanffy growthparameters are needed for the length-based mortality

analysis, however, no useful information has beenreported in the literature on the growth of the westcoast steenbras. A preliminary growth analysis was
carried out by reading ring structures on otoliths
following the method used by Buxton and Clarke(1989) and Benneth (1993) (Fig. 3). VBGE parameters for mean annual growth were estimated fromthese age-length data by non-linear least square(L170 cm, K0.083 yrÿ1 and t0ÿ2.4 yr;
Fig. 3(a)). The coef®cient of variation of length atage, CV(L|age), was estimated from the relative

136

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

K's from VBGE, i.e. in the present application,
1
L1 ÿ L1
Kln:(2)
t L1 ÿ L2
Fig. 4 shows K-values obtained in this way by
assuming that the (unknown) growth trajectories of
the individual ®sh could adequately be described by
the same L170 cm and produced a mean value of  0:072  0:006 SE yrÿ1 , (n40). Omitting threeK
possible outliers (exceeding the 99.5% probability  2:576 SD, (n37)) resulted
level estimated by K
  0:064  0:004 SE yrÿ1 (n37). Repeating
in K
the procedure for L1 values in the range 60±80 cm
resulted in mean K values that can be approximately  334 Lÿ2
described by K
1 , (n40). When individual
L1's instead were chosen at random from a normal
distribution with mean 70 cm and 0.1, we obtained  0:070 and a mean standard deviation of 0.043
K
based on 200 simulations. It should be noted that the
maximum length of steenbras recorded in 1995±1997
was Lmax80 cm (n16 174).
2.2. Estimating size-independent natural mortality(M)
Fig. 2. Results of pilot hook selectivity experiment. An equal
number of randomly selected fishermen (six per hook-size), withapproximately the same fishing experience, were fishing for two
sessions of 5 h each in the same area using the same bait. Catch innumbers of steenbras with the various hook-sizes used in theexperiment are shown; arrows indicate mean lengths. (Shownlengths refer to lower class limits; 2.5 cm classes).

residuals ( 0.116; Fig. 3(b)). Assuming L1 to be
known, the mark-recapture data were also used toobtain estimates of K independently of t0, the apparent
age of zero length. This analysis is included only as apreliminary investigation as more detailed considerations and estimation of growth are outside the scope of
this paper. The data included length when ®rst caught(L1), length at recapture (L2) and days free (t). To
avoid considerable bias on the estimation of the meanannual K due to e.g. seasonality in growth, ®sh which
had spent less than one year in freedom were excluded
from the analysis. Of the 89 recaptures recorded, 40
provided such accurate lengths at mark-recapture witht1 yr, and this data was used to estimate individual

In establishing the general basis, the impact of aconstant instantaneous rate of total mortality,
ZFM, was ®rst considered (F®shing mortality).
The traditional cohort assumption of a one-to-one
relationship between size and age (i.e. 0) furthermore implies that the number of ®sh alive may beconsidered a function of length, N(L), instead of age(or of time). Dividing the rate of change in numbers,
dN/dtÿZN, by the rate of growth, dL/dtK(L1ÿL),gives the length-based starting point,dN L
Z N L:ÿ
dL
K L1 ÿ L(3)

Integrating this (physiological) rate of population
decline over the length interval from L0 to L, assuming
constant parameters Z, K and L1, yields the lengthbased survivorship (Beyer, 1989),

L1 ÿ L Z=K;
N L  N L0 
L1 ÿ L0Z  F  M; L0  L  L1 :(4)

Thus in the case of constant mortalities independentof length, the fraction of ®sh surviving, while growing

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

137

Fig. 3. Otolith readings of steenbras (L. aureti) caught in Meob bay, Namibia, in 1996 (n282). (a) Plot with estimated von Bertalanffy mean
growth curve. Stipled curves indicate approximate 95% confidence intervals for length at age estimated as mean length multiplied by(12 ). (b) Residuals relative to mean length vs. age. The constant coefficient of variation in length at age, , is estimated as the standard
deviation of these relative residuals.

from a (constant) length L0 to some length L, isobtained as the length-ratio of potential growth spanin the square bracket to the power of Z/K and equals
the usual exponential survivorship, exp(ÿ Zt), if trepresents the time required to grow from L0 to L
(according to VBGE). L0 was obtained at the length atfull recruitment to the ®shery. The length-based beginning point for deriving the linearized catch curve isFig. 4. Single fish estimates of K from mark-recapture data of steenbras (L. aureti) with one year or more in freedom and assuming
L170 cm for all fish (n40). The stipled line shows the average omitting the three possible outliers (which are denoted by triangles).

138

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

similarly derived from dividing the catch rate, dC/
dtFN, by the growth rate, i.e.
dC F N L:
dL K L1 ÿ L(5)Considering this (physiological) catch rate at the  L1  L2 =2, of a length class to equal
mid-point, L
C(L1,L2)/L, the catch from that length class divided
by the class length (L), and inserting the survivorship, Eq. (4), gives in a ®rst approximation that Z=Kÿ1

C L1 ; L2 =L / L1 ÿ L:This proportionality captures the essence of Eq. (5)when recruitment, N0N(L0), to size L0, and allparameters of growth and mortalities can be considered constant implying a steady-state situation. In thepresent application to the closed and unexploited
population of westcoast steenbras inhabiting Meob
Bay, C and F are referred to as the catch and ®shingmortality (Ff, which is very small compared to M)
for all the surveys. Hence, considering L constantand taking logarithms gives the length-based catchcurve (Beyer, 1989),
i ;
ln Ci  constant  M=K ÿ 1 ln L1 ÿ L(6)where index i refers to length class no i. Thus,
provided that the mean annual growth parameters,
K and L1, are known, an estimate of size-independent
natural mortality for steenbras could be obtained fromlinear regression with log to catch per length class as
dependent variable and log potential growth span as
explanatory variable.

to Eq. (1):Z  F  M1 L1 =L:(7)

Inserting this expression for Z in Eq. (3), the newlength-based survivorship is again derived from simple integration (Beyer, 1989), L1 ÿ L Z1 =K L0 M1 =K;
N L  N L0 
L
L1 ÿ L0Z1  F  M1 ; L0  L  L1 :(8)

Thus the impact of natural mortality being inversely
proportional to length is that the number of ®shattaining length L is reduced by the survival factor
L0 =LM1 =K compared to the situation of a constant Z
equal to the minimum mortality level, Z1. Note againthat Z1 in Eq. (8) can be replaced by M1 as presently
there is almost no ®shing on the population. Eq. (5) forthe physiological catch rate with constant (size-independent) ®shing mortality is still valid for this or any
other case of size-dependent natural mortality as longas the appropriate survivorship is used for the survivors N(L). A similar expression of proportionality forsteady-state situations could, therefore, be obtained
but modi®ed according to the additional survival
factor, i.e.
ÿM1 =K
C L1 ; L2 =L / L M1 =Kÿ1
L1 ÿ L
ÿ1

L ÿ 1M1 =Kÿ1 :L1 =L

 taking logarithms and assumMultiplying with L,
ing constant class length, a new linear regression was
obtained,

2.3. Estimating natural mortality of infinitely old fish
(M1)

i   constant  M1 =K ÿ 1 ln L1 =L
i ÿ 1:ln Ci L(9)

In more realistic natural mortality scenarios where
smaller ®sh are gradually exposed to higher predationmortalities than bigger ®sh, the decline of numbers of®sh as they increase in length is no longer described by
Eq. (3), which is only valid for constant mortality. Itis, however, possible to relate M as obtained from theestimation procedure in Eq. (6) to M1 for old and big®sh (Section 3.2 and Appendix A). To establish acorrect basis for estimating M1 in the case of natural
mortality being inversely proportional to length, the®rst step taken was to modify total mortality according

This constitutes a very simple basis for estimating
the natural mortality of (in®nitely) old ®sh, M1, whennatural mortality increases for the younger and smaller®sh in inverse proportion to decreasing length(Eq. (1)). Assuming again the mean annual growthparameters, K and L1, are known, M1 for westcoaststeenbras could therefore be obtained from linear
regression using log to catchmidlength as dependent
variable and log to the relative potential growth spanas explanatory variable, i.e. log(max mean length/
midlengthÿ1)log(potential growth span/midlength).

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±1532.4. Simulation model of mortality regimes withindividual growth variation
A deterministic simulation programme was developed using simple (EXCEL) spreadsheets. Input parameters for growth are ®rst L1, K and t0 for specifying
mean annual VBGE, and secondly the constant coef®cient of variation in length at age, , for quantifying
the combined effect of individual growth variation andan extended spawning season. The mortality regime
considered is one of constant (size-independent) mortality, M, or one of mortality decreasing in inverse
proportion to increasing length according to Eq. (1)and speci®ed by M1. Selection is quanti®ed by astandard S-shaped logistic curve (e.g. Sparre andVenema, 1998) speci®ed by the length at 50% selection, L50, and the parameter r of selection range:
S L  1= 1  exp ÿr L ÿ L50 ;r  2 ln 3=SR;(10)where SR is the selection range.
1 , and using theStarting with the ®rst midlength, Lclass interval, L, the basic idea is to calculate theaverage length composition of the population of steenbras in a speci®c year by adding the contributions fromindividual age groups. The historical series of annual
recruitment, Ri, (i0,1,2,. . .), to age zero constitutes
the remaining input parameters. Suppose, as an example, 1996 is the year considered. If R010 000 in themodel then R1996 age 010 000 ®sh of age zero arerecruited to the population at the beginning of 1996.
The individual lengths of these ®sh are considered
normally distributed with mean length Lage 0
L1(1ÿexp(Kt0)) and standard deviation Lage 0.
At the end of 1996 the survivors of these ®sh havegrown to mean length Lage 1L1(1ÿexp(ÿK(1ÿt0)))
and their standard deviation in length has similarly
increased to Lage 1 (thereby maintaining the constant CV ). To simplify the mortality calculations
under the size-speci®c regime, the ®sh are considered
exposed to a constant rate of natural mortality
throughout 1996 which equals the rate, Eq. (1), using
the mean length of the ®sh in the middle of the year. In
the present application the model is used only tosimulate steady-state conditions so all Ri's are basically considered equal. For example, R3610 000
implies that also 10 000 ®sh were recruited at age
zero in 1960 of which only R1996 age 3610 000

139

exp(ÿCUM36) survive to age 36 years and arerecruited to the population at the beginning of
1996. Here CUM36 denotes the cumulative mortality
over the past 36 years calculated as the sum of the 36
annual levels (the ®rst of which is described above).
The recruits, R1996 age 36, are normally distributed withmean length Lage 36L1(1ÿexp(ÿK(36ÿt0))) andstandard deviation Lage 36. During 1996 thesebig ®sh will grow less than one length class, i.e. suchold cohorts will spread out over a number of lengthclasses which is determined entirely by the individual variability in size speci®ed by .
The calculations are done on a monthly basis. For
each age group the mean annual number of live ®sh ina month is derived and these ®sh are considered
normally distributed according to the common CV( ) and the mean length attained at the midpoint ofthat month. Using this normal distribution (for eachage group and month in the year considered), theexact proportion of ®sh which belongs to each lengthclass is calculated as the difference between the
cumulative normal distribution function at the upperand lower class limits of each class. Multiplying withthe mean annual number of ®sh (per age group andmonth) and adding up (over months and age groups)
then gives the complete (annual) length composition
for the population. The simulated length frequencies
for the sample are obtained by multiplying these
population frequencies with the selectivities,
Eq. (10), calculated at the class midpoints. Thesesample frequencies represent the Ci's in the linearized
length-based catch curves, Eqs. (6) and (9), and theprogramme was ®nalized by using the simulated
sample for these regressions. Fifty age groups wereused and the programme has been tested in various
ways to ensure suf®cient precision and accuracy forthe present application.
3. Results
3.1. Quantification of growth
The mean growth curve for the individuals com as a function of age
prising a cohort of steenbras, Lt, is considered described by VBGE with parameters estimated from the (non-linear) least square®t to the age-length data (Fig. 3(a)), i.e.

140

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±1531  70 cm; K  0:083 yrÿ1 and t0ÿ2.4 yr, soL  70 1 ÿ exp ÿ0:083 t  2:4:L(11)This mean growth curve did not change if estimation instead was done by minimizing the relative
residuals squared. It could not be rejected that therelative residuals were normally distributed(P0.05, 2-test) with standard deviation 0.116(Fig. 3(b)).
  0:072 yrÿ1
Assuming L170 cm resulted in K
using all 40 mark-recapture data for one year or longer
in freedom. One or two of the three possible outliers(found outside the estimated 0.5% and 99.5% probability levels; Section 2.1 and Fig. 4), however, represent outlier values in the present data. This is because
the Poisson probability, P(X3|0.4)0.8%, of getting three or more values when the expected value is400.010.4, is considerably smaller than 5%.
  0:069  0:005 yrÿ1
Omitting one outlier yielded K
SE, (n39). This estimate includes ®sh which had
spent an uneven number of seasons in freedom. Considering data for two or more years in freedom themost reliable, assuming a seasonal K-change of 30%
and, applying 10% precision criteria resulted in 112 yr
as the minimum accepted time spent in freedom. Of
the 40 recaptures previously considered (t1 yr), 19
met this criterion (t  112 yr; Table 1), including thethree outliers, and omitting one outlier yielded  0:083  0:007 yrÿ1 SE, (n18).
K
3.2. Estimation of natural mortality
Adding numbers at length from Fig. 1, the maximum frequency of the total number of westcoaststeenbras caught in 1995±1997 occurs at the(2.5 cm) length class starting at 40 cm. This pooleddata from the next class onwards (i.e. L42.5 cm) was
considered representative for the length frequencies in
the population (i.e. virtually 100% selection) and used
for the estimation of natural mortality assuming all®sh grow with L170 cm and K0.083 yrÿ1. Using
the linearized catch curve in Eq. (6) the estimate of the(maximum) size-independent natural mortality
yielded M0.41 yrÿ1 (Fig. 5(a)). Assuming sizedependent mortality we obtained M10.285 yrÿ1
(Fig. 5(b)) using the linearized catch curve in
Eq. (9). These estimates increased slightly if thelength frequencies for L42.5 cm alternatively wereTable 1Mark-recapture data for steenbras with t548 days free
L1

L2

L

t

K

27
27
28.5
28.5
31
31.5
32.5
33
33
33.5
34.5
35
35
35
35
38
45
32
36

35
35
35.5
35.5
38
37.2
38
40.5
37
38
40.5
39.5
42
41.8
42
41
53
40.5
46

8
8
7
7
7
5.7
5.5
7.5
4
4.5
6
4.5
7
6.8
7
3
8
8.5
10

704
900
929
940
848
936
937
1159
691
912
772
603
1162
847
891
888
968
581
658

0.107
0.083
0.073
0.072
0.085
0.062
0.062
0.071
0.060
0.053
0.088
0.083
0.070
0.093
0.091
0.040
0.145
0.159
0.193

Length when first caught (L1), length at recapture (L2) and growth(L) are given in cm. Single fish estimates of the VBGE parameter
K (yrÿ1) are all based on L170 cm. Possible outliers at thebottom.

pooled on a percentage basis (i.e. M0.42 andM10.294 yrÿ1 from equivalent plots (not shown)
to Fig. 5 by giving the tails (L42.5 cm) of the annual
length frequencies equal weight). Note that only ®vepoints were used for the regressions. This is mainly
because the larger the ®sh the more deviation from astraight line is anticipated due to a considerable
variation in length at age (Section 3.4). The usual
uncertainty, however, due to low catches of large ®shalso plays a role. This can easily be illustrated. Of thetotal catch of n16 174 ®sh, 272 represent the last ofthe ®ve points included (i.e. length class starting at
52.5 cm). The number of ®sh in such a length class (L)can be considered binomially distributed, Bin(n, pL),
with mean npL and variance npL(1ÿpL). We estimatepL272/16174, so the standard deviation of the number of ®sh in the length class becomes ca. 16. Thus a
95% con®dence interval for this number is approximately (240, 304). Introducing just one change due torandom chance by replacing the observed catch (272)
for the last point by the outer limit (304) will cause aslight reduction in the slope of the regression line in
Fig. 5(b) and the new estimate, M10.274, represents

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153141

speci®ed by M1, thenrepr =L1 ;M1  M L(12)

ÿ1

is the estimate of a constantwhere M0.41 yr
mortality obtained by the regression in Eq. (6) usingrepr a length range which is represented by L
48:75 cm, the midlength of that length range (Appendix A).
The results suggest the size-speci®c mortality
model, Eq. (1),
M L  20=L;

L1  70 cm; M1  0:29 yrÿ1 ;(13)

where M(L) is the instantaneous rate of natural mortality (yrÿ1) for steenbras of length L (cm).
3.3. Estimation of the resultant ogive from lengthbased catch curve

Fig. 5. Length-based catch curve analyses for the estimation ofnatural mortality for steenbras (L. aureti) in Meob Bay, Namibia,
using pooled length frequencies of rod-caught fish in 1995±1997(C in %) and assuming L170 cm. The (open circle) points used
for regression analyses represent the length range 42.5±55 cm.
Note small fish are to the right and big fish to the left in theplots: (a) constant (size-independent) mortality analysis, Eq. (6):y3.9043xÿ10.338 (n5, R20.99), i.e. M0.407 yrÿ1 for K
0.083 yrÿ1; (b) size-dependent mortality analysis, Eq. (9): y
2.4296x7.4696 (n5, R 2 0.99), i.e. M 10.285 yrÿ1 forK0.083 yrÿ1.

a reduction of 4%. Similarly the con®dence interval
for the number of ®sh representing the ®rst point (i.e.
length class starting at 42.5 cm) becomes (1627, 1783)
due to random chance. It is likely, however, that thecatch in this length class, which would truly re¯ect the
population in the sea, should be even higher due toselection (Section 3.3). With L5038 cm andr0.5 cmÿ1, Eq. (10) yields 1/S(43.75)1.056 andraising the catch by this factor gives 1800 ®sh forthe ®rst point included in the analysis. Introducing
instead this change causes a slight increase in the slopeof the regression line in Fig. 5(b) and the new estimate, M10.290, represents an increase of 2%.
It is of note that the estimates of M and M1 from
Fig. 5 are consistent with each other in the following
sense. If the method of constant M is (incorrectly) used
in a size-dependent mortality scenario, which is truly

The result of applying the Pauly (1984a) method tolength-based catch curves for a size-dependent mortality regime is a new method for estimating theresultant selection ogive, Sres(L). The idea is thatthe difference between the predicted catch (determined by the extension of the regression line towards
the smaller ®sh (i.e. to the right on Fig. 5(b)) and theactual numbers should represent the ogive resulting
from the combined effect of recruitment and gear
(hook) selectivity (Pauly, op.cit.). Denoting the intercept in the regression, Eq. (9), by `a' the predicted
catch in length class no. i (assuming 100% selection
under steady-state conditions), Cpred, i, is determinedfromi  exp a L1 =L
i ÿ 1M1 =Kÿ1 :Cpred;i L
Hence, from the condition that the actual catch
CiCpred,iSres,i the resultant ogive in the size-dependent mortality scenario is obtained as
i L1 =Li ÿ 11ÿM1 =K :Sres;i  Ci exp ÿaL

(14)

Fitting the logistic curve in Eq. (10), Sres(L), tothese class points for the resultant ogive (using standard non-linear least square) yields L5038.5 cm andr0.4 cmÿ1 for all 1995±1997 data (Fig. 6). If catch
data for 1997 is omitted L50 decreases slightly but r
increases to 0.55 cmÿ1 (Fig. 6 top). Both sets of
estimates
are
based
on
M10.285 yrÿ1,
ÿ1
K0.083 yr and L170 cm.

142

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

Fig. 6. Estimation of the resultant selection ogive, Sres(L), for therod-and-reel fishery of steenbras (L. aureti) in Meob Bay, Namibia,
from length-based catch curve analysis in a size-dependentmortality regime (M10.285 yrÿ1, K0.083 yrÿ1, L170 cm).
Points represent (resultant) selection by length obtained as the ratioof number of fish caught to the predicted number had the selection
been 100% (see text). The parameters of the fitted logistic curve are
L 50 37.6 cm and r0.55 for 1995±1996 catch data, andL5038.5 cm and r0.4 for all (1995±1997) catch data.

3.4. Robustness and bias of mortality estimation
methods
If not speci®ed otherwise results in this section referto steady-state calculations with the simulation model
based on the steenbras related parameter set
L170 cm, K0.083 yrÿ1, r0.5 cmÿ1, L50
38 cm, M10.30 yrÿ1 for the size-speci®c mortality
regime and M0.4 yrÿ1 for the constant (size-independent) mortality regime. The mortality considerations are independent of the constant annual
recruitment level since catches in numbers at length(2.5 cm classes) are calculated in percentages. Notealso that the mortality estimates are independent of
whether t0ÿ2.4 yr or say, t00 yr. The variation inlength at age as expressed by the constant coef®cientof variation, CV(L|age), constitutes a key parameter for method performance and population
dynamics.

3.4.1. Method performance in the steenbras case
The methods are producing precise and accurate
estimates of mortality, M10.30 yrÿ1 (Fig. 7(a)) andM0.40 yrÿ1 (Fig. 7(c)), when the underlying
assumption of negligible variation in length at age
is ful®lled. The ®rst (low) point on these graphs issomewhat to the left of the regression lines because of
too large class interval. These points represent highly
left-screwed distributions within the length class starting at 67.5 cm and ending at L170 cm, whichsimply are poorly represented by the class midpoint(68.75 cm) because no ®sh grow beyond L1 when
0. Moving the points ca. 0.25 cm down (toL68.5 cm) place them on the lines in the ®gures.
Using the wrong estimation method transforms theline into a convex curve (Fig. 7(b)) or a concave curve(Fig. 7(d)) depending on whether the true mortality
regime is governed by the M1-scenario or the constantM-scenario (Appendix A). This results in an overestimate (39%; M0.42, Fig. 7(b)) or an underestimate (29%; M10.28, Fig. 7(d)) of the mortality forlarge ®sh. Unbiased estimates, however, can beobtained by using Eq. (12) and Lrepr50 cm (Appendix A). Note that the concave shape of the points in
Fig. 7(d) is not very apparent because the ®rst pointshould be moved somewhat to the right.
In the steenbras case, the coef®cient of variation inlength at age is not zero ( 0; Fig. 7) but almost 12%( 0.116; Fig. 8). Catches in numbers at age are
spread out over several length classes with a considerable overlap for consecutive age groups and ®sh growbeyond L1. The ®rst points (for large ®sh) deviate themost from a straight line and convex catch curves areformed in all cases (Fig. 8). Both methods underestimate mortality (ca. 10%, M10.27, Fig. 8(a);vs. 15%, M0.34, Fig. 8(c)). Applying the wrong
method gives results (M0.37, Fig. 8(b); M10.24,
Fig. 8(d)) that are consistent with these biased mortality estimates (according to Eq. (12)).
Note that whenincreases (as from Fig. 7 toFig. 8), the fact that the mortality analyses are
restricted to ®sh<L1 will cause catch at length todecrease rather than to increase. The development of
the convex catch curves (Fig. 8) must therefore becaused by an interplay between mortality and growth(including individual variability). In the present casesof high mortality-to-growth ratios, M1/K3.6 andM/K4.8, occurring simultaneously with a large

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

143

Fig. 7. Performance of the mortality estimation methods in steady state with no variation in length at age ( 0) but otherwise based on
simulations of the steenbras case. Left panel shows results in the size-dependent mortality scenario M10.3 yrÿ1 whereas the right panel
refers to the constant mortality scenario of M0.4 yrÿ1. The four points (open circle) used for regression analyses represent the length range
45±55 cm: (a) M1-method: y2.59x7.66; (b) M-method: y4.01xÿ10.6; (c) M-method: 3.78xÿ8.91; (d) M1-method: y2.42x7.83.

variation in length at age, the decrease in numbers atmean length (from one cohort to the next) is much
faster than the decrease in numbers, within a speci®ccohort, from the mean length to neighboring classes(as described by the rather ¯at-top of a normal distribution with a CV of 12%). The increase in the catchof large ®sh in such cases (e.g. the ®rst points in
Fig. 8(a) compared to Fig. 7(a)) is therefore mainly
due to a `new' catch of big ®sh of later year-classes,
i.e. younger ®sh of which only the bigger ones havegrown to the size considered.
The model with size-dependent mortality does provide a good ®t to the tail of big steenbras in the catch
distribution (Fig. 9). In this model P[L77.5 cm]0.00010, so on average, the catch of n16 174 steenbras should comprise 1.6 ®sh of L77.5 cm. In theactual sample Lmaxca. 80 cm. The number (X) of®sh80 cm in such samples is binomially distributed,
Bin(n, p80); p80P[L80]0.00004, so the probability of catching one or more of these large steenbras
is P[X1]1ÿP[X0]1ÿ(1ÿ0.00004)16174 or ca.
50%. Similarly the catch of ®shL1 is binomiallydistributed with p700.0011, i.e. with mean
np7018 and standard deviation (np70(1ÿp70))1/2
4.2. Thus, according to the model, relating to a total
catch of 16 174 steenbras, (9, 25) is an approximative
95% con®dence interval for the number of ®sh, which
will be caught with a length greater or equal to L1. We
caught 18 (Fig. 1).
3.4.2. Sensitivity to changes and errors in the growthparameters
In steady state, with constant L1 and no variation inlength at age ( 0), the size structure of the popula-

144

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

Fig. 8. Performance of the mortality estimation methods in steady state with a considerable variation in length at age simulating the steenbras
case ( 0.116). Left panel shows results in the size-dependent mortality scenario M10.3 yrÿ1 and right panel refers to the constantmortality scenario of M0.4 yrÿ1. The four points (open circle) used for regression analyses represent the length range 45±55 cm: (a) M1method: y2.21x7.45; (b) M-method: y3.48xÿ8.89; (c) M-method: 3.08xÿ7.40; (d) M1-method: y1.93x7.49.

Fig. 9. Catch at length of steenbras (L. aureti)40 cm sampled in
Meob Bay, Namibia, throughout 1995±1997 (bars). Equivalent
catch percentages (curve) are obtained with the model understeady-state conditions in the size-dependent mortality regime(M10.285, K0.083, L170, 0.116, r0.5, L 50 38).
Lengths in 2.5 cm classes (numbers refer to lower class limits).

tion is exclusively determined by M/K and M1/K in
the constant and size-speci®c mortality regimes,
respectively. For all practical purposes simulations
show that this result is also true with a considerable
variation in length at age. For example, with 0.1,
catch length frequencies and, hence, the regression
line Eq. (9), remain unaltered when a situation of
M10.3 yrÿ1 and K0.1 yrÿ1 is changed into theextreme situation of M11.2 and K0.4 or
M10.15 and K0.05 (e.g. the deviations in estimated slopes are less than 1% (not shown)). All suchcases of M1/K3 produce the same length-based
population dynamics. If only K is doubled, however,K0.2 yrÿ1 and M10.3 yrÿ1, the mortality-togrowth rate ratio is halved, M1/K1.5, which drastically changes the length frequency distribution of thesample (Fig. 10). The ®sh grow twice as fast (as in theJ.E. Beyer et al. / Fisheries Research 41 (1999) 133±153145

Fig. 10. Steady-state catch representation with K0.2 yrÿ1 and considerable variation in length at age ( 0.1) but otherwise simulating thesize-dependent mortality scenario M10.3 yrÿ1 in the steenbras case (Fig. 8(a) and (b)): (a) percentage catches at length in 2.5 cm classes(numbers refer to lower class limits); (b) linearized catch curve: y0.508x6.52.

K0.1 case) and succeed in becoming bigger because
the mortality regime has not changed. If steenbras
were growing at this rate, we would expect
Lmax88 cm and 375 ®shL1 in the 1995±1997
sample. Despite this faster growth rate, a considerable
overlap in the length ranges of consecutive age groups
still exists. The performance of the linearized catchcurve method, however, improves considerably(Fig. 10(b)) mainly because of the impact of the
low mortality-to-growth ratio.
Another issue concerns the impact of possible errors
in the growth parameters used for the linear regressions, Eqs. (6) and (9), on the estimation of mortality.
The growth parameter K only appears in the slopes ofthese lines which implies that any relative error in K
produces exactly the same error on the mortality
estimate, (slope1)K. For example, using a 20%
underestimate of K in interpreting the slope of theline in Fig. 10(b) results also in a 20% underestimateof M1. The equivalent situation for K0.1 yrÿ1
results in a 25% underestimate of M1 because theeffect of the M1/K3 (and 0.1) scenario is alreadyan underestimate of M1/K by 6.4% (not shown). The
other growth parameter, L1, appears only in the logcalculation of the explanatory variables. An overestimate of L1 causes therefore horizontal movements ofthe points (to the right). The points will move towards
creating a concave curve with such increasing L1errors because the impact is largest on the biggest ®shclose to the true L1 (e.g. the ®rst points in Fig. 8(a)and (c)). Suppose, for example, L180 cm is (incorrectly) used for the mortality estimation in the M1scenario of Fig. 8(a) (which is truly generated with
L170 cm) then a perfectly straight line appears(Fig. 11(a)). The convex shaped (caused by0.116) is in this case balanced out by the 14%
overestimation of L1. The equivalent straight-linesituation to the M-scenario of Fig. 8(c) occurs when
L185 cm is (incorrectly) used (Fig. 11(b)). Bothsituations turn the ( -contaminated) underestimatesof mortality (Fig. 8(a) and (c)) into overestimates(8.5%, M10.33, Fig. 11(a); 34%, M0.53,
Fig. 11(b)). In general the -effect can be counteracted by an appropriate L1-error. Simulations showthat using the (incorrect) value of L1, which produces
a straight line, improves the estimate of M1 (but not
the estimation of a constant M). When 1<M1/K<3 the-effect is small (not exceeding a few percent), so onlya slight adjustment of L1 is needed for obtaining astraight line. Note that for M1/K<ca. 1.5, the -effectis a slight overestimate of M1/K so the adjustment
involves an underestimate of L1. When M1/K>3, thebias (underestimate) of M1/K caused by high -values
is considerably reduced by this concept of a straightline-L1. For example, in case of a CV of 15%, theunderestimates (due to this 0.15) increase from
ÿ11% to ÿ25% when M1/K doubles from 3 to 6.
However, the overestimates by using straight-lineL1's (81 and 93 cm, respectively) increase only from
8% to 11%. In these examinations, a straight-line-L1

146

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

Fig. 11. Apparent excellent performance of the mortality estimation methods in steady state when L1 is overestimated. The scenaria refer tothe steenbrass case as in Fig. 8(a) and (c) (L170 cm, K0.083 yrÿ1, 0.116, L5038 cm, r0.5 cmÿ1) with (a) M10.3 yrÿ1 and (b)M0.4 yrÿ1. The four points (open circle) used for regression analyses represent the length range 45±55 cm: (a) L180 cm incorrectly usedin M1-method: y2.92x6.90; (b) L185 cm incorrectly used in M-method: y5.44x17.5.

was obtained as the (incorrect) value of L1 which
made the four-points-regression line (i.e. based on
range 45±55 cm; e.g. Figs. 7 and 8) coincide withthe eight-points-regression line (i.e. based on range
45±65 cm). Using all 10 points (45±70 cm) in thesteenbras case (Fig. 5(b)) gives the best straightline-®t for L180 cm resulting in M10.304 (notshown). The equivalent case in the simulation model(M10.285, K0.083, L170, 0.116) yields
M10.307.
Simultaneous errors of opposite direction in theestimation of K and L1 tend to cancel out in their
combined effect on the mortality estimate (Table 2).
In extremes situations, however, the errors may givean additive effect. If the slope of the unbiased
linearized catch curve, Eq. (9), is negative (i.e. when
M1/K<1) then the effect of overestimating L1,(which always creates a steeper line), becomes
one of underestimating M1/K that will add to theeffect of underestimating K. An example of such asituation is M1/K0.75; the additive error effectpertains for all reasonable values of (Table 2), but
a suf®ciently high -value can turn the line down tohorizontal level which neutralizes the impact of L1errors (e.g. in the M1/K0.75 case, independently of
L1-errors, 0.25 implies virtually an unbiased estimate of M1 when K is 20% underestimated (butcreates a 56% overestimate when K is 20% overestimated)). At the other extreme shown in Table 2,
M1/K6, more than a 10% overestimate of L1 isrequired to counterbalance the impact of>0.05.

when

3.4.3. Sensitivity to recruitment variations
The issue concerning the impact of recruitment
variability on the performance of the M1-estimation
method is addressed with reference to a situation ofhigh M1/K ratio (the steenbras case; Fig. 7(a),Fig. 8(a)) and to a situation of low M1/K ratio(Fig. 10(b)). Annual recruitment to age zero (Section 2.4) is considered log-normally distributed witha CV of 50%. In the model, each realization of therecruitment process therefore consists of a series of 51
recruitment numbers which represent values of identical and independently distributed (IID) random variables. Using each of such recruitment series as input,
the model delivers length-based catch sample andrelated linearized catch curve analysis for the sizedependent mortality regime considered (M10.3).
Assuming the catch sample refers to 1996, the recruitment series refer to year-class strength for the years
1946±1996. Results are independent of the (constant)level of mean annual recruitment. Results in terms ofwhich year-classes are important for estimating M1,
however, are not independent of t0.
For clari®cation the (unrealistic) situation of novariation in length at age ( 0) is ®rst considered.
In the steenbras case (K0.083, L170, t0ÿ2.4) the
length range 45±55 cm was used for obtaining theregression line (Fig. 7(a)). These size limits areJ.E. Beyer et al. / Fisheries Research 41 (1999) 133±153147

Table 2
Bias in M1-estimation due to 10% error in the estimate of L1 and an opposite 20% error in the estimate of K
Bias in estimationM1-scenarioa10% (L1)
M1/K

ÿ1

K (yr )b

M1/K (%)

ÿ20% (K)ÿ10% (L1)20% (K)M1 (%)b

M1/K (%)

M1 (%)

c

0.025
0.025
0.025
0.025

6.0
3.0
1.5
0.75

0.05
0.10
0.20
0.40

16
8.5d
5.7
ÿ12

ÿ7.1
ÿ13d
ÿ15
ÿ30

ÿ23
ÿ19d
ÿ11
7.2

ÿ7.2c
ÿ2.7d
7.1
29

0.05
0.05
0.05
0.05

6.0
3.0
1.5
0.75

0.05
0.10
0.20
0.40

11
7.5d
6.1
ÿ13

ÿ11
ÿ14d
ÿ15
ÿ30

ÿ25c
ÿ20d
ÿ11
6.7

ÿ10c
ÿ3.4d
7.4
28

0.1
0.1
0.1
0.1

6.0
3.0
1.5
0.75

0.05
0.10
0.20
0.40

ÿ1.9
7.6
8.4
ÿ15

ÿ22
ÿ14
ÿ13
ÿ32

ÿ31c
ÿ22d
ÿ9.2
5.3

ÿ17c
ÿ6.2d
8.9
26

0.125
0.125
0.125
0.125

6.0
3.0
1.5
0.75

0.05
0.10
0.20
0.40

ÿ7.3
4.6
9.9
ÿ9.8

ÿ26
ÿ16
ÿ12
ÿ28

ÿ34c
ÿ23d
ÿ8.3
8.6

ÿ20c
ÿ8.0d
10
30

Results are based on catch curve analyses with a size-dependent mortality regime (40±55 cm length range used for regressions). Underlying
catches at length are produced by the simulation model in steady state.
aL170 cm, M10.3 yrÿ1, r0.5 cmÿ1, L5038 cm.
bThis dimensionless mortality-to-growth ratio is obtained as slope1 from regressions using (incorrectly) either 77 or 63 cm for L1 (asspecified above). Multiplying with the biased estimate of K gives M1-estimate in the next column. Only percentage deviations from true
values are shown.
c
Length range 40±50 cm used for regressions (due to shift in the mode).
dLength range 42.5±52.5 cm used for regressions.

reached at age 10 and 16 yr, so 1980±1986 are theseven recruitment years of real importance. This wascon®rmed by simulation (the slope of the line in
Fig. 7(a) did not change when recruitment varied in
the years 1946±1979 and 1987±96). Simulations withrecruitment variations for 1980±1986 yielded M1estimates (using the same four points as in Fig. 7(a))
 1  0:29. Extending
with a CV of 24% and mean M
the length range for the regressions to 10 classes, 40±
65 cm, and performing simulations with recruitment
variations in all years, 1946±1996, increased the pre 1  0:293; n50 realizacision (CV(M1)4.7%, (Mtions).
With the considerable variation in length at age forsteenbras, 0.116, later (>1986) as well as earlier(<1980) year-classes may contribute to the four-points
M1-estimation in Fig. 8(a). Considering 95%-con®dence limits to determine these contributions gives thecriteria Llower(12 )45 and Lupper(1ÿ2 )55 or
Llower36.5 cm and Lupper71.6 cm. Since on anaverage ®sh attains 36.5 cm at age 6.5 yr, year-classes
1987±1989 will contribute to the catch of 45 cm or1 , allbigger ®sh (right-tail effect) and since Lupper > L
year-classes before 1980 may potentially contribute to
the catch of 55 cm or smaller ®sh (left-tail effect). It
was con®rmed by simulation that the slope of the line
in Fig. 8(a) did not change when recruitment varied in
the years 1990±1996. Neither did recruitment variations for 1946±1979 affect the results (CV(M1)
0.5%) because of the high mortality-to-growth ratio,
M1/K3.6, i.e. negligible catches of smaller but older®sh in the 45±55 cm domain. The performance of theM1-estimation method in Fig. 8(a) is therefore determined by recruitment variations in 1980±1989. Simulations show reasonable precision, CV(M1)8.6%,
 1  0:267  0:0023 SE,and high accuracy, M

148

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153(n100). Note that 0.267 is the -affected M1(under)estimate obtained with constant recruitment(Fig. 8(a)). If the length range for the regressions
is doubled the precision improves at the cost of 1  0:253
less accuracy (CV(M1)6.4%, M
0:0023 SE, (n50), eight points (40±60 cm)).
Considering the situation of low mortality growthratio, M1/K1.5, in Fig. 10 to represent 1996, thebasic catch contribution to length range 45±55 cm
comes from only three year-classes, 1989±1991, i.e.
ages 5±7 yr due to the higher growth rate (K0.2,
t00). With 0.1 (Fig. 10(a)), however, ®sh bigger
than the average for ages 3±4 (1992±1993) and also
®sh smaller than the average for ages 8±9 (1987±1988)
are present in the population in suf®cient numbers tocontribute to the catches of 45±55 cm ®sh. The (usual
four-points) estimates of M1 change only slightly
when recruitment varies for the years 1994±1996
and 1946±1986 (CV(M1)2.0%, n40). Only theseven years, 1987±1993 (ages 3±9 yr), are importantwith respect to the impact of recruitment variability on
M1-estimation. The precision of the mortality esti 1  0:29  0:012 SE,
mates (CV(M1)29%, M(n50), four points (45±55 cm)) increases when thelength range for regression is doubled (CV(M1)
 1  0:301  0:0073 SE, (n50), eight
17%, M
points (45±65 cm)).
3.5. Application of steenbras mortality analysis tosilver kob
The silver kob (A. inodorus) stock in Namibia iscurrently being assessed using cohort analysis (e.gSparre and Venema, 1998). One of the input parameters required for a VPA is an estimate of the rate of
instantaneous natural mortality. This ®shery model isvery sensitive to changes in M when ®shing mortality(F) is relatively small, such as for silver kob in
Namibian waters. For this application it is assumed
that westcoast steenbras and silver kob, of the same
size, die at approximately the same natural mortality
rate. This seems to be a reasonable assumption, as thestudy of Lorenzen (1996) showed a good correlation
between mean weight and the natural mortality rate for®sh living in a natural ecosystem i.e. any ®sh with acertain weight will have approximately the same
natural mortality rate. Although westcoast steenbras
and silver kob are two different ®sh species whichhave different growth rates and might differ in theirsusceptibility to disease and senescence, this assumption might be feasible since, as implied by Cushing(1974) and Carpenter et al. (1985), the highest contribution to natural mortality is predation. Westcoaststeenbras and silver kob basically share the same
habitat (viz. the surfzone from Meob Bay north toCape Frio (188260 S, 128000 E)) and are therefore
exposed to the same predators such as sharks andfur seals.
Assuming that kob's natural mortality is governedby a similar natural mortality pattern to that of westcoast steenbras of approximately the same length, thenatural mortality of in®nite old kob (M1 kob) was
calculated as 0.2 yrÿ1 by using the following equation:
M1 kob  M1 steenbras  L1 Steenbras =L1 kob ;where M1 of westcoast steenbras is 0.29 yrÿ1 andFig. 12. Natural mortality at length for steenbras (L. aureti) and silver kob (A. inodorus) assuming these species are exposed to the same sizespecific predation mortality, M(L)20/L, due to predation from e.g. sharks and fur seals, which is believed to constitute the main component of
natural mortality.

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

L170 cm while L1 of kob is 103 cm (own data).
Fig. 12 shows size-speci®c natural mortality of kob
obtained in this way (i.e. Eq. (13)).
4. Conclusion and general discussion
An extremely simple size-speci®c model of natural
mortality is considered as an alternative to the usualassumption of a constant value of natural mortality for
all sizes, M. Mortality is considered inversely proportional to length and the model is completely speci®ed1 , the mean length at old
by M1, the mortality at Lage. This model can explain in detail the tailing-off in
the length frequencies of samples of mature ®sh froman unexploited population of westcoast steenbras. It
accounts for high numbers of smaller ®sh in the
population and thus provides good descriptions of
the apparent selection ogive using the Pauly(1984a) concept of resultant curve. In general themodel can explain the one in a million survival whichgoverns recruitment to most ®sh stocks (Beyer, 1989:
Example 1).
Simple, linearized and length-based catch curve
methods are presented to estimate the mortality-togrowth ratios, M/K and M1/K, which determine thelength structure of the population in the constant andsize-speci®c mortality regimes, respectively. The
methods are developed under the traditional assumptions of steady-state conditions and no variations inlength at age, CV(L|age) 0, and furthermore
1 is known. Under these conditions
require that L
the methods result in precise and accurate ratio-estimates. In general, however, lengths of individual ®shof the same age are characterized by a considerable
CV, probably around 10%, and the straight lines turninto convex curves. Considering 10% precision criteria and applying the methods with the insight gained
from simulations, the bias caused by this CV-effect is
not serious when 1<M1/K<3. Higher values of M1/K
lead to serious underestimates when CV exceeds 10%.
In general the bias can be reduced to acceptable limits
1 method, i.e. using
by applying the straight-line ± L
1 which counteracts the CV-effectthe value of L
thereby producing a straight line catch curve. Thisis important because it allows for the estimation of thefundamental length-based mortality concept, M1/K,
without prior knowledge on growth. Straightforward149

modi®cations of ``the Jones length-based (pseudo)cohort analysis'' (Jones, 1976; Jones and van Zalinge,
1981; reviewed in Jones, 1984; Pauly, 1984b; andSparre and Venema, 1998) show that knowledge ofM1/K alone is suf®cient for carrying out length-based
VPA (steady state) for exploited stocks under various
L1-scenarios with size-speci®c natural mortality.
Thus total annual catch by length and an estimateof M1/K constitute the minimum data requirement for
performing complete ®sh stock assessments and yieldpredictions. Work on this important issue for tropical®sh stock assessment and related CV-effects is inprogress.
4.1. Methods
The constant CV-growth model (Beyer and Lassen,1994) provides in a ®rst approximation a descriptionof the combined effects of an extended spawningseason and individual variability in growth. CV( ) constitutes an important parameter of population
dynamics. Unfortunately, values of CV are seldom
reported in the literature although such values arereadily obtained (and available) for most stocks either
by analyzing length at age (Fig. 3) or from Bhattacharya analyses and related methods for splitting acomposite length distribution into separate normal
distributions (Sparre and Venema, 1998, and references herein). Myers (1989) found an average CV ofca. 12% for cod but also that CV tends to increase witholder age; a phenomenon which may have generalvalidity as e.g. individual-based stochastic simulation
models of growth often reveals this feature (e.g. Beyer
and Laurence, 1980). This may imply an even greaterCV-effect than demonstrated in this study. Although
the constant CV-growth model is not intended toconsider individual growth trajectories as smooth
VBGE-curves, the implication of the model is thatan individual of length L0i at age 0 attains length t=L0 at age t where L t is the mean
Li t  L0i  L
0 similarly
length at age t in the cohort (Eq. (11)) and L
denotes the mean length at age 0. Thus, in the constant
CV model, individuals are considered to grow with thesame K but with individual values of L1 and they all
grow with the same speci®c growth rate (because theÿ1 
relationship above implies Lÿ1
i dLi =dt  L dL=dt.
Disregarding very small ®sh it should be mentionedthat mean size at age for all practical purposes is not150

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

affected by the size-selective mortality used in thisstudy.
An important aim of this paper has been to use andapply simple methods and deterministic models rather
than to optimize the various statistical estimation
procedures. It is clear that under more realistic
assumptions concerning the variation of individual
growth rates, estimation procedures for the growthparameters can be improved (Sainsbury, 1980; Francis, 1988). The linearized catch curve, Eq. (9), also
possesses simplicity at the cost of introducing someinterdependency between x and y (created through thedependency in both x and y on the class midlengths).
The effect of this type of x±y-interdependence (which
also occurs in many of the traditional methods of ®shstock assessment) can be neglected in most applications.
The (length-based) mortality estimation methods
are based on fewer approximations than the traditional
length-converted catch curve (Sparre and Venema,1998). Our methods appear robust for most species
as Beverton and Holt (1959) found that the ratio M/K
mostly lies in the range of 1.5±2.5. The sensitivity of
M1-estimates to opposite errors in the estimates of K
1 is relatively small (Table 2). This analysis isand L
based on ``the Pauly empirical phi-prime formula''(e.g. Pauly and Munro, 1984), which can be expressed
as K / Lÿ2
1 (where the coef®cient of proportionality isalmost constant within closely related ®sh species).
Applying this formula to independent trials of estimating the growth parameters based on different
length frequency samples for the same species (hadthese been available) implies that e.g. a 10% overestimate of L1 is likely to be accompanied by a 20%
underestimate of K. The considerable recruitment
variations obtained with a CV of 50% is likely torepresent the right order of magnitude for most species(Mertz and Myers, 1996). The mortality estimates are
relatively insensitive to this variation unless M1/K
attains low values in which case it is important tostabilize the estimation procedure by including thepoints representing the large ®sh (but<L1) in the
regression although these points are based on fewer®sh. For large values of M1/K the estimation procedure can instead be sensitive to the CV- or -effect but
then the straight-line-L1 method is recommended.
Note that this method does not work for the estimationof M/K in case of constant mortality.

4.2. Case study
This study provides the ®rst estimates of growth andmortality of westcoast steenbras reported in the literature. The only scienti®c ageing and growth study
done on westcoast steenbras in the past comprised 82
otolith readings by Lucks (1970). Unfortunately, theunderlying method of ageing was not documented
clearly and neither did Lucks (op.cit.) attempt toobtain estimates of the growth parameters based on
his tentative age±length key. With ®sh attaining
lengths of 40 cm and more at age 1 and ca. 75 cm
at age 6 yr, his results indicate a growth rate which is atleast three times faster than we have estimated for
Meob Bay steenbras. It is understandable, however,that the results by Lucks (op.cit.) would be somewhat
different from ours because he collected his steenbras
samples from a different population (Van der Bankand Holtzhausen, unpublished genetic data). Considering mortality, it is not possible to make a clear
differentiation between the two regimes considered(Fig. 5) although the M1-scenario appears slightly
more convincing than the M-scenario (i.e. by comparing Figs. 5 and 8). This is because the differences
between applying true and false methods happen to bevery small for both scenarios in this case (Fig. 8(a) vs.
Fig. 8(d); Fig. 8(c) vs. Fig. 8(b)). Our present knowl1 and conforms to the constantedge of M1, K, L
CV-growth model, explains the observed Lmax andprovides an adequate description of the length structure for 40 cm in the 1995±1997 total sample ofmore than 16 000 steenbras. The independent resultsfrom tagging are also in agreement with these growthestimates. As we, in the model, are considering K
constant but L1 to vary at the individual level, themark-recapture data in Table 1 could alternatively be1 assuming K ®xed (i.e.
used to estimateand L
solving Eq. (2) with respect to L1). This gives resultsin good agreement with the age±length data, e.g.
1  69 (n18) with K0.08.
0.12 and L
Several issues need further clari®cation. This
includes the relatively high (negative) value of t0(ÿ2.4 yr for a long-lived species), seasonality ingrowth as well as the bimodality in the length frequencies for 1995 and 1997 (Fig. 1). The bimodality
can be explained by recruitment variations using thesimulation model (not shown). However, westcoaststeenbras is a protandrous species and it is likely that

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

this complicated life history adds to the natural variations, in particular during immature growth and thetransient stages. The controversial issue of hook selectivity also needs further experimentation and studying
in continuation of the present pilot experiment (Fig. 2)
although many authors do use the trawl-type of selection curves (e.g. Booth and Buxton, 1997; Grif®ths,1997) as also suggested by the results from this study.
Work on these and related issues of the population
dynamics of westcoast steenbras is in progress.
The application of the steenbras mortality analysis
to other species sharing the same habitat (Section 3.5)
appears promising. Grif®ths (1997) determined
instantaneous natural mortality for the two South
African A. inodorus stocks by using relationships
among life history parameters (viz. Gunderson andDygert, 1988; Boudreau and Dickie, 1989; Hoening,1983). He assumed that M is between 0.1 and 0.2 yrÿ1,which seems to be a reasonable estimate as an input
parameter for the yield-per-recruit model, as ®shingmortalities for the South African silver kob stocks arehigh (e.g. ranging between 0.37 and 0.47). One of the
future objectives of the assessment of the Namibiankob stock would be to investigate the effect that theprotection of young ®sh would have on the total
biomass. Therefore it is the value of M at the timeof recruitment that would be important, and currently
no information on the value of M, and in particular on
size-speci®c M, is available for silver kob off the
Namibian coast.
While e.g. the mark-recapture results indicate that
the Meob Bay area has a closed population of westcoast steenbras, the same assumptions cannot be madeof the kob population. Recaptures recorded elsewhereof kob marked-and-released in the Meob Bay areasuggest that part of (if not all) kob move out of the area
for at least a certain time of the year. Mark-and-release
surveys indicate that kob move out of the area with theonset of winter in May/June, only to return with theonset of summer in November/December. Therefore,
using ``catch curve analysis'' to estimate M, could not
be applied, as the estimate of Z would contain an
unknown migration coef®cient.
In conclusion, it can be said that traditionally inmost stock assessment models, only a ``guesstimate''of average natural mortality is used currently in singlespecies assessments. Such an assumption of sizeindependent M is probably suf®cient where only the151

adult stock is considered (in particular for smaller
species), but, being able to estimate natural mortality
for a certain size class, will improve the accuracy of
the assessment, in particular when dealing with a large
range of body-sizes for larger species (i.e. slow-growing and long-lived species).
Acknowledgements
We thank Achim Lennsen of the Namibian Ministryof Environment and Tourism, the Oranjemund
Angling Club for logistic support and to their members and all the other anglers that accompanied theLine®sh section on their surveys to Meob Bay for theirendless ®shing efforts. Special thanks to our technicalteam, Stefanus and Shaun for their sampling efforts
and the Director of the Namibian Ministry of Fisheries
and Marine Resources for ®nancial support and for
permission to use the data for publishing. We are
grateful to Hans Lassen for providing the basis of
the Excel-simulation program.
Appendix A
Relationships between mortality estimates
The linearized length-based catch curves, Eq. (6)for M and Eq. (9) for M1, were derived as approximations under the assumption that the impact ofvariation in length at age can be neglected. In thisappendix these curves are considered to represent
exact relationships between catch at length (C) andlength (L) for the respective mortality regimes.First,
assuming that mortality is size-dependent as speci®edby Eq. (1), Eq. (9) statesln CL  constant  M1 =K ÿ 1 ln L1 =L ÿ 1or by taking the logarithms,
ln C  constant  M1 =K ÿ 1 ln L1 ÿ L
ÿ M1 =K ln L;
where yln C is the dependent variable andxln(L1ÿL) is the explanatory variable if the methodof constant M, Eq. (6), is (incorrectly) used. Hence
LL1ÿex, soy  constant  M1 =K ÿ 1x ÿ M1 =K ln L1 ÿ ex :152

J.E. Beyer et al. / Fisheries Research 41 (1999) 133±153

The change in the slope of this curve is given bydy M1 L1
ÿ 1;

K L
dx

L  L1 ÿ ex :The minimum slope is M1/Kÿ1 and occurs whenLL1. Thus, when the traditional constant M-method
expressed asln C  constant  M=K ÿ 1 ln L1 ÿ Lis used in the size-dependent mortality regime, theresult becomes a convex curve with slope that re¯ects
this increase in mortality for the smaller ®sh, i.e. theslope of the curve starts by producing the minimum,
MM1 for the very large ®sh, but then increases,
MM1L1/L, as L decreases (Fig. 7(b)). In practice,
however, it is not feasible to use the minimum slope
for estimating M1 because of the impact of CV>0 (forlength at age) (Fig. 8(b)) and the uncertainty in thecatch representation of the larger ®sh in the population. Instead it is better to consider the other extreme
where the impact of the unavoidable variation inlength at age (i.e. >0) is minimum and use themaximum slope (occurring for small but fully
recruited
®sh),
i.e.
M1  K 1  maximum
repr denotes the representativerepr =L1 where L
slopeL
mid-length for the small ®sh used in the M-regression.
Similar considerations apply to a constant mortality
regime if the size-dependent M1-regression is(incorrectly) used. In this case, yln(CL) and x
ln(L1/Lÿ1), the slope becomesdy M Lÿ 1;
dx K L1

L  L1 = 1  exp x;

which produces a concave curve (Fig. 7(d)). The
maximum slope (occurring for very large ®sh) represents the correct mortality, M1M, but as L decreases
the slope must also decrease to produce smaller
M1ML/L1 which re¯ects the constant (size-independent) mortality.
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