Empirical relationships to estimate asymptotic length, length at first maturity, and length at maximum yield per recruit in fishes, with a simple method to evaluate length frequency data
[Citation: Froese, R. and C. Binohlan 2000. Empirical relationships to estimate asymptotic length, length at first maturity and length at maximum yield per recruit in fishes, with a simple method to evaluate length frequency data. J. Fish Biol. 56:758773.]
R. Froese and C. Binohlan
International Center for Living Aquatic Resources Management (ICLARM)
MCPO Box 2631, 0718 Makati City, Philippines
Tel. No. +6328180466; Fax No. +6328163183; Email: r.froese@cgiar.org
ABSTRACT
Empirical relationships are presented to estimate in fishes, asymptotic length (L_{}) from maximum observed length (L_{max}), length at first maturity (L_{m}) from L_{}, lifespan (t_{max}) from age at first maturity (t_{m}), and length at maximum possible yield per recruit (L_{opt}) from L_{}_{ } and from L_{m},_{ }respectively. The age at L_{opt} is found to be a good indicator of generation time in fishes. A spreadsheet containing the various equations can be downloaded from the Internet at http://www.fishbase.org/download/popdynJFB.zip. A simple method is presented for evaluation of length frequency data in their relationship to L_{}, L_{m }and L_{opt}. This can be used to evaluate the quality of the length frequency sample and the status of the population. Three examples demonstrate the usefulness of this method.
Key words: yield per recruit; maturity; lifespan; generation time; lengthfrequency
INRODUCTION
About 7,000 species of fishes are used by humans in fisheries, aquaculture, sport fishing, or the ornamental trade (Froese and Pauly 1998). About 650 additional species are listed as threatened in the IUCN Red List of 1996 (Baillie, J. and B. Groombridge 1996). Life history information on growth and maturity, which is essential for proper management of exploited populations, is available for only about 1,200 species (Froese and Pauly 1998). This information is, however, sufficient to derive empirical relationships that can be used for management until specific data become available.
The mean length at which fish of a given population become sexually mature for the first time (L_{m}) is an important management parameter used to monitor whether enough juveniles in an exploited stock mature and spawn (e.g, Ault et al. 1998; Beverton & Holt 1959; Jennings et al. 1998). This parameter has been shown in various populations to be closely related to the mean length (L_{}) that the fish of that population would reach if they would continue to grow indefinitely (e.g., Alm 1959; Beverton 1992; Beverton & Holt 1959; Pauly 1984a; Stamps et al. 1998). The ratio of L_{m} to L_{} called ‘reproductive load’ (Cushing 1981) generally falls between 0.4 and 0.9 and appears to be relatively constant within taxa comprised of fish of approximately similar dimensions (Pauly 1984a). Mean length at first maturity is usually derived through linear interpolation, probit analysis, fitting of a logistic curve, or estimated from a plot of percent mature specimens over length (Binohlan 1998). Thus, estimation of L_{m} for any one population requires substantial data and is therefore unavailable for many species, especially in the tropics. To facilitate estimation of L_{m} in the absence of suitable data, we explored an empirical relationship linking L_{m} with L_{} and with other parameters such as sex, temperature and fecundity.
Holt (1958) pointed out that the maximum possible yield per recruit is obtained at an intermediate age t_{opt}, with corresponding length L_{opt}, where the product of the number of surviving individuals multiplied with their average weight results in the highest biomass, usually corresponding to the highest egg production (Beverton 1992). Estimation of L_{opt} requires knowledge of natural mortality (M) and of the von Bertalanffy growth function (VBGF) parameter K, two parameters that are not easily obtained. We therefore explored an empirical relationship between L_{opt} and L_{} to provide an estimation of this important management parameter.
Lifespan (t_{max}) and age at first maturity (t_{m}) are two important parameters in conservation management (e.g., IUCN 1994). We explored and present an empirical relationship between these parameters. We also suggest a method for estimating generation time in fishes.
Asymptotic length (L_{}) itself is highly correlated with the length of the largest individuals known from a population (L_{max}). We also provide an empirical equation for this relationship.
Length frequency data are routinely collected for fisheries management and as part of exploratory surveys (e.g., Asila & Ogari 1988; Piñero et al. 1997). We present a simple method that evaluates the quality of length frequency samples and the status of the respective population or fishery. The method can be applied to single species as well as to aggregated units of species of similar asymptotic length. A spreadsheet containing the equations described in this study can be downloaded from the Internet at http://www.fishbase.org/download/popdynJFB.zip.
MATERIALS AND METHODS
The symbols and abbreviations used in this study are summarized in Table I.
Table I. Symbols and abbreviations used in this study.
Parameter  Definition 
F_{t}
 Fecundity or number of offspring produced at age t. 
G
 Reproductive guild, given as bearer, guarder and nonguarder. 
K

Parameter of the VBGF, of dimension year^{1}, expressing the rate at which the asymptotic length (or weight) is approached.

L_{inf}

Asymptotic length in cm; parameter of the VBGF expressing the mean length that the fish would reach if they were to grow indefinitely.

L_{m}
 Mean length at first maturity, in cm. 
L_{max}

Length of largest individual reported for a locality, in cm.

L_{opt}

The length corresponding to t_{opt}, in cm.

S_{t}

Number of survivors to age t.

T

Annual mean surface water temperature in °C for the locality where growth study was conducted.

t_{g}

Generation time, i.e., mean age of spawners, in years.

t_{0}

Parameter of the VBGF expressing the theoretical ‘age’ in years the fish would have at length zero if they had always grown as described by the VBGF.

t_{opt}

Mean age in years at maximum possible yield per recruit. In most fishes also the age group with maximum egg production and thus the equivalent of generation time.

t_{m}

Mean age at first maturity, in years.

t_{max}

Maximum age or life span reached in a population, in years.

VBGF

Von Bertalanffy Growth Function, used to describe the growth in length or weight of fish.

For the estimation of L_{m} we derived pairs of L_{m} and L_{} from the MATURITY and POPGROWTH tables in FishBase 98 (Froese & Pauly 1998; http://www.fishbase.org). The MATURITY table holds about 2,600 records of length or age at first maturity for over 1,100 species, derived from over 400 publications (Binohlan 1998). For the purpose of this study, we only used records referring to mean or median length at first maturity or took the midpoint of a given range of values. The POPGROWTH table contains close to 4,900 records of growth parameters (L_{}, K) for over 1,200 species, derived from over 2,000 references (Binohlan & Pauly 1998a). Since we were interested in relationships applicable to wild populations we excluded growth and maturity studies done in captivity. As L_{} should not be too different from L_{max} (Pauly 1984a), we only considered estimates of L_{} that were within an arbitrary range of ± 30% of L_{max}, to exclude potentially problematic estimates of growth parameters.
About 80% of the estimates of L_{m} used here were provided by references that also contained estimates of growth parameters for the population. For the remaining 20%, we matched the best available estimate of L_{m} for a population in the MATURITY table with a corresponding value for L_{}in the POPGROWTH table, taking into consideration locality, sex, and type of length measurement (FL, TL, etc.). In some cases we converted L_{m} to match the type of length measurement used for L_{}.
Our selection criteria yielded 467 pairs of L_{m} and L_{}, encompassing a wide variety of finfish (Table II). Lampreys were excluded, despite the availability of estimates on length at maturity for some species, because they are known to shrink considerably in length after spawning (Hardisty 1986), and this makes the concept of asymptotic length difficult to apply.
Table II. Fish groups included in the L_{m}/L_{} regression analysis, in phylogenetic order.
Class
 Order 
No. of families

No. of species

Holocephali

Chimaeriformes (chimaeras)

1

1

Elasmobranchii

Orectolobiformes (carpet sharks)

1

1


Lamniformes (mackerel sharks)

1

1


Carchariniformes (ground sharks)

5

10


Squaliformes (bramble, sleeper and dogfish sharks)

1

1


Rajiformes (skates and rays)

2

8

Actinopterygii

Acipenseriformes (sturgeons and paddlefishes)

1

2


Osteoglossiformes (bony tongues)

1

5


Elopiformes (tarpons and tenpounders)

1

1


Clupeiformes (herrings)

2

27


Cypriniformes (carps)

1

2


Characiformes (characins)

3

7


Siluriformes (catfish)

5

7


Salmoniformes (salmons, pikes and smelts)

4

9


Aulopiformes (grinners)

1

2


Myctophiformes (lanternfishes)

1

3


Gadiformes (cods)

3

15


Lophiiformes (anglerfishes)

1

2


Atheriniformes (silversides)

1

3


Beryciformes (sawbellies)

3

4


Zeiformes (dories)

1

1


Gasterosteiformes (sticklebacks and seamoths)

1

1


Syngnathiformes (pipefishes and seahorses)

1

1


Scorpaeniformes (scorpionfishes and flatheads)

5

11


Perciformes (perchlikes)

35

121


Pleuronectiformes (flatfishes)

5

17


Tetraodontiformes (puffers and filefishes)

1

2

Total
 27 
88

265

We used the STATISTICA software for all the statistical analyses. The multiple regression procedure of the software was used to explore the relationship of (log) length at first maturity with log (L_{}), log (K), temperature (ºC), sex (female, male, unsexed), and reproductive guild (bearer, guarder, nonguarder) (Balon, 1975). Altogether 391 records were available with data for all five variables. Standard linear regression as well as stepwise forward and backward regressions were used to identify variables significantly correlated with length at first maturity. A simple linear regression analysis was performed using log_{10} values of the paired estimates of L_{m} and L_{}, for unsexed, male and female data. Because the values of the Xvariables are not free of error, the simple linear regressions presented in this study may only be used for predicting values of the Yvariables.
The length L_{opt }at which the total biomass of a yearclass reaches a maximum value can be calculated from L_{opt }= L_{} * 3 / (3 + M / K) (Beverton 1992). We only used records where the estimate of L_{} was within an arbitrary range of ± 30% of the species maximum length (L_{max}). We excluded records in which the estimation of M was not fully independent from growth parameters, as indicated in the POPGROWTH table and where the annual mean surface water temperature was below 2°C. We also excluded records where remarks questioned the quality of the data or reliability of the estimate, where the VBGF parameter t_{o} was smaller than –4 or larger than 0.5, where M differed by more than 150% from an estimate given by Pauly’s (1980) empirical equation, or where the VBGF parameters were outside an ellipse defining the 95% confidence interval for the species (Binohlan & Pauly 1998a). For the remaining 206 records we used the equation stated above to calculate L_{opt} and performed a linear regression analysis of log_{10}L_{opt} on log_{10}L_{}. For 76 of these records the POPGROWTH table also contained estimates of length at first maturity, which we used to estimate the relationship between L_{opt} and L_{m}.
For all records in POPGROWTH that passed the selection criteria above and had estimates of L_{}, K, and L_{m}, we calculated the corresponding ages t_{m} and t_{max}, assuming that t_{max} corresponds to the age at L_{}* 0.95 (Taylor 1958). We calculated the linear regression parameters for log_{10}t_{max} vs log_{10}t_{m}.
To estimate the relationship between L_{} and L_{max} we used data from the POPGROWTH and POPCHAR tables in FishBase 98. The POPCHAR table contains estimates of maximum length, weight, and age for over 1,500 populations (Binohlan & Pauly 1998b). We linked records from the two tables in a way that ensured that species, sex, length type, and country were identical, and the localities within the country were the same, or close to each other. That resulted in 563 pairs of L_{} and L_{max}. We excluded 12 potentially erroneous pairs for which the L_{max} / L_{} ratio was smaller than 0.5 or larger than 1.5. A linear regression analysis was done using log_{10} values of the paired estimates of L_{} and L_{max}.
RESULTS
In the multiple regression analysis of L_{m} as a function of sex (female, male, unsexed), temperature (T), reproductive guild (G: bearer, guarder, nonguarder), and growth (L_{}, K) all five variables were significantly correlated with length at first maturity (p<0.05). Some of the independent variables were highly correlated with each other (Table II). Table III presents the equations, r^{2}, and standard errors for various combinations of the five variables for predicting L_{m}.
Table II. Crosscorrelations between variables used in the multiple regression analysis of L_{m} as a function of sex, L_{}, K, temperature (T), and reproductive guild (G).

Variable

sex

L_{}

K

T

G

L_{m}

sex

1.00






L_{}

0.06

1.00





K

0.13

0.76

1.00



 T 
0.01

0.20

0.39

1.00


 G 
0.08

0.30

0.23

0.01

1.00

 L_{m} 
0.08

0.94

0.65

0.22

0.35

1.00

Table III. Regression equations estimating length at maturity from L_{} and from combinations of variables, namely growth parameters, reproductive guild, temperature and sex.
Independent variables

n

equation

r^{2}

s.e.

L_{}

467

Log_{10}L_{m} = 0.8979 * log_{10}L_{} 0.0782

0.888

0.127

L_{}female)

167

Log_{10}L_{m} = 0.9469 * log_{10}L_{} 0.1162

0.905

0.122

L_{}(male)

115

Log_{10}L_{m} = 0.8915 * log_{10}L_{} 0.1032

0.855

0.147

L_{}, T

391

Log_{10}L_{m} = 0.0431 + 0.8917 * log_{10}L_{}  0.001531 * log_{10 }T

0.882

0.129

L_{}, G

391

Log_{10}L_{m} = 5.1080 + 0.8741 * log_{10}L_{} + 0.05005 * log_{10}G

0.888

0.126

L_{}, G, T

391

Log_{10}L_{m} = 5.1761 + 0.8667 * log_{10}L_{} + 0.05115 * log_{10}G  0.00174 * log_{10}T

0.889

0.126

L_{}, K, G, T, Sex

391

Log_{10}L_{m} = 7.0351 + 0.9908 * log_{10}L_{} + 0.1682 * log_{10}K + 0.05187 * log_{10}G  0.0040 * log_{10}T + 0.01678 * log_{10}Sex

0.902

0.118

Fig. 1 shows a scatterplot of log_{10} L_{m} over log_{10 }L_{} for all 467data points with the regression lines for females and males. ANCOVA analysis shows the regression lines for females and males to be significantly different (n = 282, F = 4.3252, p< 0.05).
The relationship between L_{opt} and L_{} is shown in Fig. 2. L_{opt} can be estimated from the equation:
log_{10}L_{opt} = 1.0421* log_{10}(L_{}) – 0.2742 (n = 206, r^{2} = 0.97, s.e.= 0.073) (2)
If an estimate of length at first maturity is available, L_{opt} can be estimated from the following equation (Fig. 3):
log_{10}L_{opt} = 1.053 * log_{10}(L_{m}) – 0.0565 (n = 76, r^{2} = 0.89, s.e.= 0.139) (3)
Beverton (1992) stressed the relationship between age at first maturity (t_{m}) and lifespan (t_{max}). Our empirical relationship between t_{max} and t_{m} (Fig. 4) results in the equation:
log_{10}t_{max} = 0.5496 + 0.957*log_{10}(t_{m}) (n = 432, r^{2} = 0.77, s.e = 0.194) (4)
The relationship between asymptotic length and maximum length is shown in Fig. 5 and is described by the equation:
log_{10}L_{} = 0.044 + 0.9841 * log_{10}(L_{max}) (n = 551, r^{2} = 0.959, s.e. = 0.074) (5)
DISCUSSION
The data sets used in this study are dominated by estimates for perciform bony fishes, which might reduce the applicability to other groups. We used a routine in the beta version of FishBase 99 that highlights dots in Fig. 1 by taxonomic order. The scatter generated by perciform fishes overlapped with that of other species of similar size, and the scatter for all the orders with more than 10 species in Table II showed a more or less even distribution around the overall regression line. Thus, we suggest that the empirical equation presented in this study can be applied to all fishes for which the concept of L_{}_{}and von Bertalanffy growth are appropriate.
We decided to use the standard error of the estimate as provided by STATISTICA as measure of variance. The 95% confidence limits of the regression lines were unrealistically narrow, and there is no theory suggesting that the regression lines for all species—if we had enough data to estimate them—would fall within that confidence range. The 95% confidence limits for the estimates have about twice the range of the standard error and are thus unrealistically wide, with the upper range for the predicted L_{m} being larger than L_{}. This is probably caused by the fact that our data were drawn from a wide variety of studies using different methods, i.e., the actual variance in the relationships can be expected to be less than shown here.
The relationship between length at first maturity and asymptotic length explains 85% (males) to 91% (females) of the variance in the respective data sets (Table III), indicating that length at first maturity is foremost a function of size. This is true for fishes ranging from ancient chimaeras to derived pufferfish and spanning in asymptotic length from about 2.5 cm in the goby Mistichthys luzonensis to 14 m in the whale shark Rhincodon typus (Table II). Other variables, despite their considerable range did not contribute significantly to the explanation of the remaining variance (Table III). For example, the inclusion of temperature ranging from 2°C to about 30ºC, did not impact on the relationship, nor did the addition of reproductive guild, reflecting fecundity ranging from a few liveborn offspring in bearers to millions of eggs in nonguarders.
The difference in slope between the sexes indicates that in large fishes, females tend to mature at a slightly larger size than males (Fig. 1). The variance of L_{m} for a given L_{} seems to be higher below the regression line. This may result from estimates taken from heavily fished populations where the size structure of the population is changed by disproportionate removal of larger specimens, thus artificially reducing the observed mean length at sexual maturity. Also, a major reduction in population size may increase the relative abundance of food, which may result in faster growth, smaller asymptotic size, and smaller size at first maturity. Decrease of L_{m} over time in relation to fishing pressure has been noted for cichlids in African lakes (Lévêque 1997), for reef fishes in the Florida Keys (Ault et al. 1998) and for some heavily fished stocks in the North Sea (Jennings et al., 1998).
Pauly (1984b) and Longhurst & Pauly (1987) presented a theory, based on a limiting effect of the oxygen available for growth, which explains how speciesspecific relative gill surface area determines maximum size as well as size at first maturity. Absolute gill surface area itself is highly correlated with maximum length (Pauly 1998). Our empirical relationship allows for the replacement of the rarely available gill surface area parameter with the more widely available asymptotic length for estimation of size at maturity. For example, no length at first maturity is known for the commercially important brown surgeon fish (Acanthurus nigrofuscus) in Yap, Micronesia, where the asymptotic length (unsexed) has been estimated at 18 cm (Smith & Dalzell 1993). Applying the empirical equation for unsexed fishes (equation 1) leads to L_{m} = 10^(0.898 * log_{10} (18) – 0.0782) = 11.2 (s.e. 8.4  15.0) cm, resulting in a reproductive load of 0.62, which compares well with an estimate of 0.66 for a population in Yankee Reef, North Queensland (Hart & Russ 1996).
The empirical relationship between L_{opt} and L_{} explains 96% of the variance in the data set. It suggests L_{opt} values of 5.9 cm for fishes of 10 cm, 64 cm for fishes of 100 cm, and 706 cm for fishes of 10 m L_{}. The average ratio of L_{opt}/L_{} is 0.63 and applies well to fishes between 50 and 100 cm maximum length. If L_{opt} is estimated from L_{m}, the values for L_{opt} are 9.9 (s.e. 8.9 – 11) cm for 10 cm and 112 (s.e. 98 – 129) cm for 100 cm length at first maturity. Thus, in small fishes L_{opt} may be smaller than or equal to L_{m}, whereas in large fishes L_{opt} is usually larger than L_{m} (Fig. 3). Beverton (1992) pointed out that starting maturity at maximum biomass (L_{m} L_{opt}) would maximize egg production at the first spawning event, certainly an important factor in the reproductive strategy of small and usually shortlived fishes with high mortality rates. Interestingly, some small fishes do not seem to make use of this size for spawning, possibly because spawning season may arrive only at a later time, or other factors make it more advantageous to postpone reproduction.
Generation time (tg) is usually defined as the average age of the parents when their offspring are born and is calculated as
(6)
where is the number of survivors to age and is the average fecundity at age t (e.g., Au 1999). In fishes, fecundity is highly correlated with weight, and replacing in equation (6) with the average weight at age t would actually result in Holt’s (1958) age class with highest biomass (t_{opt}). Thus, t_{opt} as the age of maximum egg production in most fishes (Beverton 1992) appears to be a good estimation of generation time. If L_{m} is larger than L_{opt}, then the first length class that contains close to 100% spawners, as opposed to 50% at L_{m}, will have the highest spawning biomass. We suggest to arbitrarily calculate that length as , i.e., increasing by a quartile of the difference between and .
Branstetter (1997) suggested that predation risk decreases drastically above a length of about 1 m, e.g., in most sharks. Age of maximum possible yield is determined by mortality and growth, and a decrease of the M/K ratio in large species will shift t_{opt} and hence L_{opt} to older ages. Jensen (1996) presented several Beverton and Holt life history invariants and suggested that fish generally optimize their length at first maturity to coincide with the length class of maximum fecundity. He proposed an average L_{m}/L_{} and presumably L_{opt}/L_{} ratio of 0.66. This value is within the standard error range of our empirical L_{opt}/L_{} ratio. However, it fails to recognize the size dependence of the reproductive load, as pointed out by Pauly (1984a) and confirmed by our empirical equation. We suggest that instead the ratio L_{opt}/L_{} = 0.63 is a more stable Beverton and Holt life history invariant, explained by the fact that natural mortality and the rate at which the potential growth span is completed (K) are both strongly, if inversely correlated with maximum size (see also Beverton 1992 for a discussion of the M/K ratio).
For example, an asymptotic length of L_{} = 205 cm is given for the Nile perch (Lates niloticus) in Nyanza Gulf, Lake Victoria, by Asila & Ogari (1988). Our empirical equation calculates the length class with the maximum possible yield per recruit as L_{opt} = 136 (s.e. 115  161) cm, which is 8 cm larger than L_{opt} = 128 cm calculated from the VBGF and M values (see below). Asila & Ogari (1988) give an L_{m} = 102 cm estimate for females. Using this value results in L_{opt} = 114 (s.e. 83 –158) cm which is 14 cm smaller than the target value and comes with a very wide standard error range, thus suggesting that L_{} is a better predictor of L_{opt}.
If a good estimate of the age at first maturity is available, e.g., from direct observation in fishes with pronounced annual spawning peaks, lifespan can be directly estimated from equation (4). If L_{}, L_{m} and t_{m} are known, K can be estimated from the rearranged VBGF: K = ln(1 + L_{m}/L_{})/t_{m} (see also Beverton 1992).
For example, the age at first maturity for females of the critically endangered giant grouper (Epinephelus itajara) is given as t_{m} = 6.5 years in Bullock et al. (1992). Based on this value equation (4) suggests a lifespan of 21.3 (s.e. 14  33) years and from the equation K= 3/t_{max} (Taylor 1958), a rough estimate of K = 0.14 is obtained. The VBGF parameters estimated in Bullock et al. (1992) give L_{} = 201 cm and K = 0.13, resulting in t_{max} = 23 years at 0.95 L_{}, comparing well with our estimates.
The relationship between asymptotic length and maximum length (Fig. 5) explains 96% of the variance in the data set. Taylor (1958) suggested that fishes reach the end of their life span at 0.95 * L_{}. Based on this, Pauly (1984a) suggested a rule of thumb where L_{} L_{max}/0.95, i.e., asymptotic length was assumed to be about 5% longer than the maximum observed length. Our empirical equation suggests that this L_{max}/L_{} ratio changes with size, with a ratio of 0.94 for fishes of 10 cm, 0.97 for 100 cm, and 1.01 for 10 m maximum length. If we assume that the estimated asymptotic length in a lightly exploited population should be more or less equal to the maximum observed length, then it seems that either the von Bertalanffy growth function or the methods used to estimate its parameters overestimate L_{} in small fishes.
For example, only the maximum size of 150 cm is known for the pickhandle barracuda (Sphyraena jello) in South African waters (Torres 1991). Our equation suggests an asymptotic length of L_{} = 153 (s.e. 129  183) cm, which corresponds well with L_{} = 148 cm estimated for this species in the warmer waters of the Gulf of Aden (Edwards et al. 1985).
Note that our empirical equations for L_{}, L_{m,} L_{opt} and t_{max} were derived from multispecies data sets and can therefore be applied to aggregated management units of species with similar asymptotic length.
Estimates of L_{}, L_{m }and L_{opt} can be used to construct a simple framework for the evaluation of length frequency data. For example, Asila & Ogari (1988) presented length frequency data for the introduced Nile perch (Lates niloticus) in Lake Victoria. They estimated the following parameters: L_{} = 205 cm, K = 0.19 year^{1}, L_{m } = 74 cm for males and 102 cm for females, and M = 0.34 year^{1}. From these values we calculated L_{opt} = 128 cm. Asila & Ogari (1988) did not discuss the quality of their survey data or the status of the commercial fishery. Plotting their length frequency data for 1982 in a simple framework with L_{}, L_{m }and L_{opt} (Fig. 6) reveals the following:

The trawl survey performed in Nyanza Gulf, Lake Victoria failed to catch specimens above 100 cm length;

96% of the Nile perch caught by the commercial fishery were smaller than the size of maximum possible yield per recruit;

78% of the Nile perch caught in the commercial fishery were smaller than the length at first maturity of females; and

55% of the Nile perches caught in the commercial fishery were smaller than the length at first maturity of males, with a peak at about 55 cm, i.e., about two third the size at first maturity of males, and half the size at first maturity of females.
This simple analysis indicates that the survey data taken in Nyanza Gulf do not reflect the size distribution of the Nile perch population and that the commercial fishery will result in growth and recruitment overfishing if it continues its current exploitation pattern. Ochumba (1988) presents length frequency data of Nile perch from a fish kill in Nyanza Gulf in 1984, including specimens of up to 200 cm length, with 110 cm as the most common length class, thus confirming finding (1) and suggesting that the gear (=trawl) used in the Asila & Ogari (1988) study was inadequate to catch large specimens. Asila (1994) reports unstable Nile perch populations and a decline in commercial catches after 1986. This is attributed to overfishing of spawners in the preceding period, thus confirming our above suggestion of growth (2) and recruitment overfishing (34).
In another example, Piñero et al. (1997) present length frequency data from exploratory fisheries of the little studied North Atlantic codling (Lepidion eques). The largest individual caught measured 45 cm total length (TL). This maximum length agrees well with the 44 cm suggested by Cohen et al. (1990). For a maximum size of 45 cm TL, our empirical equations suggest L_{} = 47 (s.e. 40 – 56.) cm, L_{m} = 27 (s.e. 20  36) cm, and L_{opt} = 29 (s.e 25  35) cm. Our framework suggests that this is an unfished population as indicated by the presence of many large specimens above the L_{opt} range (Fig. 7). It also suggests that the trawl survey missed smaller and medium sized specimens, as indicated by the lower number of specimens in and below the L_{opt} range. Cohen et al. (1990) give the interest to fisheries of this species as “None at present” and describe the habitat and biology as: “Benthopelagic between 127 and 1850 m, with large fish living deeper”. The trawl surveys were conducted in depths of 500 – 1,250 m where smaller specimens and fishing are presumably rare, thus supporting our suggestions above.
As an example of a multispecies fishery, Lock (1986) presents length frequency data for fish caught by surface spearing in Papua New Guinea. He gives the catch composition as: 26,407 kg Tylosurus indicus, T. melanotus; 585 kg Strongylura leiura, S. incisa, 251 kg Scombridae; and 612 kg other fishes. Tylosurus indicus is a name of uncertain status for specimens collected in India (Eschmeyer 1998) and probably a misidentification of Tylosurus crocodilus, the only other large needlefish known from the area. For the identified 4 species we used maximum length estimates from FishBase and transferred them to standard length, i.e., 120 cm, 91 cm, 90 cm and 96 cm SL, respectively. We assumed that each species contributed half to the catch of its genus. We weighted the standard length for the species with their respective catches and derived a weighted average maximum length of 105 cm SL for the four species, a value that seems realistic because several specimens were caught at and above that size (Fig. 8). We then applied our empirical equations to estimate the ranges for L_{}, L_{m }and L_{opt} based on that estimate of standard length. Plotting our framework data over the length frequencies of Lock (1986) shows an apparently healthy, lightly fished population with most species above the size at first maturity and within the range of L_{opt}. This is not surprising, as spearfishing should have little impact on the oceanic, wideranging needlefishes that contributed 95% of the catch.
We hope that the equations and the simple framework presented in this study will prove useful to colleagues in speciesrich but datapoor situations.
F ig 1. Relationship between length at first maturity and asymptotic length for all records representing 265 species of fish. Regression lines are for females () and males ().
F ig. 2. Relationship between length at maximum possible yield per recruit (L_{opt}) and asymptotic length (L_{}).
Fig. 3. Relationship between length at maximum possible yield per recruit (L_{opt}) and length at first maturity (L_{m}).
F ig. 4. Relationship between lifespan (t_{max}) and length at first maturity (t_{m}).
F ig. 5. Relationship between asymptotic length and maximum observed length.
F ig. 6. Length frequency data of Nile perch catches in Lake Victoria plotted in a simple framework of L_{}, L_{m }and L_{opt}: upper graph shows commercial catch data of February 1982; lower graph shows trawl survey data taken in February 1982 in Nyanza Gulf (Asila & Ogari 1988).
Fig. 7. Length frequency data of North Atlantic codling Lepidion eques (Piñeiro et al. 1997) with ranges for L_{}, L_{opt} and L_{m} estimated from empirical equations. Note that the range of L_{opt} overlaps with that of L_{m}.
F ig. 8. Lengthfrequency data for surface spearfishing of needlefishes and barracudas (Lock 1986). Note that most specimens fall within the range between length at first maturity and maximum possible yield per recruit, indicating sustainable exploitation rates.
ACKNOWLEDGMENT
We thank Daniel Pauly, Maria L. Palomares and Michael Vakily for useful comments. We thank one of the reviewers for pointing out that age at maturity is not the same as generation time. We thank the FishBase team for compiling the data used for the empirical relationships in this study. This research was conducted under the joint Fisheries Research Initiative of African, Caribbean and Pacific (ACP) countries with the European Union and was sponsored in the framework of the capacity building project 'Strengthening of fisheries and biodiversity management in ACP countries' (7.ACP.RPR.545). ICLARM Contribution No. 1510.
REFERENCES
Alm, G. (1959). Connection between maturity, size, and age in fishes. Inst. Freshwat. Res. Drottningholm Rep. 40, 145 p.
Asila, A. A. (1994). Survival rates of Lates niloticus in Lake Victoria. In Recent trends in research on Lake Victoria fisheries (Okemwa, E., Wakwabi, E.O. & Getabu, A., eds.), pp. 5358. ICIPE Science Press.
Asila, A. A. & Ogari, J. (1988). Growth parameters and mortality rates of Nile perch (Lates niloticus) estimated from lengthfrequency data in the Nyanza Gulf (Lake Victoria). In Contributions to tropical fisheries biology: papers by the participants of the FAO/DANIDA followup training courses (Venema, S., MöllerChristensen, J. & Pauly, D., eds.) pp. 272287. FAO Fish. Rep., (389), 519 p.
Au, D.W. 1999. Protecting the reproductive value of swordfish, Xiphias gladius, and other billfishes. NOAA Technical Report NMFS 142:219225.
Ault, J. S., Bohnsack, J. A. & Meester, G. A. (1998). A retrospective (19791996) multispecies assessment of coral reef fish stocks in the Florida Keys. Fish. Bull. 96(3), 395414.
Baillie, J. and Groombridge, B. (eds.). 1996. 1996 IUCN red list of threatened animals. IUCN, Gland, Switzerland. 378 p.
Balon, E. (1975). Reproductive guilds of fishes: a proposal and definition. J. Fish. Res. Board Can. 32, 821864.
Beverton, R. J. H. (1992). Patterns of reproductive strategy parameters in some marine teleost fishes. J. Fish Biol. 41(Suppl. B), 137160.
Beverton, R. J. H. & Holt, S. J. (1959). A review of the lifespans and mortality rates of fish in nature, and their relation to growth and other physiological characteristics. In CIBA Foundation Colloquia on Ageing. Vol. 5. The lifespan of animals (Wohstenholme, G. E. & O’Conner, M., eds.), pp. 142180. London: J. and A. Churchill, Ltd.
Binohlan, C. (1998). The MATURITY table. In FishBase 98: concepts, design and data sources (Froese, R. & Pauly, D., eds.), pp.176179. Manila, ICLARM.
Binohlan, C. & Pauly, D. (1998a). The POPGROWTH table. In FishBase 98: concepts, design and data sources (Froese, R. & Pauly, D., eds.), pp. 124129. Manila, ICLARM.
Binohlan, C. & Pauly, D. (1998b). The POPCHAR table. In FishBase 98: concepts, design and data sources (Froese, R. & Pauly, D., eds.), pp. 120121. Manila, ICLARM.
Branstetter, S. (1997). Burning the candle at both ends. Sharks News, Newsletter of the IUCN Sharks Specialist Group, 9, 4.
Bullock, L.H., Murphy, M.D, Godcharles, M.F. and Mitchell, M.E. 1992. Age, growth, and reproduction of jewfish @Epinephelus itjara@ in the eastern Gulf of Mexico. Fish. Bull. 90:243249.
Cohen, D. M., Inada, T., Iwamoto, T. & Scialabba, N. (1990). Gadiform fishes of the world (Order Gadiformes). An annotated and illustrated catalogue of cods, hakes, grenadiers and other gadiform fishes known to date. FAO Fish. Synop. 125(10), 442 p.
Cushing, D. H. (1981). Fisheries biology: a study in population dynamics. 2nd ed. 295 p. Madison: University of Wisconsin Press.
Edwards, R. R. C., Bakhader, A. & Shaher, S. (1985). Growth, mortality, age composition and fishery yields of fish from the Gulf of Aden. J. Fish. Biol. 27, 1321.
Eschmeyer, W.N. 1998. Catalog of fishes. Special Publication, California Academy of Sciences, San Francisco. 3 vols. 2905 p.
Froese, R. & Pauly, D. Editors. (1998). FishBase 98: concepts, design and data sources. 293 p., with 2 CDROMs. Manila, ICLARM.
Froese, R. and Pauly, D. (Editors), Bailly, N. and M.L.D. Palomares (Translators). (1999). FishBase 99. Concepts, structure, et sources des données. 315 p., with 3 CDROMs. ICLARM, Manille, Philippines.
Hardisty, M. W. (1986). A general introduction to lampreys. In The Freshwater fishes of Europe. Vol. 1, Part 1. Petromyzontiformes (J. Holcík, ed.), pp. 1984. Wiesbaden: AulaVerlag.
Hart, A. M. & Russ, G.R. (1996). Response of herbivorous fishes to crownofthorns starfish Acanthaster planci outbreaks. III. Age, growth, mortality and maturity indices of Acanthurus nigrofuscus. Mar. Ecol. Prog. Ser. 136, 2535.
Holt, J. S. (1958). The evaluation of fisheries resources by the dynamic analysis of stocks, and notes on the time factors involved. ICNAF Special Publication I, 7795.
IUCN. 1994. IUCN Red list categories. IUCN, Gland, Switzerland, 21 p.
Jennings, S., Reynolds, J. D. & Mills, S. C. (1998). Life history correlates of response to fisheries exploitation. Proc. R. Soc. Lond. B 265, 333339.
Jensen, A. L. (1996). Beverton and Holt life history invariants result from optimal tradeoff of reproduction and survival. Can. J. Fish. Aquat. Sci. 53, 820822.
Lévêque, C. (1997). Biodiversity dynamics and conservation. The freshwater fishes of tropical Africa. 438 p. Cambridge: Cambridge University Press, Inc.
Lock, J. M. (1986). Study of the Port Moresby artisanal reef fishery. Department of Primary Industry, Fisheries Division, Port Moresby, Papua New Guinea. Technical Report 86(1), 56 p.
Longhurst, A. R. & Pauly, D. (1987). Ecology of tropical oceans. 407 p. San Diego: Academic Press, Inc.
Ochumba, P. B. O. (1988). Periodic massive fish kills in the Kenyan portion of Lake Victoria. FAO Fish. Rep. 388, 4760.
Pauly, D. (1980). On the interrelationships between natural mortality, growth parameters, and mean environmental temperature in 175 fish stocks. J. Cons. CIEM 39(2):175192.
Pauly, D. (1984a). Fish population dynamics in tropical waters: a manual for use with programmable calculators. ICLARM Stud. Rev. 8, 325 p.
Pauly, D. (1984b). A mechanism for the juveniletoadults transition in fishes. J. Cons. CIEM 41, 280284.
Pauly, D. (1998). The GILL AREA table. In FishBase 98: concepts, design and data sources (Froese, R. & Pauly, D. eds.), pp. 205208. Manila: ICLARM.
Piñero, C., Casas, J. M., Bañón, R., Serrano, A. & Calviño, A. (1997). Resultados de la acción piloto de pesca experimental en el talud de la plataforma Gallega (Noroeste de la Península Ibérica). Datos y Resúmenes No. 3, Instituto Español de Oceanografía, Madrid, 57 p.
Smith, A. & Dalzell, P. (1993). Fisheries resources and management investigations in Woleai Atoll, Yap State, Federated States of Micronesia. Inshore Fish. Res. Proj., Tech. Doc., South Pacific Commission, Nouméa, New Caledonia. 64 p.
Stamps, J.A., Mangel, M. & Phillips, J.A. 1998. A new look at relationships between size at maturity and asymptotic size. American Naturalist 152(3), 470479.
Taylor, C. C. (1958). Cod growth and temperature. J. Cons. CIEM 23, 366370.
Torres, F. S. B., Jr. (1991). Tabular data on marine fishes from Southern Africa, Part I. Lengthweight relationships. Fishbyte 9(1), 5053.
