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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


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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. 56:758-773.]


R. Froese and C. Binohlan

International Center for Living Aquatic Resources Management (ICLARM)

MCPO Box 2631, 0718 Makati City, Philippines

Tel. No. +63-2-818-0466; Fax No. +63-2-816-3183; E-mail: r.froese@cgiar.org

ABSTRACT


Empirical relationships are presented to estimate in fishes, asymptotic length (L) from maximum observed length (Lmax), length at first maturity (Lm) from L, life-span (tmax) from age at first maturity (tm), and length at maximum possible yield per recruit (Lopt) from L and from Lm, respectively. The age at Lopt 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, Lm and Lopt. 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; life-span; generation time; length-frequency

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 (Lm) 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 Lm 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 Lm for any one population requires substantial data and is therefore unavailable for many species, especially in the tropics. To facilitate estimation of Lm in the absence of suitable data, we explored an empirical relationship linking Lm 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 topt, with corresponding length Lopt, 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 Lopt 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 Lopt and L to provide an estimation of this important management parameter.

Life-span (tmax) and age at first maturity (tm) 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 (Lmax). 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


Ft

Fecundity or number of offspring produced at age t.


G

Reproductive guild, given as bearer, guarder and non-guarder.


K

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

Linf

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

Lm

Mean length at first maturity, in cm.


Lmax

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

Lopt

The length corresponding to topt, in cm.

St

Number of survivors to age t.

T

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

tg

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

t0

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.

topt

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.

tm

Mean age at first maturity, in years.

tmax

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 Lm we derived pairs of Lm 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 mid-point 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 Lmax (Pauly 1984a), we only considered estimates of L that were within an arbitrary range of ± 30% of Lmax, to exclude potentially problematic estimates of growth parameters.

About 80% of the estimates of Lm 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 Lm 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 Lm to match the type of length measurement used for L.

Our selection criteria yielded 467 pairs of Lm 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 Lm/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 (perch-likes)

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 log10 values of the paired estimates of Lm and L, for unsexed, male and female data. Because the values of the X-variables are not free of error, the simple linear regressions presented in this study may only be used for predicting values of the Y-variables.

The length Lopt at which the total biomass of a year-class reaches a maximum value can be calculated from Lopt = 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 (Lmax). 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 to 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 Lopt and performed a linear regression analysis of log10Lopt on log10L. For 76 of these records the POPGROWTH table also contained estimates of length at first maturity, which we used to estimate the relationship between Lopt and Lm.

For all records in POPGROWTH that passed the selection criteria above and had estimates of L, K, and Lm, we calculated the corresponding ages tm and tmax, assuming that tmax corresponds to the age at L* 0.95 (Taylor 1958). We calculated the linear regression parameters for log10tmax vs log10tm.


To estimate the relationship between L and Lmax 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 Lmax. We excluded 12 potentially erroneous pairs for which the Lmax / L ratio was smaller than 0.5 or larger than 1.5. A linear regression analysis was done using log10 values of the paired estimates of L and Lmax.

RESULTS

In the multiple regression analysis of Lm as a function of sex (female, male, unsexed), temperature (T), reproductive guild (G: bearer, guarder, non-guarder), 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, r2, and standard errors for various combinations of the five variables for predicting Lm.



Table II. Cross-correlations between variables used in the multiple regression analysis of Lm as a function of sex, L, K, temperature (T), and reproductive guild (G).


Variable

sex

L

K

T

G

Lm

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



Lm


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

r2

s.e.

L

467

Log10Lm = 0.8979 * log10L -0.0782

0.888

0.127

Lfemale)

167

Log10Lm = 0.9469 * log10L -0.1162

0.905

0.122

L(male)

115

Log10Lm = 0.8915 * log10L -0.1032

0.855

0.147

L, T

391

Log10Lm = -0.0431 + 0.8917 * log10L - 0.001531 * log10 T

0.882

0.129

L, G

391

Log10Lm = -5.1080 + 0.8741 * log10L + 0.05005 * log10G

0.888

0.126

L, G, T

391

Log10Lm = -5.1761 + 0.8667 * log10L + 0.05115 * log10G - 0.00174 * log10T

0.889

0.126

L, K, G, T, Sex

391

Log10Lm = -7.0351 + 0.9908 * log10L + 0.1682 * log10K + 0.05187 * log10G - 0.0040 * log10T + 0.01678 * log10Sex

0.902

0.118



Fig. 1 shows a scatterplot of log10 Lm over log10 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 Lopt and L is shown in Fig. 2. Lopt can be estimated from the equation:

log10Lopt = 1.0421* log10(L) – 0.2742 (n = 206, r2 = 0.97, s.e.= 0.073) (2)
If an estimate of length at first maturity is available, Lopt can be estimated from the following equation (Fig. 3):

log10Lopt = 1.053 * log10(Lm) – 0.0565 (n = 76, r2 = 0.89, s.e.= 0.139) (3)


Beverton (1992) stressed the relationship between age at first maturity (tm) and life-span (tmax). Our empirical relationship between tmax and tm (Fig. 4) results in the equation:
log10tmax = 0.5496 + 0.957*log10(tm) (n = 432, r2 = 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:
log10L = 0.044 + 0.9841 * log10(Lmax) (n = 551, r2 = 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 Land 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 Lm 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 live-born offspring in bearers to millions of eggs in non-guarders.

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 Lm 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 Lm 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 species-specific 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 Lm = 10^(0.898 * log10 (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 Lopt and L explains 96% of the variance in the data set. It suggests Lopt 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 Lopt/L is 0.63 and applies well to fishes between 50 and 100 cm maximum length. If Lopt is estimated from Lm, the values for Lopt 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 Lopt may be smaller than or equal to Lm, whereas in large fishes Lopt is usually larger than Lm (Fig. 3). Beverton (1992) pointed out that starting maturity at maximum biomass (Lm  Lopt) would maximize egg production at the first spawning event, certainly an important factor in the reproductive strategy of small and usually short-lived 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 (topt). Thus, topt as the age of maximum egg production in most fishes (Beverton 1992) appears to be a good estimation of generation time. If Lm is larger than Lopt, then the first length class that contains close to 100% spawners, as opposed to 50% at Lm, 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 topt and hence Lopt 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 Lm/L and presumably Lopt/L ratio of 0.66. This value is within the standard error range of our empirical Lopt/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 Lopt/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 Lopt = 136 (s.e. 115 - 161) cm, which is 8 cm larger than Lopt = 128 cm calculated from the VBGF and M values (see below). Asila & Ogari (1988) give an Lm = 102 cm estimate for females. Using this value results in Lopt = 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 Lopt.
If a good estimate of the age at first maturity is available, e.g., from direct observation in fishes with pronounced annual spawning peaks, life-span can be directly estimated from equation (4). If L, Lm and tm are known, K can be estimated from the rearranged VBGF: K = -ln(1 + Lm/L)/tm (see also Beverton 1992).

For example, the age at first maturity for females of the critically endangered giant grouper (Epinephelus itajara) is given as tm = 6.5 years in Bullock et al. (1992). Based on this value equation (4) suggests a life-span of 21.3 (s.e. 14 - 33) years and from the equation K= 3/tmax (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 tmax = 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  Lmax/0.95, i.e., asymptotic length was assumed to be about 5% longer than the maximum observed length. Our empirical equation suggests that this Lmax/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, Lm, Lopt and tmax were derived from multispecies data sets and can therefore be applied to aggregated management units of species with similar asymptotic length.

Estimates of L, Lm and Lopt 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, Lm = 74 cm for males and 102 cm for females, and M = 0.34 year-1. From these values we calculated Lopt = 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, Lm and Lopt (Fig. 6) reveals the following:



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

  2. 96% of the Nile perch caught by the commercial fishery were smaller than the size of maximum possible yield per recruit;

  3. 78% of the Nile perch caught in the commercial fishery were smaller than the length at first maturity of females; and

  4. 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 (3-4).

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, Lm = 27 (s.e. 20 - 36) cm, and Lopt = 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 Lopt 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 Lopt 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 multi-species 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, Lm and Lopt 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 Lopt. This is not surprising, as spearfishing should have little impact on the oceanic, wide-ranging 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 species-rich but data-poor 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 (Lopt) and asymptotic length (L).



Fig. 3. Relationship between length at maximum possible yield per recruit (Lopt) and length at first maturity (Lm).

F
ig. 4. Relationship between life-span (tmax) and length at first maturity (tm).

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, Lm and Lopt: 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, Lopt and Lm estimated from empirical equations. Note that the range of Lopt overlaps with that of Lm.


F
ig. 8. Length-frequency 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.



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