Breeding, growth, mortality and yield of the mudskipper Periophthalmus barbarus (Linneaus 1766) (Teleostei: Gobiidae) in the Imo River estuary, Nigeria

Fisheries Research 56 (2002) 227–238

Breeding, growth, mortality and yield of the mudskipper Periophthalmus barbarus (Linneaus 1766) (Teleostei: Gobiidae) in the Imo River estuary, Nigeria Lawrence Etim*, Richard P. King, Mfon T. Udo Faculty of Agriculture, Department of Fisheries and Aquaculture, University of Uyo, Akwa Ibom State, Nigeria Received 24 February 2000; received in revised form 5 January 2001; accepted 15 June 2001

Abstract We studied the reproductive biology, growth, mortality, recruitment pattern and yield-per-recruit to the fishery of the mudskipper Periophthalmus barbarus (¼ P. papilio) in the intertidal swamps of the Imo River estuary in the south-eastern Nigeria. Monthly variation in gonadosomatic index reveals that the males spawn once in a year from February to May and the females from March to May. The size at which 50% of the specimens matured was 10.2 cm (total length) for females and 10.5 cm for males while the median size at spawning was 10.8 cm for females and 11.9 cm for males. We collected and analysed 12 consecutive months length–frequency data using FISAT software. Fitting the seasonalized von Bertalanffy growth function to these data gave the following results: L1 ¼ 21:6 cm total length, K ¼ 0:55 yr1 , C ¼ 0:9, WP ¼ 0:2 of the year. Using the seasonalized catch curve procedure, the estimated instantaneous total mortality coefficient Z ¼ 4:21 yr1 . The instantaneous fishing mortality coefficient F ¼ 2:86 yr1 , and the instantaneous natural mortality coefficient M ¼ 1:35 yr1 . Our computed exploitation rate E ¼ 0:68 showed that fishing pressure on the stock is high. This was supported by the results of our relative yield-per-recruit analysis which showed the predicted maximum exploitation level Emax ¼ 0:51 which was lower than the current exploitation rate ðE ¼ 0:68Þ. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Mudskipper; Periophthalmus barbarus; Breeding; Growth; Mortality; Yield; Imo River estuary; Nigeria

1. Introduction Periophthalmus barbarus (¼ P. papilio) is the only representative of the genus Periophthalmus in West Africa. It is found in the entire West African region from Senegal to Angola (Fischer et al., 1981), where it inhabits burrows in the swampy intertidal banks of estuaries. It occupies a unique niche in its habitat being the only example of resident intertidal fish. Literature search (including ASFA) reveals that except * Corresponding author. E-mail address: [email protected] (L. Etim).

for the work of Etim et al. (1996) and King (1996) there is no other published work on the mudskipper in the West African region. Berti et al. (1992, 1994) and Colombini et al. (1995) studied the behaviour, zonation and activity patterns of a mudskipper population in Kenya (East Africa). A recent and comprehensive review of literature on the biology of mudskippers is given by Clayton (1993). The Imo River is located in the tropical rainforest belt with an equatorial climate regime. There are two seasons—the rainy season (May to October) characterized by moist winds and heavy precipitation and the dry season (November to April) with hot humid

0165-7836/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 7 8 3 6 ( 0 1 ) 0 0 3 2 7 - 7

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winds and scanty precipitation. There is a brief 2 weeks dry period in August called ‘‘August break’’. The intensity of rain before the August break is higher than that after. The intertidal zone of the Imo River estuary is fringed by mangrove/nipa palm swamps. The sediment consists of peaty fibrous mud. In the Imo River, the mudskipper is exploited for use as fish bait and human food. We studied aspects of the breeding biology and spawning season of the species. We also quantified the growth parameters and mortality rates and elucidated the recruitment pattern and yield of the species to the fishery.

2. Material and methods From April 1993 to March 1994, we obtained monthly samples of P. barbarus from the intertidal zone of the Imo River estuary at Ikot Abasi in southeastern Nigeria (Fig. 1). Fishing was done by the artisanal fishers using non-return valved basket traps (Marioghae, 1980). Each trap measured about 50 cm in length, 15 cm in big-end diameter with mesh sizes of 0.2–0.5 cm. We sexed each specimen based on differences in external genitalia and determined the stage of sexual maturity using the guidelines in Bruton (1976). We measured the total length and total weight of each specimen before dissecting it to weigh its gonad and liver separately. We computed the gonadosomatic index (GSI) (weight of gonad expressed as a percentage of body weight less gonad weight), the condition index (CI) (weight of the specimen expressed as a percentage of its length cube), and the hepatosomatic index (HSI) (weight of the liver expressed as a percentage of the body weight minus gonad weight). The monthly mean of each of these indices was plotted against time of sample collection to obtain the pattern of the seasonal variation. We estimated the length at which 50% of each sex in the population matured and spawned. This entailed fitting logistic curves to data on fish length versus proportion of fish in maturity stages IV–VI. Total absolute fecundity was estimated according to the method of Bagenal and Braum (1978). We used FISAT (Gayanilo et al. (1996) to analyse the length–frequency data, and ELEFAN procedure contained in FISAT software, to sequentially arrange

and restructure the length–frequency data. We then fitted the seasonalized von Bertalanffy growth function (VBGF) (Eq. (1)) proposed by Pauly and Gaschutz (1979) and later modified by Somers (1988) to our length–frequency data.


 CK Lt ¼ L1 1  exp Kðt  t0 Þ þ 2p 
 CK  sin2pðt  ts Þ  (1) sin 2pðt0  ts Þ 2p where Lt is the length at age t, L1 the asymptotic length, K the von Bertalanffy growth coefficient, C the amplitude of growth oscillations, t0 the age of the mudskipper at zero length, ts the time between birth and onset of the first growth oscillations. WP is the time when growth is slowest and is substituted for ts since WP ¼ ts þ 0:5. Eq. (1) reverts to the original VBGF if C ¼ 0, i.e. if the effect of changing season on growth is not considered. We used the Powell–Wetheral procedure (Powell, 1979; Wetherall, 1986) as modified by Pauly (1986) to obtain initial estimate of L1. In this method, the Beverton and Holt (1956) length-based Z-equation was rearranged to a linear regression model of the form L  L0 ¼ a þ bL0

(2) 0

Here L ¼ ðL1 þ L Þ=fð1 þ ðZ=KÞÞg is the mean length of all mudskippers L0 and L0 the smallest length of fully recruited mudskipper. From Eq. (2), we computed L1 as a/b, and Z/K as ð1 þ bÞ=b. We then seeded this initial value of L1 into ELEFAN to obtain optimized values of the seasonalized VBG coefficients. The seasonalized length-converted catch curve (Pauly, 1990; Pauly et al., 1995) was used to estimate the instantaneous total mortality coefficient Z of the single negative exponential mortality model Nt ¼ N0 ezt

(3)

Here N0 is the initial number and Nt the number at time t. Seasonalized Z was then computed from the regression equation lnðNÞ ¼ a þ bt0

(4)

where N is the number of mudskippers in pseudocohorts sliced by the growth curves (in Fig. 5), t0 the relative age of mudskipper in that pseudo-cohort, b

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Fig. 1. Map of the Imo River estuary showing the sampling location at Ikot Abasi, south-eastern Nigeria.

with sign changed gives Z with seasonality. We also computed non-seasonalized Z from the equation:
 Ni (5) ln ¼ a þ bti Dti Here Ni is the number of mudskippers in length class i, Dti ¼ ð1=KÞ lnfðL1  L1 Þ=ðL1  L2 Þg the time needed for mudskipper to grow through length class i, ti ¼ ð1=KÞ lnfð1  ðLt =L1 ÞÞg the relative age corresponding to the mid-length of class i, L1 and L2 are the lower and upper limits of length class i, and b with sign changed gives the value of non-seasonalized Z. Additionally, the Beverton and Holt (1956) method was used to estimate Z from the mean length of mudskipper in the catch thus Z¼

KðL1  LÞ L  L0

(6)

where L and L0 are as defined for Eq. (2). We used the model of Pauly (1980) to calculate the instantaneous natural mortality coefficient M log M ¼ 0:006 þ 0:27 log L1 þ 0:654 log K þ 0:463 log T

(7)

where T is the mean annual surface water temperature in the estuary in degree centigrade (here 29 8C). The Pauly and Munro (1984) relationship was used to calculate the index of overall growth performance F0 F0 ¼ 2 log L1 þ log K

(8)

and the potential longevity tmax of the mudskipper was estimated from the relationship (Taylor, 1958; Pauly, 1980) tmax ¼

3 K

(9)

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The method of Pauly (1987) was used to analyse the probability of capture P of each size class i using the ascending left arm of the length-converted catch curve. This entails dividing the numbers of mudskippers actually sampled by the expected numbers (obtained by backward extrapolation of the straight portion, i.e. the descending part of the catch curve) in each length class of the ascending part of the catch curve. Plotting the cumulative probability of capture against mid-length, gives a resultant curve from which the length at first capture Lc was taken as corresponding to cumulative probability at 50%. We traced the seasonal recruitment pattern of the mudskipper using our entire restructured length–frequency data set. This involves projecting backward, along a trajectory defined by our computed VBGF, all the restructured length–frequency data onto an arbitrary 1-year time scale (Pauly, 1987). Then using the maximum likelihood approach, we separated the recruitment pattern into its Gaussian component using the NORMSEP (normal separation) method of Hasselblad (1966). The model of Beverton and Holt (1966) as modified by Pauly and Soriano (1986) was used to predict the relative yield-per-recruit.        Y0 3U 3U 2 U3 ¼ EU M=K 1    1þm R 1 þ 2m 1 þ 3m (10) where E ¼ F/Z is the exploitation rate, i.e. the fraction of mortality caused by fishing activity, F the instantaneous fishing mortality coefficient, U ¼ 1  ðLc =L1 Þ the fraction of growth to be completed by the mudskipper after entry into the exploitation phase, m ¼ ð1  EÞ=ðM=KÞ ¼ K=Z. The relative biomass per recruit was estimated as B0 Y 0 =R ¼ F R

(11)

Then, we estimated Emax (exploitation rate which produces maximum yield), E0.1 (exploitation rate at which the marginal increase of Y0 /R is 0.1 of its value at E ¼ 0), and E0.5 (the value of E under which the stock is reduced to 50% of its unexploited biomass) through the first derivative of the Beverton and Holt (1966) function. Additionally, we plotted the yield contours to assess the impact on yields of changes in E and Lc/L1.

3. Results and discussion In 1013 specimens of P. barbarus, there were more females (61.9%) than males (38.1%) (w2 ¼ 57:34, 1 d.f., P < 0:001). The pooled rainy season (May to October) sample ðn ¼ 468Þ showed a sex ratio of 1.00 male to 1.67 females (w2 ¼ 29:8, 1 d.f., P < 0:001), and the dry season (November to April) sample ðn ¼ 545Þ also showed a sex ratio of 1.00:1.56 in favour of females (w2 ¼ 27:76, 1 d.f., P < 0:001). The numerical preponderance of females over males may indicate sex-related differences in mortality or longevity. Conversely, it may be that males spend more time in the burrows guarding eggs as is the case with most gobiids (Cole, 1982; Miller, 1984). For both sexes, the length–weight relationship took the form: weight ¼ 0:15 (total length)2.9 and fecundity–length relationship was fecundity ¼ 4:8 (total length)3.29. The monthly variation in gonadosomatic, condition and hepatosomatic indices for both sexes are shown in Fig. 2. For males, peak values of GSI occurred from February to May, for females they occurred from February to April. For males, there was a gradual decline in GSI from June to October. In females this decline was steep during the first 2 (May to June) months. The trend in mean testis and ovary weight followed this same general pattern. The temporal location of peak GSI values between February to May (for males) and February to April (for females) means that the species breeds during these periods. Thus, it spawns once in a year towards the end of dry season/beginning of rainy season, so the fry stand to profit from the attendant abundance of detritus and benthic algae during the ensuing rainy season months. The mudskipper Boleophthalmus dussumeri in Bombay (latitude 198N) also spawns once in a year but from July to September (Mutsadi and Bal, 1970) as does B. dentatus (January to February) on the Jodia coast (228N) (Soni and George, 1986). But B. dussumeri (Hoda, 1986) and B. dentatus (¼ B. dussumeri) (Hoda and Akhtar, 1985) spawn twice in a year (April to May, July to September). The smallest male and female P. barbarus with matured gonad in our sample were 8.9 and 7.7 cm, respectively. The largest ripe male and female measured 15.6 and 13.5 cm. The median size at maturity (that is the size at which 50% of the specimen are matured) for P. barbarus is 10.2 cm for females and 10.5 cm for males, while

L. Etim et al. / Fisheries Research 56 (2002) 227–238

Fig. 2. Monthly variation in different indices for the mudskipper P. barbarus (: males; *: females).

231

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Fig. 3. (A) Size at 50% maturity (or median size at sexual maturity) of males (10.5 cm) and females (10.2 cm) P. barbarus. (B) Median size at spawning of male (11.9 cm) and female (10.4 cm) P. barbarus (: males; *: females).

the median size at spawning is 10.8 cm for females and 11.8 cm for males (Fig. 3). Thus, females matured and spawned at a slightly smaller length from the males. This compares with B. dussumeri from Korangi

creek which attained 50% maturity at about 7.0 cm (Hoda, 1986), while in Bombay they reach 50% maturity at a larger length of 9.6–11.0 cm (Mutsadi and Bal, 1970). The difference between the mean maximum and mean minimum GSI for females (2.89) is greater than that for males (0.249). Thus, the reproductive build-up and subsequent drain on the females is greater than that on the males. The variation in mean CI showed that except for April when there was a peak, CI was stable year round. If an increase in GSI occurs when there is a decrease in HSI, then energy reserves from the liver are being utilized for gonad regeneration. As shown in Fig. 2, such phenomenon does not occur in P. barbarus. Growth of fish could be studied using annuli on hard parts like otoliths, marked recapture experiment or by analysing length–frequency data. Here, we used length–frequency analysis because annuli on hard parts of these specimens were rather unclear and because of logistic constraints we were unable to examine daily rings or conduct mark-recapture experiments. Many workers (e.g., Etim et al., 1996; King, 1996; Pauly and Morgan, 1987) have successfully used length–frequency data to study the growth and mortality of mudskipper and other fish and the results are comparable to annuli studies. Table 1 presents the monthly length–frequency data of 1016 specimens of the mudskipper collected during the 12 months study. Analysis of these data by the Powell–Wetherall method gave the following results: L1 ¼ 14:82 cm and Z=K ¼ 2:62 (Fig. 4). After seeding this initial value into ELEFAN, we obtained the optimized seasonalized growth curves (L1 ¼ 21:60 cm, K ¼ 0:55 yr1 , C ¼ 0:9, WP ¼ 0:2) which we have superimposed on both the normal and restructured length– frequency histograms (Fig. 5). Here t0 is absent because ELEFAN cannot estimate its value from length–frequency data. t0 is useful as a location parameter, i.e. in locating the origin of the curve, so its absence does not compromise the accuracy of the computed VBG coefficients. Asymptotic length is the maximum theoretical average length that a species could attain (granted it grows throughout life) in its habitat given the ecological peculiarities of that environment, and the K parameter indicates the speed at which the species grows towards this final size. Our computed values (L1 ¼ 21:60 cm, K ¼ 0:55 yr1 ) are

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Table 1 Monthly (April 1993 to March 1994) length–frequency data of P. barbarus (ML: mid-length of length class, N ¼ 1016) ML/date 1.45 2.45 3.45 4.45 5.45 6.45 7.45 8.45 9.45 10.45 11.45 12.45 13.45 14.45 15.45 P

April 1993

May

June

3 8 23 6 9 5 0 1

1 4 14 9 11 6 2 4 2

2 8 9 8 10 10 4

55

53

51

July

August

3 1 8 5 8 10 6 5 2 4

2 4 8 18 19 16 9 11 4 0 0 1

52

92

September

October

2 1 12 11 17 18 10 14 15 13 10 3

7 26 29 13 6 7 5 1

126

94

within range for those computed by Etim et al. (1996) (L1 ¼ 19:39 cm, K ¼ 0:51 yr1 ) and King (1996) (L1 ¼ 17:8 cm, K ¼ 0:36 yr1 ) for the same species in the Cross River estuary. Growth of fish is not linear, so direct comparison of these coefficients does not make much biological sense as one species or stock can grow faster than the other when young and slower when old. Thus growth comparison must be taken as a

November

December

January 1994

1 8 36 17 11 19 6 2

3 14 17 25 23 17 9 2 3 5 1

9 9 8 10 6 4 3 1 0 1

100

119

51

February

March

1 10 8 22 19 27 14 10 5

2 4 9 19 22 27 15 9

116

107

multivariate problem in which both the asymptotic length and growth rate are simultaneously considered. Within this context, the index of Pauly and Munro (1984) (F0 ) is suitable for computing and comparing the overall growth performance of different species or fish stocks. Moreau et al. (1986) compared other alternative indices (e.g. o ¼ KL1 ) in 100 different tilapia populations and found that F0 was the best as it

Fig. 4. Powell–Wetherall plot for P. barbarus. The original length–frequency data of Table 1 are pooled and the frequency plotted against midlength of class interval (left graph). Points to the right of the arrow were selected for use in fitting the Powell–Wetherall regression plot (right graph) whose equation is Y ¼ 4:090:276X, r ¼ 0:974. L1 ¼ 14:816 cm, Z=K ¼ 2:621.

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Fig. 5. Seasonalized von Bertalanffy growth curves (L1 ¼ 21:60 cm total length, K ¼ 0:55 yr1 , C ¼ 0:9 yr1 , WP ¼ 0:2) of P. barbarus as superimposed on the (A) normal length–frequency histograms and (B) the restructured length–frequency histogram. The black (¼ pseudocohorts) and white bars in (B) are positive and negative deviations from the ‘‘weighted’’ moving average of three length classes.

exhibits the least variance. In this study, the F0 is 2.28. This is the same as obtained by Etim et al. (1996) and compares well with a value of 2.06 obtained by King (1996) for the mudskipper population in the Cross River. The longevity of this species is about 6 years. This is similar to tmax ¼ 6 years (Etim et al., 1996) but lower than 8 years obtained by King (1996) for the Cross River population. In this study, the amplitude of growth oscillation is 0.9 and the Winter point (period when growth is slowest) is 0.2, i.e. between February and March. Growth seasonality in aquatic animals is a well-known phenomenon especially in temperate zones where growth could be quite slow or completely retarded during winter. In tropical West Africa, the environmental water temperature is high all year. For example, in the Imo River estuary, the difference between the mean maximum temperature ( 30 8C) and the mean minimum temperature ( 24 8C) (Enplan Group, 1974) could be less than 5 8C. Thus temperature

changes may not be a major contributor to growth retardation in this environment. Probably, seasonal growth oscillations of the mudskipper in the Imo River estuary may be linked to spawning stress or to trade-off between growth and breeding especially as February/March period is the time for heightened breeding activity. The non-seasonalized length-converted catch curve gave a Z value of 4.59 yr1, while the seasonalized length-converted catch curve gave a comparable value of Z ¼ 4:21 yr1 (Fig. 6). However, this seems to have been underestimated by the Beverton and Holt (1956) method ðZ ¼ 3:2 yr1 Þ. A Z of 2.208 yr1 was determined for the Cross River population by Etim et al. (1996) using the non-seasonalized catch curve procedure. In this study M ¼ 1:35 yr1, giving an F (instantaneous fishing mortality coefficient) of 2.86 and an exploitation rate E (¼F/Z) of 0.68. The fishing pressure on this stock is greater than that in the Cross River estuary ðE ¼ 0:393Þ. The species is not a food

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Fig. 6. Length-converted catch curves for P. barbarus. (A) Seasonalized catch curve from where the slope of the descending right arm (black dots) of the line with sign changed gives an estimate of the seasonalized Z. Parameters of the regression line: a ¼ 12:87 (standard deviation ¼ 1:59, confidence interval ¼ 8:9816:76), b ¼ 4:21 (standard deviation ¼ 0:81, confidence interval ¼ 2:106:32), r ¼ 0:89, n ¼ 8. Estimated seasonalized Z ¼ 4:21 yr1 . (B) Non-seasonalized length-converted catch curve. Parameters of the regression line: a ¼ 14:22 (standard deviation ¼ 0:55, confidence interval ¼ 12:8815:56), b ¼ 4:59 (standard deviation ¼ 0:35, confidence interval ¼ 3:75 to 5.46), n ¼ 8, r ¼ 0:98. Other computed statistic: cut-off length L0 ¼ 7:95 cm, mean length (from L0 ) ¼9.91 cm. Estimated Z ¼ 4:59. Points on the ascending left arm of the curve (open dots) were used in calculating the probability of capture of each size class (see Fig. 7). (open dots ¼ numbers actually sampled, open squares ¼ expected numbers). Dividing Ni by Dti, serves to correct for the non-linearity in the growth of the mudskipper.

Fig. 7. Probability of capture of each size class of P. barbarus as derived from the ascending left arm of the catch curve (Fig. 6B). The estimated length at first capture L50 or Lc ¼ 7:692 cm; L25 ¼ 6:858 cm; L75 ¼ 8:527 cm. Lc is one of the inputs in computing the relative yield-per-recruit and relative biomass per recruit (Eqs. (10) and (11)) and the yield isopleths in Fig. 9.

Fig. 8. (A) Relative yield-per-recruit and (B) relative biomass per recruit for P. barbarus using the knife edge selection procedure. Summary statistics: Emax ¼ 0:623, E0:1 ¼ 0:57, E0:5 ¼ 0:32. (C) Relative yield-per-recruit, (D) relative biomass per recruit using selection ogive procedure. Summary statistics: Emax ¼ 0:512, E0:1 ¼ 0:465, E0:5 ¼ 0:296.

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fish cherished by the peoples of the Cross River area, so the population is only lightly exploited for use as bait. The Imo River population is exploited both for human food and for bait, so there is a greater fishing pressure on this stock. Our analysis of probability of capture of each size class show that the length at first capture Lc is 7.69 cm (Fig. 7) which is below the estimate of 9.3 cm for the Cross River population (King, 1996). The knife edge procedure for the analysis of relative yield-per-recruit gave an Emax ¼ 0:62, but the selection ogive method gave an Emax ¼ 0:51 (Fig. 8). The knife edge procedure assumes that specimens 1, then the population is mortality-dominated; if less than 1, then it is growthdominated. In a mortality-dominated population, a value of Z=K ¼ 2 indicates a light level of exploitation. Here the Z/K value of 2.6 indicates a more than light level of exploitation. The yield isopleths in Fig. 9

237

demonstrate the response of the mudskipper to both variation in E (exploitation level) and Lc/L1 (a proxy for mesh size). Yield contours based on Eq. (10) with Lc =L1 0:5 usually consist of four quadrants (Pauly and Soriano, 1986) each with its characteristics. Our yield isopleths with Lc =L1 ¼ 0:356 and E ¼ 0:68 belong to quadrant C, which implies that large specimens are caught at high effort level, fishing is eumetric and the fishery is developed. Considering that open-access fisheries stands the risk of being overcapitalized (or overexploited) if not properly managed, effort should be stabilized or reduced if possible. The instantaneous natural mortality coefficient M, apart from indicating the fraction of death caused by all possible causes of death except fishing, is a necessary input in the computation of many models in fish population dynamics study, e.g. the Beverton and Holt’s relative yield-per-recruit and relative biomass per recruit. However, direct and reliable estimate of M in an exploited population is difficult to obtain. Pauly (1980) derived his model (Eq. (7)) for M estimation from data obtained from 175 different fish stocks distributed in 84 species, both freshwater and marine, and ranging from polar to tropical waters. Our estimated M ¼ 1:35 yr1 compares with that of Etim et al. (1996) ðM ¼ 1:34 yr1 Þ for the same species in the Cross River estuary. Recruitment pattern shows the variation in intensity of recruitment into the fishery with time. In reconstructing the recruitment pattern, it was found that there is one pulsed recruitment peak in a year. This agrees with our analysis of the monthly variation in GSI which suggests that the species spawns once a year. Thus, the recruitment window opens once in a year for the juveniles which were also spawned once a year.

Acknowledgements

Fig. 9. Yield isopleths for P. barbarus. The yield contours predicts the response of the relative yield-per-recruit of the fish to changes in Lc (length at first capture) and E (exploitation rate). Lc/L1 values represent varying scenarios equivalent to a change in mesh size and E corresponds to changing levels of F/Z. The dotted line is the actual computed value of the critical ratio Lc =L1 ¼ 0:351.

This is part of a European Union/International Centre for living Aquatic Resource Management (EU/ICLARM) supported project—‘‘Strengthening fisheries and biodiversity management is ACP countries’’. Dr. Etim is grateful to EU/ICLARM for support and to Prof. D. Pauly and Dr. Vakily for their encouragement.

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