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+{"metadata":{"id":"09217b523059981fb1ebbdc0f8dc0e0a","source":"gardian_index","url":"https://digitalarchive.worldfishcenter.org/bitstream/handle/20.500.12348/3196/Pub%20TR4%2027.pdf"},"pageCount":93,"title":"The curacy of Some Length-Based Methods / for Fish Population Studies","keywords":["Q '\"\",J5lJ t/","t","C ,,:\" r,,~i .\" c \":, jj'\" 'i ': '! t' 4:H)O '?11'V' '.,'c \" }c\"\" }\"\"\"\" '\" \"', n\" '\" \" \"'n \"xu,,) CL>,),' ,:,:JJ ',","I\",f",", Irl,x'\" \",'11M :RhJO d KJ \",:,51) 43,2 V \", ,'V'\", j \"r \" r-\" ,'. . c I \",.\"'\" \" , r .~"],"chapters":[{"head":"","index":1,"paragraphs":[{"index":1,"size":486,"text":".5. Average parameters and percentage of bias obtained with the P-W method in Series II experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Table 3.6. Average parameters obtained for the experiments of Series Ill, in which growth was assumed to oscillate seasonally and the coefficient of variation of L, and K was 0% and 20%. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Table 3.7. Total length of fishes with I%, 50% and 100% probability of being caught, when parameter a of the logistic selection curve , wasvaried. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Table 3.8. Average parameters and percentage of bias obtained with each method for samples with variable size-dependent selection effects, without and with 10% individual variability of the growth parameters. . . . . . . . . . . . . 31 Table 3.9. Bias of growth parameters obtained in a set of data from each population type before and after the correction of the frequencies for selection effects via the left ascending side of a length-converted catchcurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Table 3.10. Average parameters and percentage of bias obtained with ELEFAN, SLCA and P-W method for the data created for Experiment V. . . . . . . . . . . 33 Table 3.1 1. Average parameters and percentage of bias obtained with each method in the Series VI experiment, with varying width of length classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Table 3.12. Results obtained with a set of length data and the ELEFAN method, when including length-at-age data in the adjusting procedure; . . . . . . . . . . . 36              3.1 0 (Series V). . . . . . . . . . . . . . . . . . . . . . . .                Allen's (1966) method. . . . . . . . . . . . . . ."}]},{"head":"LIST OF FIGURES","index":2,"paragraphs":[]},{"head":"FOREWORD","index":3,"paragraphs":[{"index":1,"size":59,"text":"While the first methods for estimating growth from length-frequency data were proposed nearly 100 years ago, it is only in the last decads that these methods have began to be based on rigorous algorithms, rather than on subjective interpretation of hand-drawn curves. This transition was accelerated, obviously, by the wide accessibility of computers, particularly those of the personal kind."},{"index":2,"size":40,"text":"This Technical Report presents a study of the sensitivity of three methods, developed in the 1980s for the analysis of length-frequency data, to one of their key assumptions, namely that the variability of the growth of individual fishes is naligible."},{"index":3,"size":70,"text":"As is shown here, this assumption is not valid, for either of the three investigated methods, and indeed, all three collapse when individual gnowth variability becomes too high. This problem had been previously studiedalthough in less detail than herefor only one of the three methods. That it is shown to also be the case for the other two methods, is a new finding, and only in part a discouraging one."},{"index":4,"size":75,"text":"The reason for continued optimism is essentially that it is better to know one's enemies, as it allows one to take countermeasures. In this case, this possibly involves (i) estimating, from one's data set, individual variability of length about relative age, to infer the degree to which one's growth parameter estimates are affected by individual growth variability and (ii) adjusting one's estimates in the appropriate directions, by an amount determined by the results in (i)."},{"index":5,"size":59,"text":"I expect, in any case, that investigations on the reliability of length-based methods will continue, hand in hand with the development of new approaches, and that these studies will eventually lead to methods much more robust than those now in use, but still straightfoward to implement. The present contribution is, I believe, a big step in the right direction."},{"index":6,"size":14,"text":"A standard foreword would end here, perhaps after some perfunctory praise to the author."},{"index":7,"size":73,"text":"What makes this specialto me at least -is the fact that Dr. Isaac, who wrote the thesis upon which this document is based, was my first PhD student, which certainly added to her problems not to speak of the fact that I developed, and heme became particularly attached to, one of the methods she was submitting to such cruel tests. As her text shows, she withstood all that, and is to be congratulated."}]},{"head":"ABSTRACT","index":4,"paragraphs":[{"index":1,"size":37,"text":"Length-based methods have lately found widespread use for the estimation of growth in fish populations, especially in tropical areas, because of the various disadvantages presented by 'ageing' techniques such as otolith or scale examinations or tagging experiments."},{"index":2,"size":70,"text":"Monte-Carlo simulations of fish populations with different biological characteristics were generated to test the accuracy of some recent methods for the assessment of growth in fishes on the basis of length data. Three methods were investigated: D. Pauly and N. David's Electronic Length-Frequency Analysis (ELEFAN). J.G. Shepherd's Length Composition Analysis (SLCA) and the method derived by J.J.-Wetherall from the general model of D.G. Powell referred to as the \"P-W method\"."},{"index":3,"size":43,"text":"The effects of different growth strategies; variability of growth between individual fishes; seasonal oscillations of growth rates; size-dependent gear selection; recruitment variability; variable width of the length classes in the data; and combination of size-atage and length-frequency data of fish populations were analyzed."},{"index":4,"size":119,"text":"The simulated populations were sampled at random, and the resulting length-frequency distributions were used to estimate the parameters L, and K of the von Bertalanffy growth equation. To determine the magnitude of individual growth variability, the variance of the parameters L, and K between fishes of Lebistes reticulatus and of tilapia hybrids were calculated. Additionally, length data sets for 13 species of fish of the family Sciaenidae from various parts of the world were used to test the length-based methods on field data. A sensitivity analysis of the lengthanverted catch curves, used for estimation of total mortality rate (Z) was performed, and the implications of the input of biased growth parameters for the derivation of mortality rates are discussed."},{"index":5,"size":36,"text":"The principal results and conclusions of this investigation are: h e ELEFAN and P-W methods are more ad4uate for fast-growing and short-lived fishes than the SLCA method, which is more suitable for slowgrowing and long-lived fishes."},{"index":6,"size":17,"text":"In most of the experiments a general tendency to overestimate L, and to underestimate K was observed."},{"index":7,"size":146,"text":"All methods give accurate estimates of L , and K (or UK), if individual variability of growth parameters, recruitment variability and selection effects are small. Bias attains unacceptable levels when individual variability of growth parameters is 20% or more. When the individual variability of growth parameters is high, ELEFAN provides more accurate estimates of L , than SLCA, which provides more accurate estimates of K. Also, in such cases, the estimates of L, obtained by the P-W method are strongly biased. When size-dependent selection or long recruitment periods occur, the estimates of L , obtained using the SLCA or P-W methods are more accurate than those obtain4 with ELEFAN. Seasonal growth oscillations, the presence of two recruitment pulses per year and the width of the length classes used to represent the samples have little effect on the bias of estimates of L, and K (or ZIK)."},{"index":8,"size":66,"text":"The three methods investigated here appear useful for the study of growth in sciaenid fishes. However, the quality of the results depends strongly on two factors: a) the representativeness of the samples and b) the growth strategy of the species in question. Slow-growing fishes were more difficult to analyze. A knowledge of the biology of the species is of considerable help in the interpretation of results."},{"index":9,"size":58,"text":"Lengthanverted catch curves underestimate Z when individual variability affects the structure of the length data, but overestimate this parameter when sizedependent selection affects the samples. However, the bias was small when both effects occur simultaneously. Length-converted catch curves tend to have a stronger bias when applied to fishes with a strategy of slow growth and low mortality rate."},{"index":10,"size":32,"text":"Estimates of Z obtained from lengh-mnverted catch curve have a positive correlation with the parameters and K, but the effects of changes in K are stronger than those of changes in L."}]},{"head":"Chapter 1","index":5,"paragraphs":[{"index":1,"size":50,"text":"Growth studies are an essential instrument in the management of fisheries resources because these studies contribute to estimates of production, stock size, recruitment and mortality of fish populations. The estimation of growth parameters may be based on absolute or relative age of the individual fishes or derived from length-frequency data."},{"index":2,"size":43,"text":"Ageing fishes through the identification of periodc marks on hard structures (otoliths, scales, vertebrae, etc.) and tagging experiments are expenske and time-consuming procedures. In many aquatic animals (e.g., squids, crustaceans, shrimps &id various tropical fishes) age determination is very difficult or even impossible."},{"index":3,"size":39,"text":"Moreover, random and systematic errors in age determination occur with the existing ageing techniques (Lee et al. 1983) and bias in growth rate astimates resulting from these techniques may be introduced by the particular statistical procedre used (Ricker 1969)."},{"index":4,"size":79,"text":"At the end of the 19th century the Danish biologkt J. Petersen (1891) developed the first technique to assess the growth of fishes on the basis of length data. After the erroneous interpretation of the age of North Sea herring by D'Arey Thompson at the beginning of the 20th century (Went 1972), these techniques were regarded with suspicion, and until 1970, growth studies were fundamentally based on age determinations from the analysis of otoliths, scales, vertebrae, etc. (Pauly 1987)."},{"index":5,"size":95,"text":"Since then, however, the above mentioned disadvantages of age-based methods have led to the development, in the past decades, of new methods for analysis of length data for growth and stock assessment. Length data can be collected rather cheaply, and generally do not require specialized staff. Moreover, such data qre frequently available in government fishery departments or laboratories. According to Pauly (1987), length-frequency data are probably the most underutilized information on fish resources. The increasing use of microcomputers in fisheries science, even in developing countries, now permits the application of techniques involving sophisticated computing procedures."},{"index":6,"size":63,"text":"Moreover, many biological and fishery processes, 8.g. fecundity, predation, selection by gear, etc., are better correlated with size (length or wdight) than with age. Many characteristics of marine ecosystems are, broadly speaking, functions of the size of the organisms (Caddy and Sharp 1986). It is therefore being recognized that thae are good theoretical justifications for preferring length-based over age-based methods (Gdland 1987a;Pauly 1987)."},{"index":7,"size":63,"text":"Because most of the present-day length-based techniques are recent developments, few investigations have been done on their accuracy and sensitivity, or their limits in practice. Several scientists have cautioned against the so-called 'finger methods' (in analogy to the finger applied to the computer keyboard) (Gulland 1987b) and therdore, accuracy studies are very important to warn users of the danger associated with these metbds."},{"index":8,"size":62,"text":"This report, based on a doctoral dissertation predented to the University of Kiel (Federal Republic of Germany) analyzes the accuracy of threellength-based methods: ELEFAN (Electronic Length Frequency Analysis; Pauly and David 1981), $LCA (Shepherd's Length Composition Analysis;Shepherd 1987) and the regression technicpe derived by Wetherall (1986) from the general model of Powell (1979) and referred to here I s the P-W method."},{"index":9,"size":105,"text":"A description of the theoretical foundations and the practical implementation of each method is presented in Chapter 2. To study the accuracy of the methods, Monte-Carlo simulations of various fish populations were produced and length samples from these simulated populations were analyzed with the three methods (Chapter 3). The bias in the estimates of the growth parameters of the different populations was related to the differences in growth strategy, individual variability, seasonal growth oscillations, recruitment variability, size-dependent selection, and width of length classes in the samples. Also, the effects of adding age information to the length data for the calculations with ELEFAN were also investigated."},{"index":10,"size":35,"text":"To determine the magnitude of the individual variability in growth, the variance of the growth parameters L, and K was calculated based on length-at-age data obtained from tilapias and guppies kept in aquaria (Chapter 4)."},{"index":11,"size":109,"text":"The application of the three abovementioned methods on real data was undertaken in Chapter 5. Twenty-three sets of length-frequency data from various stocks of croakers (Family Sciaenidae) were analyzed. This family was selected because of its economic importance and the relatively large amount of biological information available. Moreover, length-based methods appear particularly promising for the estimation of growth patterns in this group, because it is difficult to age (Isaac 1988); the difficulty in ageing stems from the fact that the otoliths (sagitta) in this fish are usually larger and thicker than in most Perciformes (Chao 1978), and the scales of older sciaenids often present very narrow or indistinguishable rings."},{"index":12,"size":25,"text":"Finally, some implications of the use of biased growth parameters for the estimation of total mortality (2) using length-converted catch curves were examined (Chapter 6)."}]},{"head":"Chapter 2 LENGTH-BASED METHOB FOR GROWTH STUDIES","index":6,"paragraphs":[{"index":1,"size":60,"text":"Length-based methods for stock assessment m y be classified into two groups: a) analytic and b) synthetic (Shepherd et al. 1987). Analytic methods a e used to estimate vital parameters which determine the structure of a stock. Synthetic methods use the lrngth data and the information obtained from analytic methods to perform assessments of a stock, e.g., yidd-and biomass-per-recruit computations."},{"index":2,"size":33,"text":"Analytic methods may be subdivided into those used to determine growth parameters, those used to estimate mortality, and those used to estimate both. Several such methods have been developed in the last decades."},{"index":3,"size":76,"text":"However, the principle involved in methods for the estimation of growth parameters from length data is not new. Petersen (1891) developed their basic prhciple by attributing successive ages to the most pronounced modes of mixed distributions. This triggered the development of a variety of graphical and other methods for the separation of mixture of distribution into their components, assumed to be normal distributions (Harding 1949;Cassie 1954;Hasselblad 1966;Bhattacharya 1967;Abramson 1971 ;MacDonald and Pitcher 1979;Pauly and Caddy 1985)."},{"index":4,"size":120,"text":"Recently, Schnute and Fournier (1980), Fournier and Breen (1983), Sparre (1987) and Pope (1987) presented sophisticated improvements of these techniques. To follow the progression of a cohort through time, samples weighted by catch per effort are linked by a von Bertalanffy growth curve. Assumptions on mortality rate and initial cohort strength are used to calculate the location of each cohort in the next sample and how large its contribution to the mixture distribution should be. Expected frequencies and observed frequencies are then compared through a statistical criterion, such as chi-square or a maximum likelihood estimator. These methods require a large number of assumptions, and the number of parameters which must be optimized is very high, making the computation very time-consuming."},{"index":5,"size":40,"text":"Moreover, these \"mixture methods\" require the lengths of the fishes of a cohort to be normally or log normally distributed, the number of cohorts in each sample to be specified, and the length frequencies to be proportional to the population."},{"index":6,"size":83,"text":"Different approaches, perhaps not so rigorous frbm a statistical point of view, were presented in the form of the ELEFAN I (Electronic Length Frequency Analysis; Pauly and David 1981) and SLCA (Shepherd's Length Composition Analysis;Shepherd 1987) computer programs. The principle of these techniques is simple: given a set of growth parameters and a growth equation, an index of the coincidence between observed and expected modes of the available length-frequency data is computed and used to indicate the adequacy of the assumed growth parameters."},{"index":7,"size":37,"text":"Finally, the P-W method constitutes a very simple and quick procedure to estimate the asymptotic length and the ratio UK of a population, based on the structure of a single length sample representative of the (steady-state) population."},{"index":8,"size":64,"text":"The principal advantage of the three latter mettmds (ELEFAN, SLCA and P-W) is that they are relatively simple in their application and require very few assumptions to be met. This provides a strong incentive for their use. This investigation is intended 10 contribute to the knowledge about these techniques, the risks associated with their use and the precautions to be taken when using them."}]},{"head":"The ELEFAN Method","index":7,"paragraphs":[{"index":1,"size":84,"text":"The ELEFAN system was initially developed by Pauly andDavid (1980, 1981) and Pauly (1982) for the estimation of growth parameters and mortality in fish populations, and later improved by Brey and Pauly (1986) and Brey et al. (1988). Most of its implementations are in BASIC and are designed to be used on microcomputers. The system has recently been revised, expanded and presented as a comprehensive software package which incorporates various new routines for length-based fish stock assessment (Gayanilo et al. 1988;Gayanilo and Pauly 1989)."},{"index":2,"size":34,"text":"For the present study, a FORTRAN-77 version of ELEFAN I, which includes all routines of the original ELEFAN I and ELEFAN I 1 programs of Brey and Pauly (1986) and Brey et al. (1988)  "}]},{"head":"Estimation of growth parameters","index":8,"paragraphs":[{"index":1,"size":130,"text":"The first part of the program (ELEFAN I) fits a seasonally oscillating version of the von Bertalanffy growth function (VBGF), Lt = L, (l -exp(-K(t-to)+CK/2n sin 2x(t-t,))) ... The identification of modes (or peaks) is obtained through a so-called \"restructuring\" procedure, performed for each sample via the following steps: Calculation of the adjusted frequencies, by dividing the observed frequencies of each class by the corresponding moving average; , Computation of the relative adjusted frequencies by dividing the adjusted frequencies by the average of all adjusted frequencies within a $ample, then subtracting 1 ; A procedure to avoid the attribution of extreme values to isolated frequencies (adjacent to zero frequencies), generally at either end of the bstributions; A procedure to obtain equal sums of positive and negative values within a sampleb."},{"index":2,"size":22,"text":"After restructuring a sample, either a positive value (peak), a negative value (trough) or a zero value corresponds to each length class."},{"index":3,"size":41,"text":"Fig. 2.1 shows an example of the effect of restructuring the data in a hypothetical sample (from Pauly 1987, based on Goeden 1978). In this context1 groups (\"runs\") of adjacent length intervals with positive values are assumed to potentially represent cohorts."},{"index":4,"size":28,"text":"Standard length (cm) The Available Sum of Points (ASP) is the sum, for all samples, of the points with a maximum value in each \"run\" of positive values."},{"index":5,"size":54,"text":"To fit the growth model (i.e., VBGF), ELEFAN Iltraces numerous growth curves through the restructured data according to a set of growth parameters chosen by the user. For a given combination of growth parameters, the Explained Sum of Points (ESP) is the sum of all points (negative and positive) over which each curve funs."},{"index":6,"size":76,"text":"The best combination of parameters will produoe a curve which hits most peaks, avoids most troughs and thus obtains the highest ESP value. The relation ESPIASP may range from a negative value to unity (depending on the data), and higher values Indicate better fit. If a curve hits a single \"run\" repeatedly, the recent versions of ELEFAN I, including the FORTRAN-77 version used here, add the corresponding points only once to the ESP score (Pauly 1986)."},{"index":7,"size":100,"text":"This treatment of the length-frequency data may produce undesirable effects, particularly at the end of the distribution. Table 2.1 shows the results of the restructuring process on a simulated sample with 5 cohorts. As a consequence of the algorithm, the frequency of the fourth cohort (originally 3 fishes) is converted to a negative value (-0.483), i.e., to a t r o a h located between two \"cohorts\" (see arrows in Table 3). Consequently, the program will try to avoid this point.  Because the actual age of each cohort is unknown, length data alone do not permit the calculation of to."},{"index":8,"size":51,"text":"To fix the curve to a point of the abscissa, a \"starting point\" must be determined, at the base of any length class. From this point, the curve will be projected backward and forward. ~h u s , the Starting Point (SP) becomes the parameter of the model which replaces to."},{"index":9,"size":41,"text":"Early versions of ELEFAN I were unable to estimate values of L , smaller than L, , (i.e., the largest fish in the data), but this problem has been overcome in later versions, including the FORTRAN-77 version used in this study."},{"index":10,"size":168,"text":"This FORTRAN-77 version of ELEFAN I allows the input of any number of samples, without a limitation as to the number of classes. The curve fitting procedure can calculate values of ESPIASP for an unlimited number of parameter combinations, but the necessary calculations would be time-consuming. A run of the program using a file of 12 samples with 31 classes, using 10 different values each for L , , K, C and WP and a fixed Starting Paint took over 12 hours CPU time on a VAX 8550 minicomputer! For all calculations done with ELEFAN I in this study, the ESPIASP ratio was determined for a very wide range of parameter combinations using a 'response surface' procedure. To guarantee objectivity, the combination with the highest value ~f.~oodness-of-fit (ESPIASP) was always chosen, even when more than one maximum was found. When the same highest value of ESPIASP corresponded to several adjacent parameter combinations, the combination closest to the simulation input parameters of the simulation was selected over the others."},{"index":11,"size":82,"text":"Another important feature of the ELEFAN I approach is that independent information on the age of the fishes, i.e., tagging data or length-at-age data can be included and combined with the lengthfrequency data (Morgan 1987). Due to the mathematical problem related to the estimation of to on the basis of length data (see above), a procedure to convert the length-at-age data into length increments with a structure similar to tagginglrecapture data is used, thus avoiding the confusion of absolute and relative ages."},{"index":12,"size":150,"text":"Given a set of individual ages, lengths and dates of sampling, the data are arranged in ascending order according to the age they pertain to. Combinations of lengths between adjacent ages are randomly selected to represent the size increments. With such a set of increments (L1 , t l ; L2, t2) and any combination of growth parameters, it is possible to calculate the theoretical length that an animal with length L1 at time t i would have at time tn. It is then possible to calculate the difference existing between the theoretical and the observed length increments, and the program searches for the growth parameter combination that minimizes the variance of these differences. Thus, it becomes possible to combine length-frequency data with age data. A new value of goodness-of-fit (GA = goodness-of-fit of age data; GT = goodness-of-fit of tagging-recapture data) for the length increment data procedure is calculated:"},{"index":13,"size":24,"text":"where V, = variance of empirical length increments Vd = variance of the difference between empirical and theoretical length increments, and similarly for GT."},{"index":14,"size":22,"text":"Finally, the program computes the average goodness-of-fit of both sets of data (i.e., of the length increment data and the length-frequency data)."}]},{"head":"Estimation of total mortality (Z)","index":9,"paragraphs":[{"index":1,"size":46,"text":"The first part of the ELEFAN II program includes a routine for estimation of total mortality (Z) using a length-converted catch curve. Additionally, probabilities of capture by length and the seasonal pattern of recruitment are estimated from the left, ascending arm of the length-converted catch curve."},{"index":2,"size":50,"text":"A set of samples representing the structure of a stable age-distributed population is required. A \"pooled\" sample is created with all or part of the length data, the aim here being to simulate a steadystate population. \"Length-converted\" catch curves a n created by plotting In (NiIAt) against relative age ti."},{"index":3,"size":25,"text":"A first estimate of Z is obtained when the following function is adjusted to the points of the right descending arm of the catch curve:"},{"index":4,"size":28,"text":"where Ni = number of fish in the i-th length class At = time required for the fish to grow through length class i and Z = -b"},{"index":5,"size":82,"text":"The program includes an iteration procedure (Sprre, pers. comm. to Pauly in 1984) to correct this estimate of Z for the nonlinearity of the growth model and for the mortality which occurs within each length class. However, this new estimate was not userl in the present investigation, because in most of the cases it resutted in a higher bias than the linear regression estimate. The estimation of Z requires estimates of the g r M h parameters and involves the following assumptions:"},{"index":6,"size":16,"text":"Z is constant over all sizes classes included in the calculation; Recruitment varies little and randomly."},{"index":7,"size":50,"text":"The underlying selection curve is of the %awl type\" (see below). The selection of the points to be included in the estimation of Z is probably the most sensitive part of the whole procedure. Usually the first point included 36 the point immediately to the right of the highest point."}]},{"head":"Calculation of probabilities of capture","index":10,"paragraphs":[{"index":1,"size":129,"text":"Under the assumption of a trawl-type selection, h e left arm of a length-converted catch curve consists of fisheswhich are too small to be retained by the gear. If natural mortality (M), acting on the lowest length classes and total mortality (Z), acting on the fully recruited classes are known, the mortalities between the first and the last class of the left, ascending arm of the catch curve can be interpolated. Consequently, one can calculate the number of fishes that should have been caught in each length class if the effect of selection did not exist. The corresponding probability of capture can then be obtained from the ratio between observed and expected frequencies (Pauly 1987). Fig. 2.2 illustrates the principle of this method. The method has the following assumptions:"},{"index":2,"size":15,"text":"The gear in question is a trawl or has the selection curve of a trawl;"},{"index":3,"size":10,"text":"The smallest fish caught are fully recruited to the fishery;"},{"index":4,"size":38,"text":"The mortality values used for the calculation are accurate. If the second of these assumptions is violated, the computed probabilities correspond to a resultant curve, i.e., to the product of a selection and a recruitment curve (Gulland 1983)."},{"index":5,"size":41,"text":"The probability of capture at lower sizes can be used to correct a length-frequency data set for selection effects, and the ELEFAN packages include routines which allow for easy implementation of this approach (Brey and Pauly 1986;Pauly 1987;Gayanilo et al. 1988)."}]},{"head":"Recruitment pattern calculation","index":11,"paragraphs":[{"index":1,"size":62,"text":"The recruitment of a natural fish population does not generally represent (even in the tropics) a continuous introduction of young fishes into the exploitable stock, but rather corresponds to a seasonal pattern with one or more (usually two) pulses during an annual cycle (Longhurst and Pauly 1987). This mechanism is responsible for the existence of peaks representing cohorts in the length distributions."},{"index":2,"size":61,"text":"Thus, if we know the cohort structure and the growth@arameters of a population, it becomes possible to reconstruct the pulses of the annual recruitment. Pady's (1982) implementation of this approach assumes the same growth parameters for all fish in Smples used to derive a given recruitment pattern. This assumption is known to increase the width of apparent recruitment pulses (Pauly 1987)."},{"index":3,"size":8,"text":"The resulting recruitment pattern has the followiRlg features:"},{"index":4,"size":57,"text":"The absolute position of the recruitment frequency on the time axis is not known, because the true value of to is unknown; For procedural reasons, the output is standardized to give zero recruitment in one month. Given a set of growth parameters and assuminglto = 0, the derivation of the recruitment pattern can be summarized as follows:"},{"index":5,"size":52,"text":"a. Backward projection of each length class, edimating the \"month\" in which length would have been zero; b. Because the accuracy of the calculations deareases with age, the frequency of each length class is weighted by dividing it by the time (At) needed by the fish to grow through a length interval;"},{"index":6,"size":10,"text":"c. All values obtained for one \"month\" are addecl up;"},{"index":7,"size":47,"text":"d. The lowest monthly score is subtracted from every monthly score; e. The relative \"monthly\" recruitment values are expressed as percentages of total annual recruitment. Fig. 2.3 illustrates this procedure to estimate the recruitment pattern on the basis of length data and a set of growth parameters. "}]},{"head":"Relative time","index":12,"paragraphs":[]},{"head":"The SLCA Method","index":13,"paragraphs":[{"index":1,"size":55,"text":"Shepherd's Length Composition Analysis (SLCA) is conceptually similar to ELEFAN I in that it is based on the optimization of the goodness-of-fit index obtained by comparing the position of the modes of one or more length compositions with the location of expected modes (from a VBGF). SLCA has the same data requirements as ELEFAN I."},{"index":2,"size":58,"text":"A test function is calculated for a growth curve as goodness-of-fit criterion. The algorithm is known as \"complex demodulation\" and is similar to that used in time series analysis. The growth parameters L , , K and to' are estimated (the latter is conceptually not t k same as 10 values that can be obtained from sizeat-age data)."},{"index":3,"size":32,"text":"The original FORTRAN-77 program of Shepherd (1987) was slightly modified for the present study. Length-frequency data must be entered to the program in the same format as for the ELEFAN I program."},{"index":4,"size":39,"text":"Score values can be obtained in a matrix of up to 100x1 00 combinations of L , and K. The criteria used to choose the best combination of growth parameters were the same as given above for ELEFAN I."},{"index":5,"size":45,"text":"The SLCA method uses the growth function of von Bertalanffy (VBGF) and uses as score function where Lt= predicted length at age t L , = asymptotic length K = growth constant td= \"age\" at Lt = 0 tsd= fraction of year until sampling date."},{"index":6,"size":26,"text":"The published version of the method does not include a parameter for seasonal oscillations in the growth model, although it would be possible to do so."},{"index":7,"size":73,"text":"Given values for L , , K and to, the predicted modal lengths L1, L2, L3, etc, for t = 1,2,3, etc. can be calculated using the growth function. The observed frequencies occurring at or near these predicted lengths may be interpreted as confirming the adequacy of the current parameters. On the other hand, the current parameters do not explain the observed frequencies occurring near the predicted intermodal lengths L0.5, L1.5, L2.5, etc."},{"index":8,"size":9,"text":"The method uses a test function defined as follows:"},{"index":9,"size":3,"text":"sin c (tmatmin)"},{"index":10,"size":33,"text":"where T(I)= test function tmin = age at the lower limit of a length class tmax = age at the higher limit of a length class ta = average of tmax and tmin"},{"index":11,"size":15,"text":"The function is positive near the predicted modal lengths and negative near the intermodal lengths."},{"index":12,"size":108,"text":"The first term of this equation becomes small when there is more than one mode in an interval, and in this case the weight of such observations is reduced, especially for length classes near L , . The sum of the values of the test function multiplied with the square root of the number of individuals observed in each class, is used as criterion of goodnessof-fit: where S = score 1 = index for the length class i = index for the sample N = number of individuals Since tmin, t , , and T(l) are periodic in to (period=l), the procedure also allows the estimation of b'."},{"index":13,"size":29,"text":"Thus, for given values for L and K, where A = value of S obtained with to = 0 B = value of S obtained with to = 0.25"},{"index":14,"size":18,"text":"However, the relationship of the parameter to' to \"real\" to values is not discussed further in this contribution."}]},{"head":"The P-W Method","index":14,"paragraphs":[{"index":1,"size":71,"text":"Wetherall (1 986) and Wetherall et al. (1 987), based on Powell (1 979) developed a technique from the principle that the shape of a representative size distribution of a population is determined by the value of the asymptotic length ( L, ) and the ratio between the total mortality rate and the growth constant (i.e., by UK). These parameters are then estimated by means of a relatively simple regression calculation."},{"index":2,"size":8,"text":"Requirements for the application of the method me:"},{"index":3,"size":39,"text":"The sample is representative of a steady-stde population, i.e., recruitment and mortality are constant; Recruitment is continuous; Growth follows the von Bertalanffy model (without seasonal oscillations); Growth is deterministic, i.e., there is no indivklual variability in the growth parameters."},{"index":4,"size":48,"text":"Because a steady-state population is difficuh to find in nature, the length samples available from a population with discontinuous recruitment are pooled into one sample, which will usually lead to a reasonable approximation of a steady-state distributbn. Moreover, the fishes that are not fully selected are not considered."},{"index":5,"size":20,"text":"The P-W method is based on the method of Beverton and Holt (1956) for estimating Z from mean length (L)."},{"index":6,"size":31,"text":"where L , = asymptotic length K = growth constant L = mean length of the fishes above L, L' = a length upward of which the fishes are fully selected."},{"index":7,"size":26,"text":"Rearranging this equation and considering L atxi L' as variables, which implies that the mean length (L) is a linear funQion of the cutoff length (L')."},{"index":8,"size":62,"text":"The idea of the method is to partition the length-frequency sample using a specified sequence of L' values. Thus, for a series of arbitrary cutoff lengths (Li), it is possible to calculate the corresponding Li, i.e., the mean length of all fishes longer than the actual L'. In practice, L'i values are taken as the lowest limits of each length class (i)."},{"index":9,"size":61,"text":"A regression analysis of such a data series prouSdes an estimate of the intercept (a) and of the slope (p) of the linear function. With and which can be solved for the parameters L, and Z/K as: and It is possible to calculate the variance of the estimates, but such calculations were not included in the program used in this study."},{"index":10,"size":70,"text":"The method was slightly modified by Pauly (1986) and included as a subroutine in the ELEFAN package, as an option to obtain a preliminary estimate of L. Thus, instead of plotting successive mean lengths (Li) against their corresponding L'i, the difference (Li -L'i) can be plotted against L'i. Thus, the parameters being, and This modification permits graphic visualization of L , as the point where the line intercepts the abscissa."},{"index":11,"size":43,"text":"Because the results obtained with the P-W method depend on the length classes included in the regression, only the points belonging to the right side of the mode of the underlying distribution were used, beginning with the point corresponding to the mode itself."}]},{"head":"Discussion","index":15,"paragraphs":[{"index":1,"size":50,"text":"Many questions concerning the methods appropriate for stock assessment in developing countries have been raised during the last few years. Evaluations of the strengths and weaknesses of numerous recently developed techniques are of particular interest (Csirke et al. 1987), and this was one of the objectives of the present investigation."},{"index":2,"size":24,"text":"The three methods chosen for this purpose are simple in their application and require few preliminary Length-at-age data or tagging data can be incorporated."},{"index":3,"size":17,"text":"The program part of a comprehensive system which also permits other analyses of the same data set."}]},{"head":"SLCA","index":16,"paragraphs":[{"index":1,"size":12,"text":"No preliminarytreabnent of the original data is needed to identify the peaks."},{"index":2,"size":13,"text":"Gives less weight to length intervals with more than one expected age mode."}]},{"head":"P-W method","index":17,"paragraphs":[{"index":1,"size":3,"text":"Rigorous, but eimple."},{"index":2,"size":10,"text":"Allows the computation of the variance of L, and UK."}]},{"head":"ELEFAN-I","index":18,"paragraphs":[{"index":1,"size":10,"text":"Preteatment d the data may produrn changes in their structure."}]},{"head":"SLCA","index":19,"paragraphs":[{"index":1,"size":18,"text":"Growth is calculated on the basis of the von Bertalanffy model, but does not consider seasonal growth oscillations."}]},{"head":"P-W method","index":20,"paragraphs":[{"index":1,"size":13,"text":"The parameter K is not estimated, i.e., only the ratio ZfK is calculated"},{"index":2,"size":17,"text":"All three methods require representative sample of a population but catch and/or effort data are not needed."},{"index":3,"size":108,"text":"Recently, some studies have attempted to deternine possible sources of error in these and other length-based methods (Hampton and Majkowski 19Wb; Damm and Herrrnann 1986; Basson et al. l988), but a comprehensive investigation has still been IacHng. Sophisticated techniques developed in industrisdized nations are not immediately available in tropical countries, mainly due to lack of communication (Codes 1987;Csirke et al. 1987), even though fishery scientists in these countries have a great demand for reliable stock assessment methods which permit the management of the fish resources. Because of their simplicity, the methods analyzed here can be of considerable help in growth studies, but their limitatians must also be considered."}]},{"head":"Chapter 3","index":21,"paragraphs":[]},{"head":"ACCURACY OF LENGTH-BASED METHODS","index":22,"paragraphs":[]},{"head":"Introduction","index":23,"paragraphs":[{"index":1,"size":61,"text":"Length-based methods have lately come into widespread use for determining vital parameters in exploited aquatic stocks, especially in tropical countries (Venerna et al. 1988). Investigations on their accuracy, sensitivity and applicability, however, are scarce, and the theoretical and practical problems associated with these methods were the topic of an international workshop held in 1985 in Sicily, Italy (Pauly and Morgan 1987)."},{"index":2,"size":48,"text":"To determine the accuracy of vital parameter estimates obtained with a given growth assessment method, we should know the actual or theoretical value of those parameters in the population. Then we can calculate the difference between their real value and those obtained by applying the method in question."},{"index":3,"size":64,"text":"However, when we consider a natural fish population, we never know the true values of vital parameters. Therefore, a straightforward procedure to analyze the efficiency of any method is to create (or simulate) a hypothetic \"population\", with known characteristics as similar as possible to those of natural populations. Then we can extract a set of data (for example length data) for the desired analysis."},{"index":4,"size":39,"text":"The difference between simulated and calculated values (in this case the growth parameter values) provides a measurement of the accuracy of the method, i.e., the bias of the method. This approach belongs to the so-called Monte-Carlo methods (Halton 1970)."},{"index":5,"size":56,"text":"An advantage of such artificial \"populations\" is that we can create as many sets of data as we need. A wide range of population \"types\" can be obtained by varying biological features of the model, i.e., the input parameters of the simulation. Today's wide accessibility'of computers makes the application of Monte Carlo techniques a standard tool."},{"index":6,"size":48,"text":"In summary, a Monte-Carlo procedure can test the ability of certain methods to describe the underlying structure of any simulated data set, and in this way, it becomes possible to indicate under which conditions a method will or will not perform acceptably in the study of natural populations."}]},{"head":"Materials and Methods","index":24,"paragraphs":[]},{"head":"Generation of stochastic variates","index":25,"paragraphs":[{"index":1,"size":33,"text":"To implement the simulation of the samples, several stochastic variates must be generated to determine the structure of the simulated data. These stochastic variates may correspond to one of the following probability distributions:"},{"index":2,"size":85,"text":"For the generation of exponentialvariates the mthod described in Ahrens and Dieter (1972) was used, based on the premise that a value x can be sampled from its probability distribution f(x) by using the inverse of the function f, i.e., f-1 and a random vatiable u [0,1], so that For the simulation of normally distributed variates, the approximation technique to obtain standardized normal variates (N[0,1]) described in Bluknecht et al. (1976) was used, in which where Ui = successive uniformdistributed random variates from interval [0,1]."},{"index":3,"size":14,"text":"A normal variate x, with mean p and standard &viation a is obtained as:"},{"index":4,"size":110,"text":"The procedure used to generate gamma-distribllted variates with parameters a and P implies a complex succession of procedures, which are descrbed in Jdhnk (1964). Gamma distributions constitute a family of very flexible statistical distributions ranging from slightly skewed bell-shaped to J-shaped distributions, which include both the exponential and the chi-square distributions. Gamma-distributed variables are always positive. When the parameter a approaches =, the distribution approaches a normal distribution. The parameters a and P control the shape and the relative position of the curve. The mean and variance of a given variable were defined as folbws (Fisz 1980): Fig. 3.1 shows some gamma distributions with succtssively varying parameters a and P."},{"index":5,"size":96,"text":"The quality of the random variate generators described above was tested for both normal and gamma variates. A set of 2,000 normal-and gammadistributed variates were generated in each of 8 different experiments, with means varying from 0.01 to 1.2, and coefficients of variation varying from 10 to 40%. Averages, standard deviations and the gamma parameters a and j3 were then computed from each set of data. The differences between these computd values and those initially assumed in each experiment give a measure of the bias of the procehres to generate normally and gamma-distributed variates (Table 3.1)."},{"index":6,"size":22,"text":"The results show that the technique works very well for normally distributed variables; low bias occurred, even when high vaTabilities are simulated."},{"index":7,"size":33,"text":"The method for the gamma variates has low bias when low variabilities are simulated. With coefficients of variation of 30 to 4OoA of the mean, tlm bias is negative and higher than 5%.  "}]},{"head":"Simulation model","index":26,"paragraphs":[{"index":1,"size":41,"text":"The size and structure of an exploited fish population is basically regulated by four processes (Russell 1931  The cumulative probability distribution of time between recruitment and death due to either natural causes or encounter with the fishing gear is defined as:"},{"index":2,"size":83,"text":"... The exponentially distributed variable was gemrated using the inverse function method explained above (Equation 3.4). Theoretically, 1 can assume vdues ranging from To to =, but for computational convenience the upward limit was set to T , , = 40 years; The rates of natural and iishing mortality (M 8nd F) are assumed to be constant for all fishes; The probability of fish death due to natural auses is and the probability of death due to fishing c a be estimated correspondingly."},{"index":3,"size":52,"text":"Size-dependent probabilities of capture werd simulated. Thus, if a fish encounters a fishing gear, it must be decided whether the fish is retains by the gear, or is too small and escapes. A logistic curve (Pope et al. 1975) was used to simulak selection, P, = 1/(1 + exp (-(a + bLt)))"},{"index":4,"size":237,"text":"... In the case of escape, the probability of future encounters is calculated; Each fish is followed during its entire \"life\", and the procedures which calculate the time of natural deathtencounter with the gear are repeated until: a. the fish dies from natural causes; b. it is caught; c. is greater than T , , , (i.e., 40 years). After all fishes of a cohort are treated in this way, the program begins with a new cohort. A maximum of 40 cohorts are simulated. Total length and time of capture are stored in memory, and monthly length-frequency series are created for each cohort; Length data, which are integrated in the form of 12 monthly length-frequency samples of the entire population, are extracted and wtitten to an ASCII file with the format required for the input data in the ELEFAN and SLCA programs; Four samples of length-at-age data are derived. The size and the month of capture of up to 40 fish per age group are stored in another file. The program to implement this simulation model was written in FORTRAN-77 and runs on a VAX 8550. A seed value is required to begin the generation of pseudo-random numbers. This value was coupled to the actual time, and was therefore different for each simulation. A fixed value, however, could also have been used. A list of the input parameters required by the program is given in Appendix A."},{"index":5,"size":89,"text":"Each run of the program produces 5 sets of 12 samples from a given population with identical input parameters. A run of the simulation program generating 5 such sets of 12 samples of a population with the following settings: two recruitment peaks; 10% individual variability in growth parameters; fishing without size-dependent selection; approximately 25 length classes takes approximately 37 minutes CPU time on a VAX 8550. Length-at-age data were simulated independently on the basis of the VBGF, assuming a set of growth parameters L , , K and to."}]},{"head":"Simulated population types","index":27,"paragraphs":[{"index":1,"size":21,"text":"The bias that occurs when using length-based methods to estimate vital parameters can be produced by two different sets of factors:"},{"index":2,"size":28,"text":"1. Bias produced by external factors during sampling or during preliminary treatment of the data, giving a false picture of the real population. In this category, we have:"},{"index":3,"size":10,"text":"a. Gear selection acting on a part of the population;"},{"index":4,"size":87,"text":"b. Samples which are too small or too infrequent; c. Systematic errors in length measurements, or nonrandom selection of the measured fishes; d. Errors in the method for grouping the length measurements. 2. Bias produced by intrinsic features of the population, such as: a. Variation in growth rates among individual fishes; b. Variation in time of recruitment among individual fishes; c. Seasonal variations of population growth rates. Additionally, some methods may be better suited for investigation of certain population types, such as slow-growing fishes, or fast-growing fishes."},{"index":5,"size":75,"text":"In the present study, different \"populations\" were created with the objective of investigating the effects of some of these factors and the magnitude of the bias that they produce when length-based methods are used to estimate growth parameters. First, a standard or control population was constituted, and then consecutive populations (population types) were created, in which only one or two input parameters or factors were varied systematically. Overall, seven series of such experiments were conducted."},{"index":6,"size":46,"text":"For all experiments, the cohort strength (N,) waq assumed to be 10,000 fishes with a standard deviation of 1,000 fishes. The parameter to of the VBGF was always assumed to be 0. Fishing mortality (F) was always assumed to be equal to natural mortality rate (M)."},{"index":7,"size":19,"text":"The growth performance index (q) was calculatl)d according to Pauly and Munro (1984) and Moreau et al. (1986) as:"},{"index":8,"size":31,"text":"For each population type, five sets of length da@ were simulated, each one containing twelve samples. Growth parameters were calculated with the three methods already described, for each set of data."},{"index":9,"size":45,"text":"In the case of ELEFAN and SLCA, the goodnes6-of-fit of a wide range of parameter combinations was calculated ('response surface' procedure) and the combination of parameters with the highest score was always chosen as the final result. The P-W method gives only a single solution."},{"index":10,"size":34,"text":"Following estimation of growth parameters, a masure of the bias was obtained for each case by computing the O h difference between the simulation input parameters and the results estimated with the methods. Thus."},{"index":11,"size":5,"text":"(Estimated parameterlnput parameter) * 100"}]},{"head":"Bias = lnput parameter","index":28,"paragraphs":[{"index":1,"size":65,"text":"It should be noted that the differences in the estimated parameters, as occurred frequently between the results for the five separate datasets of a given population type, are due to random effects. Although this random component increased the calculated b i a s in practice it was assumed to be of minor importance and its magnitude was not computed bemuse of the small sample size."},{"index":2,"size":24,"text":"However, the average of the estimated paramewrs and of the bias was calculated for each group of five length datasets constituting a population type."},{"index":3,"size":106,"text":"The features of each of the seven experiments are described below. 1 year-1 0.0 year 0 % 2.0 cm not operating It shduld be remembered that only the ELEFAN method can fit a seasonally oscillating version of the VBGF. For this reason the results obtained when C+O were initially tested with that method, and the same method was then used assuming G O , to permit comparison with the results obtained by SLCA and the P-W method. Fig. 3.2. Theoretical gamma probability density function for the parameter K when the mean i s 0.5 year-1 and the coefficients of variation (C.V.) are lo%, 20% and 30%."}]},{"head":"Growth coefficient ( K;","index":29,"paragraphs":[{"index":1,"size":190,"text":"In addition, the effect of variability of L and K among individuals, combined with an oscillatory pattern of the growth rate, was investigated. For that purpose, coefficients of variation of 0% and 20% were assumed, alternating for both parameters. To evaluate whether the ELEFAN II routine is able to correct length-frequency data for selection effect, only one set of data of each population type was used. Probabilities of capture for each length class were calculated using the growth parameters previously estimated with ELEFAN I and the true value of M. The original data were corrected by dividing the frequencies of each length class by the corresponding probability of capture. New parameters were estimated once more with each method, and the results were compared with the results obtained before the correction. Although the two groups of fishes are simultaneously recruiting into the adult stock (but at different ages, 0.5 and 0.8 year), the resulting length distributions are comparable with those produced by a natural population with two different recruitment periods, or spawning twice a year, in which the recruits join the adutt stock at equal ages, but at two different times."},{"index":2,"size":34,"text":"Series VI. Effect produced by increasing the width of the length classes. Initially two groups of data were regrouped after sampling into length classes of 2, 3 and 4 cm, respectively. Fixed input parameters:"},{"index":3,"size":38,"text":"Using the editing facilities of the ELEFAN program, the length-frequency samples were then regrouped after sampling into length classes of 2,3 and 4 cm, respectively. Fixed input parameters: 1 year-1 0.0 year 0 % 2.0 cm not operating"},{"index":4,"size":33,"text":"In addition, length-at-age data of 120 fishes we* obtained through the simulation program, and three sets of hypothetical length-at-age data for 20 fishes were simulated independently, using the VBGF and the following parameters:"}]},{"head":"Effects of differences in growth strategy","index":30,"paragraphs":[{"index":1,"size":72,"text":"The average parameters obtained from five set8 of simulated data for each population type in the Series I experiment, and the corresponding percentae of bias for the ELEFAN, SLCA and P-W methods are presented in Table 3.2. A complete table with all.lvalues is given in Appendix B (Table B.1). Fig. 3.3 shows the magnitude of the bias as a function of thdtype of population, i.e., of the growth strategies of the populations."},{"index":2,"size":82,"text":"The ELEFAN I method proved to be more adeqbate for populations of small fishes with faster growth and shorter life span. However, the parameter K wag always underestimated, and L , was always overestimated. The bias was strongest when L , war high and K was low (1 10 and 0.1, respectively), attaining 24% and 12%, respectively. The growth performance index (a'), as a combination of L , and K was less affected and only a positive bias of 4% waa observed."},{"index":3,"size":43,"text":"The SLCA method showed a relatively high vanlability in the estimates. As opposed to ELEFAN I, the bias was smaller for fishes with slow growth rates greater for fishes with fast growth rates; the resutts are inconclusive for populations with intermediate gkwth strategies."},{"index":4,"size":40,"text":"The P-W method showed a clear tendency to derestimate both L , and ZIK. This is more pronounced for fishes with slow growth rate and Ion# life span, reaching 16% and 25%, respectively (when L , = 1 10 crn). "}]},{"head":"Effects of individual variability in the pammeters L, and K","index":31,"paragraphs":[{"index":1,"size":77,"text":"Table 3.3 and Fig. 3.4 (right) show the results abtained by applying ELEFAN I to populations with increasing individual variability in growth parameters. These results are the average estimates of K, Lw and $' for five data sets and the corresponding bias. vable B.2 (in Appendix B) presents the complete results. Accurate estimations of all parameters are obtained only when the underlying length data were derived from a distribution without any individual variability in the growth parameters."},{"index":2,"size":174,"text":"When variability was generated only for K, and L, was assumed constant for all individuals, the maximum lengths (Lmax) in the data were always smller than the true L , (see Table 8.2); the values of L , and a' were slightly underestimated, and the und~restimation of K increased with increasing variability.  On the other hand, when variability was generated for Lonly, ELEFAN I showed a tendency to overestimate this parameter, and produced a strong negative bias for the estimates of K, which reached 41% when the coefficient of variation of L, was assumed to be 30%. The growth performance index t)' was also underestimated. Moreover, the bias of L seems to be linked to the longest length occurring in the samples (Lm& (see Table 8.2). When the coefficient of variation of L , was assumed to be 3076, the bias of this parameter varied from 0.3% to 17% for different data sets with maximum lengths of 67.5 crn and 75.5 cm, respectively. The magnitude of bias in K estimates was always quite high."},{"index":3,"size":75,"text":"When both L and K varied among individuals, it seems that a compensatory effect occurred, the positive bias of L , attaining a maximal value of 6.5% against 8.6% obtained when only L , varied. No such effect was observed for the estimates of K. This parameter was underestimated even more, the negative bias varying from 5% to 45%, according to the magnitude of the individual variability. The parameter $ was also underestimated by 7%."},{"index":4,"size":32,"text":"Multiple peaks of the ESPIASP ratio were frequently found in the response surfaces, particularly when individual variability was high. However, the absolute maximum was always identifiable (see example in Table 8.1 3)."},{"index":5,"size":69,"text":"The averages of the estimated parameters and the corresponding bias obtained with the SLCA method are shown in Table 3.4 and Fig. 3.4 (center). Table 8.3 (Appendix B) shows all the values obtained by this method with the data of the Series II experiments. As was the case with ELEFAN I, the \"control population\" was analyzed by SLCA with high accuracy, and L, and K were reproduced without error."},{"index":6,"size":30,"text":"When K varied among individuals, the bias in this parameter was initially negative, becoming positive with increasing coefficients of variation of K; L , was ahvays overestimated by approximately 8%."},{"index":7,"size":18,"text":"When the variability was simulated only for L , , an overestimation of this parameter and of t)'"},{"index":8,"size":16,"text":"resulted. The parameter K, initially overestimated, was underestimated when the variability in L , was 30%."},{"index":9,"size":49,"text":"When both parameters L and K varied among individuals, the bias of L , was very strong, attaining more than 50% in the extreme. The parameter K was always underestimated, but by no more that 8%, and the growth performance index $' was overestimated by as much as 10%."},{"index":10,"size":49,"text":"The SLCA method showed a strong tendency to produce multiple peaks of the score function in the response surface procedures, particularly when variability was high. In several cases the parameter combinations yielding the best results were extremely different, and the choice of the best combination was difficult (Table 8.13)."},{"index":11,"size":66,"text":"Individual variability of the growth parameters affected the accuracy of the results estimated with the P-W method much more than was the case with the other two methods. A complete table with all the results obtained for the series II experiments with thd P-W method is given in Appendix 6 (Table B.4). Average parameters and the corresponding bias arwkhown in Table 3.5 and Fig. 3.4 (left)."},{"index":12,"size":64,"text":"Bias increased with increasing coefficients of vdiation of L , and K, especially in the case of L , . Bias was under 10% only when the coefficient of variatiorwof the growth parameters among individuals was also 10%. When both parameters varied; the bias atibined very high values, more than 100% when the coefficient of variation was assumed to be 30% for blrth parameters. "}]},{"head":"Effects of seasonal oscillations on growth","index":32,"paragraphs":[{"index":1,"size":150,"text":"A complete list of the results obtained in the experiments of Series Ill can be found in Appendix B (Table 8.5). Average parameters obtainod for the fiw data sets of each population type and the percentage of bias of the estimated parameters are presented in Table 3.6 and in Fig. 3.5. All the populations simulated for these experiments had a seasonally oscillating pattern in growth, and the magnitude of the parameter C of the VBGF was always 0.65. Estimation of C is only possible by the ELEFAN I method. The bias obtained varied from 6% to -80h (Table 3.6).  To appreciate the reciprocal effect of oscillation in growth and variability of the growth parameters among individuals, the results obtained in experiments II and Ill should be compared in those cases where both parameters, L a n d K, had coefficients of variation of 20°h (see Tables 3.3, 3.4 and 3.5)."},{"index":2,"size":61,"text":"Among the parameter sets estimated using ELEFAN I, results were best when C was assumed variable and no individual variability was present in the samples. Second best were the estimates assuming C=O, with practically no difference in the bias between absence or presence of individual variability. Inclusion of C for the data with individual variability gave the poorest result (Table 3.6)."},{"index":3,"size":74,"text":"With the SLCA method, the existence of seasonal growth oscillations does not seem to influence the estimates of L , and very much, compared to those obtained in experiment Ill producing an overestimation of K of under 10%. This overestimate is compensated by the tendency to underestimate K when variability among individuals is assumed, explaining the decrease of the positive bias to 2.4% when the coefficient of variation was 20% for the growth parameters."},{"index":4,"size":43,"text":"The P-W method initially estimated L and Z/K with 7% and 15% negative bias, respectively, but the bias became positive when individual variability was simulated. However, the magnitude of this bias did not reach the values observed under similar circumstances in experiment II."}]},{"head":"Effects of size-dependent selection","index":33,"paragraphs":[{"index":1,"size":201,"text":"The use of trawl,gear to sample fishes results in the escape of those individuals small enough to pass through the mesh, meaning that they will not be fully represented in the samples. The proportion of fishes of each size that escape is a function of the mesh size. lri the present simulation model the effect of different mesh sizes was controlled through the parameter a in the logistic selection curve (Pope et al. 1975; see Equation 3.1 I ) , while b was left unchanged at 0.667. Table 3.7 shows the effects of three values of a on the probability of capture of fishes with a determined total length (experiments of Series IV). When a was assumed to be -1 0, almost no fishes smaller than 8 cm occurred in the samples, 15 cm long fishes had a 50% probability of being captured, and almost all fishes longer than 23 cm were retained by the gear. The lower the value of a is, the less representative are the samples in relation to the population structure. When a was assumed to be -20, more than half of the length range occurring in the population failed to be correctly represented in the samples."},{"index":2,"size":58,"text":"The combined effect of biased sampling due to size selection, and individual variability of the growth parameters was also investigated. Table   For all methods, the estimates of growth paramters were again very accurate for the \"control populations\" without selection and variability. Bias becomes evident when selection effects were stronger and individual variability was assumed in the growth parameters."}]},{"head":"and","index":34,"paragraphs":[{"index":1,"size":59,"text":"When only selection effects were simulated, bath ELEFAN I and SLCA always overestimated L , and underestimated K, but the estimation by ELEFAN I were more strongly biased than those obtained with SLCA. Bias increased as the absolute value of the Wrameter a increased, i.e., when a higher number of length classes was not fully represented in the samflles."},{"index":2,"size":26,"text":"The additional effect of the variability of growthibarameters between individuals increased the positive bias of L , and the negative bias of K in all abses."},{"index":3,"size":21,"text":"Considering the cumulative effects of selectionQnd individual variability, L was calculated better with ELEFAN I, and SLCA estimated K more accur~ely."},{"index":4,"size":20,"text":"The growth performance index $' was generall$estimated accurately (maximal bias was 5%) by both ELEFAN I and SLCA (Table 3.8)."},{"index":5,"size":130,"text":"The P-W method does not seem particularly sensitive to the decrease of parameter (i.e., increase mesh size), the magnitude of the bias depending mlinly on the degree of individual variation of the growth parameters. When no variability was assumel, the estimates of L, and ZIK were very satisfactory. Table 3.9. shows an evaluation of the ability to *orrect for selection effects in samples using the approach outlined on p. 11 and referred to as the \"dEFAN II\" procedure. The corrected length frequencies were analyzed with all three methods, dmparing the output with and without the correcting procedure. Results were slightly but consistently be#er for ELEFAN I and in most cases for SLCA, while the results obtained with the P-W method did not prdfit at all from the correction (Fig. 3.7).  "}]},{"head":"Effects of variation in recruitment pattern","index":35,"paragraphs":[{"index":1,"size":43,"text":"Young fishes are not subject to fishing until they join the exploited stock, and the effect of recruitment on length-frequency samples is somewhat comparable to the selection produced by a trawl, i.e., the smaller individuals will not be fully represented in the samples."},{"index":2,"size":52,"text":"However, the input parameters of the Series V experiments and the gamma probability distribution assumed for the mean age-at-recruitment (t,), led to a higher frequency of individuals in the lower length The population types correspond to the different combinations of simulated parameters, as indicated in Table 3.9 (Note the differences in scale)."},{"index":3,"size":214,"text":"classes of the samples than was the case in the samples simulated in Series IV, in which the symmetrical logistic distribution was used to simulate selection. Thus, the simulated samples had a slightly better coverage over the size range. Table 3.1 0 shows the average parameters and the corresponding percentage of bias obtained with the three methoddwhen tr was varied. A complete list of the results of this experiment, including the range of lengths occlltring in each set of data, can be found in Appendix B (Table B.9). When only one recruitment peak was simulated and the age at recruitment (t,) was assumed to be the same for all recruits (0.5 years), the samples cdhtained fishes varying from 11 to 49 cm in length. The assumption of variability in tr led to the occurrence Qf smaller fishes (down to 7 cm in length), but the maximal lengths ( L , , ) did not change, never excMding L , (50 cm). In the second part of the experiment, two annual recruitment peaks 3 months apart were assumed. Due to the decrease in growth rate with age, the Wdal lengths of these two peaks can be distinguished only in the first cohort. Each mode corresponds to the 0.5 and 0.8 year old recruiting fishes, respectively."},{"index":4,"size":27,"text":"Older cohorts showed a unimodal distribution. The range of lengths occurring in the samples was similar to that described above for the first part of this expqriment."},{"index":5,"size":69,"text":"When age at recruitment was fixed and constdnt for all individuals, the ELEFAN method underestimated K and overestimated L , . Howeveril it is to be expected that the magnitude of this bias should be correlated with the value assumed for tr.phus, the older the fishes are when they join the adult stock, the less representative the samples will be 4 the population, intensifying the tendency of the bias."},{"index":6,"size":151,"text":"When variability in the mean age at recruitmemt was assumed, the ELEFAN I estimates improved, most probably because of the presence of smallerdishes in the samples. In populations with two annual recruitment peaks, the ELEFAN method permits adjustment of a growth curve across the length class corresponding either to the first or to the second peak. This is implemented by changing the \"starting point\" of the curve. Fig. 3.8 illustrates the results of this procedure on a set of 12 samples with two recruitment peaks and a standard deviation for tr of one month. For each population type with two recruitment peaks, two possible results for the population were calculated, corresponding to the adjustment for either peak. The values for the second peak are marked with asterisks in Table 3.1 0; a comparison shows that the estimates of L, and K were lower in the adjustment for the second recruitment peak."},{"index":7,"size":68,"text":"The SLCA method produced rather accurate eqimates of the growth parameters when t , was assumed constant, but had an increasing tendency rb underestimate L, and to overestimate K in all other cases. The bias of K was relatively high (41%) whedtwo recruitment pulses with variable tr were assumed (Fig. 3.9). The estimates of the growth performance index@' were always very accurate with both ELEFAN I and SLCA."},{"index":8,"size":33,"text":"As in the experiment with selection effects, the P-W method was not very sensitive to the part of the population lacking in the samples. Bias remained low and constant in all the cases."},{"index":9,"size":71,"text":"The efficiency of ELEFAN I1 in determining the necruitment pattern was investigated on a set of data from each population type, using the growth paramders previously estimated with ELEFAN 1. The resutts (Fig. 3.10) show that the procedure reproduced the peaks adequately in relation to the number of pulses and the distance between them. The temporal distribution of the calculated pulses was wider than in the original input data, as expected."}]},{"head":"Population type","index":36,"paragraphs":[{"index":1,"size":53,"text":"Bias in Z/K @ Bios in Lm Bios in K Fig. 3.9. Percentage of bias obtained with ELEFAN, SLCA and P-W method on populations with one and two annual recruitment peaks. Asterisks indicate the results of ELEFAN when using a 'starting point\" fixed in a length class corresponding to the second recruitment peak."}]},{"head":"Effects of length class width","index":37,"paragraphs":[]},{"head":"Month Month","index":38,"paragraphs":[{"index":1,"size":20,"text":"Fig. 3.10. Recruitment patterns obtained'using ELEFAN II and a data set from each population type in Table 3.10 (Series V)."},{"index":2,"size":70,"text":"The objective of the experiments of Series VI M s to estimate growth parameters for the same data sets, but grouping the frequencies in wider length cl4sses. The average results and the corresponding percentage of bias for each method are given in ~atble 3.1 1 and in Fig. 3.1 1. A list of all estimates can be found in Appendix B (Table B.lO, B . l l and 8.1 2)."},{"index":3,"size":24,"text":"Table 3.1 1. Average parameters and percentage of bias obtained with each method in the Series VI experirnerlt, with varying width of length classes."},{"index":4,"size":1,"text":"-----."},{"index":5,"size":1,"text":"--."},{"index":6,"size":1,"text":"--"},{"index":7,"size":65,"text":"Without individual variability, ELEFAN I showed an increasing tendency to overestimate L , and underestimate K as the width of the length classes increased. Bias was relatively low for the data with length class intervals of 1 cm and 2 crn (48 and 24 classes, respectively), but attained 20% for the 4 cm intervals. The estimates of the parameter V always had a low bias."},{"index":8,"size":47,"text":"When the effect of individual variability of the growth parameters was combined with the increase in length class width, the bias in L of ELEFAN I increased proportionally with the width of the length classes, and the bias in K was relatively high for all four cases."},{"index":9,"size":57,"text":"When no individual variability of the growth parameters was assumed, the increase in length class width did not influence the estimates of SLCA strongly; bias was,always low. However, when individual _ variability was assumed, the bias of SLCA for both parameters L and K increased with the length class width, L , being overestimated and K underestimated."},{"index":10,"size":31,"text":"The R-W method had a slight tendency to produce improved estimates of L , and Z/K when length class width was increased, but the differences were too small to be conclusive."},{"index":11,"size":67,"text":"When individual variability of the growth parameters was combined with the increase in length class width, the bias for L, of the P-W method increased with increasing length class width. Z/K was reproduced with 60% bias independently of the length class width, this value being similar to that found in the experiments of Series II, when individual variability of the growth parameters was 20°h (see Table 3.5). "}]},{"head":"Effects of the addition of length-at-age d a a","index":39,"paragraphs":[{"index":1,"size":113,"text":"Table 3.12 shows the parameters estimated for the data in the experiment of Series VII. Probably because of its similarity to the length data, the additional length-at-age data, which had been generated together with the simulation model, did not improve the estimates of the growth parameters obtained with ELEFAN I. Bias was -7% for K and 1.4% for L, whictl are the same as before the addition of the lengthat-age data. When length-at-age data were simulated separately, but with the same assumed growth parameters, ELEFAN I still reproduced the same estimates as before. The inclusion of data corresponding only to a particular year class, omitting the others, did not affect the estimates either."},{"index":2,"size":91,"text":"On the other hand, when the length-at-age data originated from populations with different parameters as those of the length-frequency data, the magnitude of the bias changed, because the estimates tend to approximate the parameters assumed for the length-at-age data. The influence on K was stronger than for L , . While a value for L, of 60.0 cm in the length-at-age data produced an estimate only 1% greater than the control, a value for K of 0.4 year-1 reduced the estimates by 8% in relation to the control (Table 3.1 2)."}]},{"head":"Discussion","index":40,"paragraphs":[{"index":1,"size":42,"text":"The von Bertalanffy equation, still the most frequently used model for describing growth in fishes, was derived by considering growth as the balance between anabolic and catabolic processes in an animal's body (von Bertalanffy 1934(von Bertalanffy , 1938(von Bertalanffy , 1957;;Pauly 1980)."},{"index":2,"size":38,"text":"The deterministic nature of the von Bertalanffy equation is the primary problem when individual variability in growth exists, each fish in a group being considered to grow according to the model, but with its own L, and K."},{"index":3,"size":106,"text":"Individual variability is probably the most arguable point in fitting the VBGF to average values, since one should expect that individual variability of the growth parameters is a general feature of natural populations. Every individual organism is a unique result of heredity and environment, so that no two organisms in a population will grow at precisely the same rate and attain the same size at a given age (DeAngelis and Mattice 1979). The present study uses a simulation model which considers each fish individually, in contrast to most simulation experiments found in the literature, in which different overlapping cohorts are simulated (Jones 1987;Rosenberg and Beddington 1987)."},{"index":4,"size":119,"text":"Additionally, some authors have already shown that if a deterministic age-length-key is used to determine the age frequency of catches on the basis of length data, biased results are to be expected (Kimura 1977;Westrheim and Ricker 1978). Bartoo and Parker (1983) incorporated a stochastic element in von Bertalanffy's relationship to improve this approach. Sainsbury (1980) developed a stochastic version of the VBGF for size increment data and affirmed that K will be underestimated when data obtained from populations with different individual growth parameters are analyzed with the classic deterministic equation. Schnute (1981) developed a new growth model, which includes von Bertalanffy's, Gompertz's and other models as special cases, and in which an error component for the size-at-age is incorporated."},{"index":5,"size":32,"text":"Given that all three methods tested in the present study assume a deterministic model of growth, it is not surprising that they were highly sensitive to the individual variability of growth parameters."},{"index":6,"size":60,"text":"The ELEFAN I program has been rather widely disseminated since 1980, and used on a relatively large number of fish and invertebrate stocks (see e.g., Table 3 of Pauly 1987or Venema et al. 1988). SLCA and the P-W method are more recent, and therefore, only a few critical applications have been found (Damm and Herrmann 1986;Lozano 1987;Basson et al. 1988)."}]},{"head":"The ELEFAN method","index":41,"paragraphs":[{"index":1,"size":154,"text":"In the present study, ELEFAN I always overestimated L, and underestimated K when individual variability of growth parameters was assumed. Because the bias on L , and K compensate each other, at least partially, the estimates of 6' were generally very accurate. The bias in K was only acceptable (510%) when the coefficient of variation of the parameters L , and K did not exceed 10%. Bias increased strongly when variability was high. This may be partially attributed to the procedure used for the generation of the stochastic variate for K, which had an intrinsic tendency towards negative bias. The magnitude of this bias, however (under 8%; see Table 3.1), is quite snrlall compared to the bias resulting from variability in the growth parameters (more than 40% in some caas). Additionally, it was demonstrated that a coefficient of variation of L, greater than 10°/~ also Moduced an important bias in K (see Table 3.3)."},{"index":2,"size":88,"text":"The tendency of ELEFAN I to underestimate K Way also be partially due to the fact that the identification of peaks (or modes) is quite difficult w t h the cohorts overlap, especially in older age groups. Moreover, the occurrence in the samples of 'fishes longer that L , leads to an overestimation of L , and underestimation of K, since both parameters ard strongly correlated. Hampton and Majkowski (1987b) showed that the elimination of the largest length classes from the original length data slightly improves the estimates."},{"index":3,"size":41,"text":"Additionally, as mentioned above, the deterministic nature of the VBGF is certainly the principal source of error in K, and the solution to this problem will be the implementation of a stochastic model for all the methods used in growth studies."},{"index":4,"size":92,"text":"Factors such as seasonal changes in growth rde, variable recruitment period, size-dependent selection, or data grouped in greater length class inlervals did not essentially change the tendency of the bias of L , and K in ELEFAN I. Because seasonal oIlYcillations in growth are expected to be very frequent in natural populations, the oscillating version of the VBGF can be used in conjunction with the ELEFAN I method. However, the results of this investigation are inconclusive with regard to the effects of such inclusion on the accuracy of L , and K."},{"index":5,"size":77,"text":"Variation in the growth rates due to seasonal effects, and variation in time of recruitment did not have a great influence on ELEFAN I results, and even the presence of two annual recruitment peaks produced a bias of less than 10%. The ELEFAN II vocedure to determine the recruitment pattern is useful to estimate the number of peaks per year, bul their temporal spread was wider than in reality, as has already been suggested by Pauly (1987)."},{"index":6,"size":58,"text":"The combination of growth variability and the dfect produced by size-dependent selection reduced the accuracy of the growth parameter estimates (patticularly K) obtained with ELEFAN I. The estimates of L , were not strongly biased (always less than 12% by the influence of these factors, but the bias of K was in these cases always greater than 12%."},{"index":7,"size":65,"text":"The size range of fishes not fully sampled due lo the selection of the fishing gear must not exceed 50% of the value of L, , if bias is to be kept near 10%. The correction of the frequencies by the ELEFAN II procedure produced a slight improvement of the estimates, but a reasonable estimate of natural mortality (M) should be used in this case."},{"index":8,"size":64,"text":"Size-dependent selection effects and recruitment processes eliminate slow-growing fishes (i.e., the smallest ones) from the first cohort in the samples. Therefore, the difference between the modal lengths of the first and second cohorts is smaller in the samples than the true size difference in the natural population. This leads to the computation of a smalkr annual growth rate and therefore an underestimation of K."},{"index":9,"size":81,"text":"The same applies when two annual recruitme* peaks occur, generating lower values of K when ELEFAN I is used for fitting the second recruitment peak. This must be taken into account when populations with two annual recruitment pulses are analyzed, in order to avoid the attribution of a slow growth patterr: to the fishes correspmding to the seond recruitment peak. Therefore, if two recruitment peaks are evident, a length class corresponding to ihe first peak should be used as starting point."},{"index":10,"size":50,"text":"On the other hand, a bimodality in the length-fkquency distribution of the first cohorts can also be caused by other ecological or biological circumstances (DeAngelis and Coutant 1982), and a good understanding of the biology of the species studiedlis needed in order to interpret the results obtained with length-based methods."},{"index":11,"size":39,"text":"In simulation studies, Rosenberg and Beddinglon (1987) and Hampton and Majkowski (1987b) investigated the combined effect of variable recruitment time and individual variation of growth parameters, and their results had the same tenden#y as those of the present study."},{"index":12,"size":50,"text":"The way in which length classes were grouped was another source of error, particularly in the ELEFAN I method. A reduction of the number of leggth classes resulted in \"aliasing\", i.e., hiding some cohorts, thus increasing the bias. In practice, 25 to185 classes are generally adequate for all three methods."}]},{"head":"Introduction","index":42,"paragraphs":[{"index":1,"size":77,"text":"Most published growth data on fishes refer to mean length-at-age values for entire populations. However, as shown above, the impact of individual growth variability on the growth parameter estimates is considerable. Therefore, data with individual observations of length and age over a considerable period of time were required, in order to calculate the growth parameters for each individual fish and the variability between fishes, i.e., the variance of L , and K occurring in real fish population."},{"index":2,"size":68,"text":"To assess the individual variability of growth parameters in a population, it is necessary to follow the growth of individual fishes during the course of their lives, and to compute the parameters for each individual from its length at various points in time. This type of data is very scarce. The present section is an attempt to estimate the magnitude of this variability with two differea data sets."}]},{"head":"Materials and Methods","index":43,"paragraphs":[{"index":1,"size":33,"text":"The first data set was collected by Ursin (1967), who reared seven newborn females and four males of Lebistes reticulatus (guppy) individually for 58 weeks under experimental conditions, and periodically recorded their lengths."},{"index":2,"size":38,"text":"In the same way, length-at-age data from Dr. R. Doyle (pers. comm.; see Doyle and Talbot 1989) were also obtained for 70 young hybrids of Oreochromis mossambicus and 0. hornorurn (tilapia) also reared individually for approximately 25 weeks."},{"index":3,"size":51,"text":"In both cases, the VBGF was fitted to the data of each fish using the nonlinear method of Allen (1966). Additionally the data of tilapia were also fitted with a nonlinear method developed by Soriano et at. (1990) which allows the fitting of a two-phase growth curve. The equation used was:"},{"index":4,"size":23,"text":"and h = measure of the strength of the deviation from the standard VBGF th = age at which the deviation is strongest"},{"index":5,"size":53,"text":"The analysis of the length-at-age data of guppy and tilapia permitted the approximate estimation of the individual variability of the growth parameters within these populations. The parameters K, L and to were estimated for each individual fish. Average, variance and coefficient of variation between the individual sets of growth parameters were also calculated."}]},{"head":"Results","index":44,"paragraphs":[{"index":1,"size":232,"text":"Theoretically, the variability of the growth pattern of a cohort can only be produced by individual variations in the growth parameters. However, the didribution pattern of the length-at-age values and their variance can be very useful to make inferences an the growth parameters variability. Fig. 4.1 shows growth curves with varying K and/or L, and constan&. When only L , varies, the variance of length-atage increases with age and length. When only K varids, the younger and intermediate age classes represent the greater variation in length. If both parameters vary, a combination of both patterns of variation is observed. Table 4.1 shows the results obtained with Allerd(1966) method on the average values of the growth parameters and their variation. Both parameters (Lland K) vary among individuals. The variation was stronger in the tilapias and in K, with a maximum of &O%,  The exclusion of some outlying points for young tilapias reduced the coefficient of variation of L to 20% but did not change the corresponding value for K. When the same data were fitted for the two-phase growth, a reduction of the coefficient of variation of L , to 15% was observed. However in this case, K had a coefficient of variation of 44%. The fit of a two-phase growth model seems to explain better the change in growth pattern produced by the transfer of the fishes to larger tanks (Fig. 4.4)."},{"index":2,"size":50,"text":"Table 4.2 summarizes the coefficients of variation of L and K obtained with the different fitting methods applied to the tilapia length-at-age data. These coefficients of variation ranged from 15% to 44%, but independently of the method used, the individual variability of K was always stronger than that of L."}]},{"head":"Discussion","index":45,"paragraphs":[{"index":1,"size":88,"text":"The questions underlying this chapter were: how and how much do L, and K vary in natural populations?   The results presented above demonstrate that hdividual variability in fishes can be quite large. Indeed, the coefficients of variation of K were as high as 40°h, always higher than those of L , . Different methods indicate differences in the variability of growth parameters. However, the coefficients of variation of L , and K seem to be inversely correlated, i.e. when L, varied more, K varied less and vice versa."},{"index":2,"size":85,"text":"The results on the individual variability of growth parameters are only an approximation, because they were gained under experimental conditions. InMrferences during the experiments due to population density, size of the tanks (Yoshihara 1952), type of Mod, temperature, etc., probably affected individual growth rates. However, considering the results obtabed with these experimental populations suggests that the coefficients of variation of lo%, 20% and 34% assumed in the simulation model of Chapter 3 were probably realistic and should probably include the true values for natural populations."},{"index":3,"size":71,"text":"Rosenberg and Beddington (1987) presented compilation of several values for the coefficient of variation of L , between years or between populatiols for a number of species. The differences are smaller for the data between years, never exceedind lo%, but the estimates were made by taking the average of the mean size at age in the oldest age gtbup, or by averaging several estimates of L, and could therefore be biased."},{"index":4,"size":107,"text":"Differences in growth pattern, caused by intrinsic or extrinsic factors, between different populations or for different time periods, are amply documented (e.g., Bannister 1978;Craig 1978;Anthony and Waring 1980;Mollow 1984). These differences refl-, in average, modifications in the budget of catabolism and anabolism, and are expressed by th& parameters L, (or W, ) and K (Beverton and Holt 1957). According to these authors, changes in the r#te of food consumption probably directly affect the rate of anabolism, whereas catabolism should be af&cted to a greater extent by the amount of body material available to be broken down, i.e., the weigtd of the organism and the general metabolic activity."},{"index":5,"size":72,"text":"The parameter L of the VBGF is proportional to the ratio of anabolism and catabolism (Hlk), and the parameter K is proportional to the coefficient of catabolism (k). Thus, factors which affect the food consumption rate should produce changes in the coefficient of anabolism and therefore in L , . Other differences in general metabolic activity should affect more the rate of catabolism and therefore the parameter K (Beverton and Holt 1957)."},{"index":6,"size":72,"text":"It is reasonable to suppose that the differences between the individuals of a population, which live under similar external conditions, should mostly be caused by genetic factors and affect the general metabolic activity of the organism, and probably indirectly both parameters L and K. The proportion of the variability of each parameter probably differs according to the species in question, but this preliminary investigation suggested that K varied more strongly than L."},{"index":7,"size":127,"text":"In many fishes the variance of length-at-age increases with increasing age (see e.g. Steinmetz 1974;Westrheim and Ricker 1978). This has led some authors to suppose that L, constitutes the major source of variation betweev individuals (Jones 1987;Rosenberg and Beddington 1987). However, in other fishes (mostly pelagic and fast-growing species) and in many molluscs, variance in length-at-age first increases and then decreases (Wolf and Daugherty 1961;Feare 1970;Poore 1972;Bartoo and Parker 1983), suggesting that it is the variance of K which is high. Moreover, it could also be argued that bias in the determination of age or sampling errors are the cause of such patterns in the data. Natural variability and sampling bias are probably combined in real data, and therefore, further investigation is needed to clarify these questions."},{"index":8,"size":87,"text":"The pattern of variation of length-at-age (see Fig. 4.1) may be used to gain an idea of the variation of the growth parameters between individuals of a species (Sainsbury 1980) until better methods are developed for the purpose, unveiling the underlying ecological and physiological relationships. Experimental research must nevertheless be intensified in the future, if we are to learn more about individual variations of growth within populations. This will be essential, in order to permit at least a partial correction of the bias resulting from high variability."}]},{"head":"Chapter 5 LENGTH-BASED METHODS APPLIED TO SClAENlD FISHES","index":46,"paragraphs":[{"index":1,"size":76,"text":"The sciaenids, commonly known as croakers or drums are a large family of mainly coastal demersal marine fishes inhabiting all tropical and most ternper4te oceans, comprising approximately 200 species (Wheeler 1979). Many are found in brackish waters, tat least seasonally, and some are endemic to fresh waters. Many species use estuarine environments a6 nursery grounds during their juvenile phase and as feeding grounds during the adult phases. Others are estuarine inhabitants throughout their lives (Fischer 1978)."},{"index":2,"size":51,"text":"For this part of the present study, biolo~ical infarmation was compiled on several species of Sciaenidae from different regions of the world. Most of the sciaenids have large and thick otoliths (which are difficult to read), and therefore I was interested t6 investigate the application of length-based methods on these fishes."},{"index":3,"size":34,"text":"The objective of the present study was to apply the length-based methods tested in Chapter 3 to natural populations, in order to examine their usefulness in practice and to compare simulated and real data."}]},{"head":"Materials and Methods","index":47,"paragraphs":[{"index":1,"size":13,"text":"For the present study, length data for the following sciaenid species were analyzed:"}]},{"head":"Species Area","index":48,"paragraphs":[]},{"head":"Umbrina canosai Micropogonias furnieri Cynoscion striatus Cynoscion jamaicensis Macrodon ancylodon","index":49,"paragraphs":[]},{"head":"Cynoscion regalis Leiostomus xanthurus Cynoscion nobilis","index":50,"paragraphs":[]},{"head":"Johnieops vogleri Protonibea diacanthus Pseudosciaena coibor","index":51,"paragraphs":[]},{"head":"Pseudotolithus senegalensis Umbrina canariensis","index":52,"paragraphs":[{"index":1,"size":4,"text":"Southwest and West-Central Atlantic"}]},{"head":"Northwest Atlantic and Northeast Pacific","index":53,"paragraphs":[{"index":1,"size":3,"text":"Northern Indian Ocean"}]},{"head":"East-Central Atlantic","index":54,"paragraphs":[{"index":1,"size":71,"text":"The length data used in the present investigatihn and general information on the samples are summarized in Table 5.1. The data were taken from tables when available, but in some cases they had to be read off from graphs. Some original length frequencies were pooled by month or year, when more than one sample per month or year was available from the same source. These cases are indicated in Table 5.1."},{"index":2,"size":27,"text":"ELEFAN I, SLCA and the P-W methods were amlied to the length data in the same way as done previously for the simulated data (see Chapter 3.4)."},{"index":3,"size":32,"text":"I selected the results of the calculations exclusiWy according to the best goodness-of-fit, regardless of whether they agreed with my personal knowledge on the biology or growth of the species in question."},{"index":4,"size":53,"text":"In the case of ELEFAN I, when the maximum value d ESPIASP could be attributed to several adjacent values of K and L , ('response surface procedure'; seg Chapter 2), the combination with the lowest L , and the highest K was preferred, because of the demonslrated bias in ELEFAN (see Chapter 3)."},{"index":5,"size":60,"text":"Although seasonal oscillation of growth rates is R be expected in natural populations, the parameters corresponding to a non-oscillatory (\"ELEFAN\") and to an oscillatory (\"ELEFAN-C\") curve of VBGF were calcula?ed for each set of data, to obsewk the differences between the estimates and to compare the results with the other methods which docnot consider seasonal oscillation in the growth equation."},{"index":6,"size":19,"text":"The correction of the length frequencies for sizelrselection with the ELEFAN-II procedure was not applied to the Sciaenidae data."}]},{"head":"Results","index":55,"paragraphs":[{"index":1,"size":50,"text":"Table 5.2 displays the growth parameters obtaimd with ELEFAN, ELEFAN-C (i.e., when the parameter CsO), SLCA and the P-W methods. Length distributions of the species with slow growth rates were more difficult to analyze due to the occurrence of multiple maxima of the score functions of ELEFAN I and SLCA."},{"index":2,"size":23,"text":"A statistical examination of the growth parameters estimated for all the length data sets (except C. nobilis) is summarized in Table 5. 3."},{"index":3,"size":218,"text":"ELEFAN generally computed the highest values for central tendency and measures of deviation of estimates of L, , and the lowest values for central tendency of K, followed by ELEFAN-C, SLCA and the P-W method. SLCA and the P-W method led to very similar values. The measures of deviation of K were lowest in ELEFAN. The central tendency of I $' shows that errors in L , and K compensate each other at least in part. Measures of deviation of q were highest in SLCA. The range of variation (interquartile range) shows a wide overlapping region of the estimates of the growth parameters, and a slight tendency of decreaslng L , and increasing K from left to right (Fig. 5.1).  However, no systematic differences between the methods could be shown for the estimates of K and @' by means of a Friedman rank test (Table 5.4). There were, however, significant differences (5% level) between methods in the estimates of L. The highest values were produced by ELEFAN, followed by ELEFAN-C, SLCA and the P-W method (Table 5.5, top). A test for multiple comparison (Conover 1980) demonstrated that the estimates of the two ELEFAN methods on one hand, and those of SLCA and the P-W method on the other are significantly different at the 5% level (Table 5.5, top)."}]},{"head":"Discussion","index":56,"paragraphs":[{"index":1,"size":252,"text":"The results of the present section demonstrated that ELEFAN tends to overestimate L , more than the SLCA or P-W methods. In the test on the samples simulated in Chapter 3, ELEFAN estimates of L , were more positively biased than those of SLCA and P-W only when the populations had no individual variability in growth parameters, and size-dependent selection or when variable recruitment were assumed (see Tables 3.9 and 3.1 1). Assuming that a comparison between simulated and real data is valid, it follows that, at least for the sciaenid data analyzed here, selection effects and recruitment variability influenced the samples more than the individual variability of growth parameters.   The number of small Sciaenidae discarded by the commercial trawlers may reach more than 60% of the catch in some areas, such as in the USA and southern Brazil (Chittenden and McEachran 1976;Haimovici and Maceira 1981). The data sets with the higher estimates of L, in ELEFAN or ELEFAN-C (rank=4 in Table 5.4, e.g., CORV1, CAS1, CAS2, PESC) are those with strong selection effects, generally obtained from a commercial trawl fishery. On the other hand, data sets such as LXANT, JVOG or CJAM3, in which small fishes are well represented, have higher estimates of L in SLCA than in ELEFAN. This indicates that the effect of selection is more important for the ELEFAN method than has been assumed, and strongly suggests that the ELEFAN II procedure should be used to correct this bias (at least partially) whenever \"real\" length-frequency data are analyzed."},{"index":2,"size":94,"text":"In summary, the estimates of L, obtained with ELEFAN or ELEFAN-C will be more biased than those obtained with SLCA and the P-W method when selection effects are important and cannot be corrected. When selection is negligible, only the individual variability in growth parameters should affect the estimates, and the estimates of L, will be least biased with ELEFAN. The estimates of I$' , being a combination of both L , and K, will compensate the opposing tendencies of bias, and therefore, this index can indeed be considered a useful indicator of growth performance."}]},{"head":"ACCURACY OF TOTAL MORTALITY ESTIMATES","index":57,"paragraphs":[{"index":1,"size":43,"text":"Several of the commonly-used methods for estihation of mortality rates, cohort strength and fishery yields require previous estimates of the von Bertalar#y growth parameters. Therefore, the accuracy of the estimates of mortality are related to the magnitude of the bias in growth parameters."},{"index":2,"size":35,"text":"In this section, the bias of the estimates of Z owined from a length-converted catch curve is investigated. Additionally, a sensitivity analysis of thd procedure in relation to uncertainties in L, and K is perf~rmed.~"}]},{"head":"Materials aJld Methods","index":58,"paragraphs":[{"index":1,"size":71,"text":"Some simulated length-frequency data of Seriel I, II and IV (see Chapter 3) were selected to investigate the sources of bias in the lengthconvert&l catch curves. Such curves can be created by the ELEFAN II program from a pooled data set of lengthifrequencies and values for L, and K. Total mortality is calculated from a regression between In(Nt/At) and relative age ti. (For more details on the procedure, see Chapter 2.)"},{"index":2,"size":25,"text":"The following effects were investigated: selection of the points included in the regresion; different growth strategies; individual variability of the growth parametem and size-dependent selection."},{"index":3,"size":27,"text":"The effects of the exclusion of some points of the catch curve on the estimates of Z were analyzed on a control population with the following parameter's:"},{"index":4,"size":5,"text":"Asymptotic length ( L, )"},{"index":5,"size":181,"text":"Coefficient of variation of I , The December 1990 issue of Fishbyte (ICLARM, M ila) presents papers by P. Sparre and 0. Paul y which discuss the biasing effect of seasonal growth on catch curve estimate# f 2, and a simple modification of the standard length converted catch curve which eliminates this bias, respectively.  Samples of the population with low asymptotic length (L,=30) and high value of K (=1.8) generated negative bias in Z. The estimates obtained with all other population types showed a tendency to produce positive bias in Z. This bias is higher with increasing I-, and decreasing K (Table 6.2). The bias was not excessive far the intermediate populations, but exceeded 35Oh for fishes with a high asymptotic length (L=110 cm) and low value of K (=0.1 year') (Table 6.2).  When individual growth variability was simulated in the samples, the effect of the underestimation of Z already described in last section predominated. However, the positive bias produced by size-dependent selection partially compensated for this effect, and the underestimates of Z were only between 1 and 11% (Table 6.4)."}]},{"head":"Sensitivity analysis of the length-convemd catch curve method for estimation of Z","index":59,"paragraphs":[]},{"head":"I","index":60,"paragraphs":[{"index":1,"size":53,"text":"The bias in estimates of Z resulting from a wide lange of input values for L, and K is shown in Fig. 6.3. The samples are from a control population without individual variability or size-dependent selection effects. The lines represent points with the same Z values, expressed as percentage of the true value."},{"index":2,"size":62,"text":"The estimates of Z are positively correlated with both L and K. Thus, overestimations of L, will produce an overestimation of Z and underestimations of K will produce an underestimation of Z. Since L, and K are inversely correlated, the bias tend to compensate 2, but the effects of changes in K are stronger than those of changes in L, (Fig. 6.3)."}]},{"head":"Bias in La, ( ' %I","index":61,"paragraphs":[{"index":1,"size":53,"text":"Asymptotic length ( Laol cm) Length-converted catch curves are obtained fro* the length frequencies and from growth parameters. Changes in the structure of the samplesbr in the input growth parameters will alter the shape of the curve and thus the estimates of I , which are derived from the slope of the curve."},{"index":2,"size":42,"text":"When size-dependent selection is not operating and no individual variability affects the growth parameters, the population is adequately represented in the samples and the points do not deviate significantly from the calculated regression line. Thuq very accurate estimates of Z are obtained."},{"index":3,"size":35,"text":"Variability in growth produces outlying points in datch curves. That is particularly critical in the case of the older fishes, when individuals are scarce and *ed a long time to grow through the length classes."},{"index":4,"size":43,"text":"The inclusion of all the points of the catch curve genhally produces a decrease in the slope of the regression and therefore an underestimation of 2. TI& estimates can be improved by eliminating some of the points corresponding to the oldest (largest) fishea"},{"index":5,"size":33,"text":"The effects of variability in growth parameters has already been investigated by Laurec and Mesnil (1987) for the estimates of Z obtained with the method of Beverton and Holt (1956) (see Equation 2.9)."},{"index":6,"size":89,"text":"They reported a moderate bias of 1.5% in Z estimates when the C.V. of K was 20%. These authors also recornmended the use for cohort analyses of a value of L not higher than 70% of the estimated value, in order to improve the estimates of fishing mortality. This procedure would probably be useful for the calculation of the length-converted catch curve as well, although the bias of Z resulting from growth parameter variability is greater for length-converted catch curves than the results obtained by Laurec and Mesnil (1987)."},{"index":7,"size":51,"text":"The results of this investigation suggest that a compensation of the bias in Z will occur if both sizedependent selection and individual variability affect the samples. In these cases (see Table 6.4, last two boxes) negative bias was moderate (only 3%) when the last points were not included in the calculation."},{"index":8,"size":45,"text":"Similarly, such a compensation should be expected if the parameter L, is overestimated and K underestimated, which is the general pattern for ELEFAN I. However, that compensation is only partial, because the method is more sensitive to changes in K than in L , ."},{"index":9,"size":137,"text":"Moreover, according to this investigation, bias in K is stronger that in L, increasing the tendency to underestimate 2. This combination of effects must be taken into account when evaluating the accuracy of Z estimates. As an example, let us consider the average bias produced by ELEFAN I in Table 3.3, obtained from the analysis of populations with 20% individual variability in both growth parameters; the average bias of L , estimates was 6.5% whereas the average bias of K was -40%, and according to Fig. 6.3, the input of those parameters will produce a negative bias of Z of 50%. In the same experiment SLCA produced an average bias of +33% and -6% for L and K, respectively (Table 3.4). This combination of biased input growth parameters will produce an overestimation of Z of approximately 20%."},{"index":10,"size":94,"text":"Although the present sensitivity analysis of Z is based on a single simulated control population, its results can serve to evaluate the magnitude of possible bias in the estimates of Z obtained with lengthconverted catch curves. Appendix Table B.9. Simulated and estiifiated parameters and percentage of bias obtained with ELEFAN, SLCA and the P-W method on the length-frequency data created for the Series V experiments. P =proportion of recruits in the first peak. trl = mean age (year) at peak i of recruitment. m1 = standard deviation of the mean age at recruitment i. "}]}],"figures":[{"text":"Fig. 2 Fig. 2.1. a) Original length-frequency data and runlring average frequencies over 5 length classes; b) data after the restructuring process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 2 Fig. 2.2. a) Derivation of a length-converted catch anre based on growth parameters and a pooled length-frequency file; b) Estimation of the probability of capture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig 2. 3 . Fig 2.3. Example of the recruitment pattern obtained with ELEFAN 11. "},{"text":"Fig. 3 . Fig. 3.1. Gamma distributions with different values b r the parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .aandp. "},{"text":"Fig. 3 .Fig. 3 . 3 . Fig. 3.2. Theoretical gamma probability density function for the parameter K when the mean is 0.5 year*1 and the coefficients of variation are lo%, 20% and 30%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 3.3. Percentage of bias in the estimates of growth parameters with ELEFAN, SLCA and the P-W methods, applied on four populations with increasing L and decreasing K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 3 Fig. 3.4. Bias in L , , K, @' (and UK where appropriate) as a function of three methods (ELEFAN, SLCA and P-W) and of coefficient of variations of L , andlor K ranging from 0 to 30%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 3 . Fig. 3.5. Bias in the esthates of the parameters obtained with ELEFAN-C, ELEFAN, SLCA and P-W methods for populations with seasonal growth oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 3 . 6 . Fig. 3.6. Percentage of bias in the estimates of growth parameters obtained with ELEFAN, SLCA and P-W method applied to data with increasing size-dependent selection effects, without and with 10% individual variability of the growth parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 3 . Fig. 3.7. Comparison of the bias obtained with the three methods before and after correction of the frequencies for selection effects by ELEFAN II routine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 3 . Fig. 3.8. Growth curve estimates of ELEFAN with data from a population of type 4 of the experiment of Series VI with two annual recruitment peaks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 3 . Fig. 3.9. Percentage of bias obtained with ELEFAN, SLCA and P-W method on populations with one and two annual recruitment peaks. . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 3 . Fig. 3.1 0. Recruitment patterns obtained using ELEFAN II and a data set from each population type in Table3.1 0 (Series V). . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 3 . Fig. 3.1 1. Percentage of bias in growth parameters estimated with ELEFAN, SLCA and P-W methods in the experiments of Series VI. . . . . . . . . . . . . . . . . . "},{"text":"Fig. 4 Fig. 4.1. Theoretical growth curves of the fish of a cohort with individual variability in growth parameters L, andlor K. . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 4 Fig. 4.2. Mean length-at-age and standard deviation of 7 young female guppies reared individually during 58 weeks under experimental conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 4 . 3 . Fig. 4.3. Mean length-at-age and standard deviation of 70 young hybrids of tilapia reared individually during 58 weeks under experimental conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 4 Fig. 4.4. Two-phase growth curve fitted for the length-at-age-data of an individual tilapia and estimated parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":"Fig. 5 Fig. 5.1. Box-and-whisker plot for L , , K and +' estimated with ELEFAN, ELEFAN-C, SLCA and P-W method on 22 length data sets of sciaenid fishes. . . . . . . . . . . . "},{"text":"Fig. 6 Fig. 6.1. Catch curve obtained with ELEFAN II using length data sampled from a control population with the true growth parameters I , = 50 and K = 0 . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 6.2. Catch curve obtained from populations with size-dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . effects and individual growth variability. "},{"text":"Fig. 6 Fig. 6.3. lsolines of estimates of Z obtained with ELEFAN II and varying input values of the growth parameters L , and K from a control population. . . . . . . . . . . . . . "},{"text":" Simulated and estimated parameters, and percentage of bias with the P-W method on the length-frequency data created for the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series II experiments. "},{"text":" 10. Simulated and estimated parameters, and percentage of bias with ELEFAN on the length-frequency data created for the Series VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . experiments. 73 Table B . l l . Simulated and estimated parameters, and percentage of bias . . . . . . . . . . with SLCA on the length-frequency data created f M the Series VI experiments. 74 viii Table B.12. Simulated and estimated parameters, and percentage of bias with the P-W method on the length-frequency data created for the Series VI experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "},{"text":" a. Computation of a moving average over 5 length classes; this was originally written, and the corresponding simulations performed, equation 2.1 has been shown to generate a bias in the estimation of b (see e.g., Sorners 1988 (Fishbyte)); this conbibution does not deal with to and hence is not affected by this bias. The most recent version of ELEFAN include a growth equation which overcomes this problem. "},{"text":"Fig. 2 Fig. 2.1. a) Original length-frequency data and running average frequencies over 5 length dasses. Peaks are repesented by the shaded areas above the running average. b) Data after the restructuring process. Arrows show the pomts used in the computation of ASP (modified from Pauly 1987). "},{"text":" Fig. 2,2, a) Derivation of a length-converted catch curve based on growth parameters and a pooled length-frequency file; the backward projection of the catch curve is used to estimate the number of fish that would have been caught in the absence of selection. b) Estimation of the probability of capture (modified from Pauly 1987). "},{"text":"Fig. 2 . 3 . Fig. 2.3. Example of the recruitment pattern obtained with ELEFAN II. The numbers on the abscissa only cornspond to successive months and not to actual months of the yew. "},{"text":" assumptions. All are based on the von Bertalanffy growth model and can analyze one or more irregularlyspaced length data samples.The following advantages and disadvantages have generally been attributed to these methods:ELEFAN-I Allows the andysis of seasonal growth oscillations. "},{"text":"Fig. 3 . Fig. 3.1. Gamma distributions with different values for the parameters a and P. "},{"text":" of natural deathtencounter with the fishhg gear F = fishing mortality M = natural mortality To= time of recruitment "},{"text":" 3.11)where P, = probability of retention Lt = length at the time of encounter a = parameter which shifts the curve on the x (Jength) axis b = parameter indicating the steepness of the curve "},{"text":" Series I. Populations with different growth strategies. The following input parameters were fixed:Coefficient of variation of L , ( t 2 . V . ~) Coefficient of variation of K (C.V.Effect of the variability of the parameters K and L among individual fish. Fixed input parameterscoefficient of variation of the parameters K and L-was varied as follows: "},{"text":" Series IV. Effect of size-dependent selection on the samples. Input parameters: Asymptotic length ( L, ) Growth constant (K) Amplitude parameter (C) Natural mortality rate (M) Recruitment peaks (R,) Age at recruitment (tr) Coefficient of variation of t, Width of length classes Inflection point (b) of the selection curve 50parameter a of the selection curve (Equation 3.1 1) and the coefficients of variation (C.V.) of the parameters K and L , were varied as follows: "},{"text":" Series V. Populations with different recruitment patterns. Input parameters: Asymptotic length (L,)Coefficient of variation of L, (C.V.L,) Growth constant (K) Coefficient of variation of K (Crecruitment peaks per year (R,), the mean age at each recruitment peak (trl), the standard deviations of these means (s.d.i) and the proportion of recruits belonging to the first recruitment peak (P) were varied as follows: "},{"text":" Asymptotic length ( L, ) Growth constant (K) Amplitude parameter (C) Natural mortality (M) Recruitment peaks (Rp) Age at recrulment (tr) Coefficient of variation of t , Size selection (Sel) Effect of the addition otlength-at-age datato the estimates obtained from the growth parameters with the ELEFAN method. Input parameters: Asymptotic length ( L, ) Coefficient of variation of L , "},{"text":"Table 3 . 2 .Fig. 3 . 3 . Fig. 3.3. Percentage of bias in the estimates of growth parameters with ELEFAN, SLCA and P-W methods, applied on four populations with increasing L, and decreasing K. "},{"text":" Fig. 3.4. Bias in L , , K, $' (of ZIK where appropriatd) as a function of three methods (ELEFAN, SLCA and P-W) and of coefficient of variations of L , andlor K renging from 0 to 30% (note the difference in scale). "},{"text":"Fig. 3 . 5 . Fig. 3.5. Bias in the estimates of the parameters obtained with ELEFAN-C (i.e., Ca), ELEFAN (C=O), SLCA and P-W methods, for populations with seasonal growth oscillations. a) without individual variability of growth parameters. b) with 20% individual variability of growth parameters L, and K. (Note the differences in scale). "},{"text":" Fig. 3.6 show the average parameters estimated with the three methods and the corresponding percentage of bias. Complete tables with the results obtained in this experiment may be found in Appendix B (Tables B.6, B.7 and 8.8). "},{"text":" Fig. 3.6. Percentage of bias in the estimates of growth "},{"text":" without and with 10% individual variability of the growth parameters. Bias of growth parameters obtained in a set of data from each population type before (b) and after (a) the correction of the frequencies for selection effects via the left ascending side of a length-converted catch curve. "},{"text":"Fig. 3 . Fig. 3.7. Comparison of the bias obtained with the three methods before (b) and after (a) correction of the frequencies for selection effects by the ELEFAN II routine. The population types correspond to the different combinations of simulated parameters, as indicated in Table3.9 (Note the differences in scale). "},{"text":"Table 3 . 10. Average parameters and percentage of bias obtained with ELEFAN, SLCA and P-W method for the data created for experiment V. trl and tr2 are the simulated mean ages (in years) at the corresponding standard deviations. Asterisks indicate the results of the ELEFAN method using a length dass belonging to the ,second recruitment peak as \"starting point\". P = proportion of recruits induded in the first peak. "},{"text":"Fig. 3 . Fig. 3.8. Growth curve estimates of ELEFAN with data from a population of type 4 of the experiment of Series VI (see 3.2.3) with two annual recruitment peaks. a) \"Starting point\" fixed at the second recruitment peak. b) 'Starting poinr fixed at the first recruitment peak. "},{"text":"Fig. 3 . Fig. 3.1 1. Percentage of bias in grow* parameters estimated with ELEFAN, SLCA and P-W methods in the expehents of Series VI. Left: data without individual variability. Right: data with a coefficient of variation of 20% for individual growth parameters. (Note the differences in scale). "},{"text":" Fig. 4.1. Theoretical growth curves of the fish of a cohort with individual variability in growth parameters "},{"text":"Fig 3 . Fig. 4.2. Mean length-at-age and standard deviation of 7 young female guppies reared individually during 58 weeks under experimental conditions. "},{"text":"/Fig Fig. 4.4. Two-phase growth curve fitted for the lengthat-age data of an individual tilapia and estimated "},{"text":"Fig. 5 . Fig. 5.1 shows a box-and-whisker plot of the results. The central box covers the central 50% of the values, between the lower and upper quartiles. The vertical lines ('whiskers') extend out to the minimum and maximum values and the central line represents the median. The points represent outliers (more than 1.5 times the interquartile range).The range of variation (interquartile range) shows a wide overlapping region of the estimates of the growth parameters, and a slight tendency of decreaslng L , and increasing K from left to right (Fig.5.1). "},{"text":" Fig. 5.1. Box-and-whisker plot for I-, K and 9' estimated with ELEFAN, ELEFAN-C, SLCA and P-W method on 22 length data sets of sciaenid fishes (see explanation in text). "},{"text":"Fig. 6 . 1 . Fig.6.1. Catch curve obtained with ELEFAN II using length data sampled from a control population with the true growth parameters: k 5 0 and Kz0.5. "},{"text":" Fig. 6.2. Catch curve obtained from populations with size-dependent effects and individual growth variability. "},{"text":"Fig. 6 . 3 . Fig. 6.3. Isolines of estimates of Z obtained with ELEFAN-II and varying input values of the growth parameters L, andfK for samples from a control population. Estimates of Z are expressed as percmtage of the true value. PR = point of referenm, calculated with the true growth parameters, L, = 50 and K = 0.5. "},{"text":"Table 2 . 1. Example of restructuring effects on a hypothetical sample with 17 classes and five cohorts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  "},{"text":"Table 3 . 1. Comparison of the computed and assumed parameters in 8 experiments to generate normally and gamma-distributed variates. . . . . . . . . . . . . . 17  "},{"text":"Table 3 . 2. Average parameters and percentage of bias obtained with each method in the Series I simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25  "},{"text":"Table 3 . 3. Average parameters and percentage of bias obtained with ELEFAN in Series Il experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26  "},{"text":"Table 3 . 4. Average parameters and percentage of bias obtained with . the SLCA method in Series I1 experiments. . . . . . . . . . . . . . . . . . . . . . . . . . ' . 27 Table Table  "},{"text":"Table 4 . 1 . Means and coefficients of variation gf the parameters $, . . . . . . . . . . . . . . . . . . . . . and K in experimental guppy and tilapia populdtions. 43 . . . . . . . . . . . . . . . . . . . . . . . .Table 4.2. Coefficients of variation of L, and K. 5.1. Sources of length-frequency data used in the present study. 47 Table 5.2. Growth parameters estimated for 231 sets of length data on Sciaenidae with ELEFAN, ELEFAN-C, S L C ~ and P-W methods. . . . . . . . . . . . . 47 Table 5.3. Mean, median, mode, standard devbation, standard error, minimum and maximum of the estimates of I -, K and 9' obtained with ELEFAN, ELEFAN-C, SLCA and P-W method bn 22 length data sets of sciaenid fishes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  "},{"text":"Table A . 1. Input parameters used in this simulation Table 6.1. Simulated and estimated parameters, and percentage of bias with ELEFAN, SLCA and P-W method on the length-frquency data created for the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series I experiments. Series I experiments.  "},{"text":"Table 8 . 2. Simulated and estimated parameters, and percentage of bias with . . . . . . . . . . . ELEFAN on the length-frequency data created for the Series II experiments. Table B.3. Simulated and estimated parameters, and percentage of bias with . . . . . . . . . . . . . SLCA on the length-frequency data created for the Series II experiments.  "},{"text":"Table 6 .4.   "},{"text":"Table 8 . 5. Simulated and estimated parameters, afid percentage of bias with ELEFAN (C=O), ELEFAN (C+O), SLCA and P-W methods on the length-  "},{"text":"Table 5 . 8. Simulated and estimated parameters, and percentage of bias with the P-W method on the length-frequency datalcreated for the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series IV experiments. Series IV experiments.  "},{"text":"Table B . 9. Simulated and estimated parameters, and percentage of bias with ELEFAN, SLCA and the P-W method on the length-frequency data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . created for the Series,V experiments. created for the Series,V experiments.  "},{"text":"Table 8 . 13. Response surfaces of the goodness-of-fit criterion calculated with ELEFAN and SLCA for a set of simulated length data with individual variability in growth parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  "},{"text":"Table B . 15. Individual growth parameters (L, , K and to) of 7 females and 4 males of Lebistes reticulatus calculated from Ursin's (1 967) data with Allen's (1 966) method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  "},{"text":"Table 8  .1 7. lndividual growth parameters ( L , , K and to), variances and .1 7. lndividual growth parameters ( L , ,K and to), variances and variation coefficients for 70 individuals of  variation coefficients for 70 individuals of  "},{"text":"Oreochromis mossambicus-homorum calculated   "},{"text":" was developed by J. Sommer (pers. comm.) for a VAX 780 computer. The listing of the program is available at the Department of Fisheries Biology of the Institute of Marine Research of the University of Kiel or from ICLARM.  "},{"text":"2.l)a  where where L, = predicted length at age t L, = predicted length at age t L , = asymptotic length L , = asymptotic length K = growth constant K = growth constant C = amplitude of the seasonal growth oscillations C = amplitude of the seasonal growth oscillations to = \"age\" at L, = 0 to = \"age\" at L, = 0 t , = age at the onset of first growth oscillation, t , = age at the onset of first growth oscillation, to one or more length-frequency distributions, estimating the parameters L , , K, C, and Winter Point (WP to one or more length-frequency distributions, estimating the parameters L , , K, C, and Winter Point (WP = t s + 0.5, or the time of the year at which growth is slowest). It should be noted that when only one = t s + 0.5, or the time of the year at which growth is slowest). It should be noted that when only one sample is available, the seasonally oscillating version of the VBGF cannot be applied. sample is available, the seasonally oscillating version of the VBGF cannot be applied. Requirements of the method are: Requirements of the method are: Samples must be representative of the structure of the population; Samples must be representative of the structure of the population; Growth must follow the von Bertalanffy model modified for seasonal growth; Growth must follow the von Bertalanffy model modified for seasonal growth; Recruitment must occur in seasonal pulses. Recruitment must occur in seasonal pulses. Not required are: Not required are: Regularly spaced samples; Regularly spaced samples; Catch and/or effort data; Catch and/or effort data; Normality of the distributions of lengths about successive ages; Normality of the distributions of lengths about successive ages; Knowledge of the number, position and standard deviation of successive mean lengths-at-age. Knowledge of the number, position and standard deviation of successive mean lengths-at-age.  "},{"text":"Table 2 .1. Example of restructuring effects on a hypothetical sample with 17 classes and five cohorts. Arrows indicate the mode of the fourth cohon. Moving Adjust FREQUENCY average fiiMAi FAiIFA-1 for 0 All fi M Ai FAi F'i frequencies  -1 -0 ~eutralization Results -1 -0~eutralization Results  "},{"text":"SUM(-) = -0.846; RATIO = 1.603   "},{"text":"Table 3 .1. Comparison of the computed and assumed parameters in 8 experiments to generate normally and gamma-distributed variates. Bias are expressed in percentage of the assumed parameters   PamfiwWl 1 2 3 4 5 6 7 0 PamfiwWl12345670  Mean 0.01       Mean0.01  S.D. 0.001       S.D.0.001 Simulated         Simulated  C.V.(%) 10       C.V.(%)10 Values Alpha 1 M       ValuesAlpha1 M Edmemd Mean am (%) 0,0100 -0.12       EdmemdMean am (%)0,0100 -0.12 Values S.D. 0.0010       ValuesS.D.0.0010  Bias (%) 0.40       Bias (%)0.40 (N-) Minimum Maximum 0.0067 0.0135       (N-)Minimum Maximum0.0067 0.0135  Mean 0.0088       Mean0.0088  Elan (%) 1.23       Elan (%)1.23 Eetlmred S.D. 0.0010       EetlmredS.D.0.0010  Bas(%) 4.70       Bas(%)4.70 Values Alpha 97.43m       ValuesAlpha97.43m  Bias p) 2.57       Bias p)2.57 (Gamma) Beta 9.W.12       (Gamma)Beta9.W.12  Bias (%) 1.36       Bias (%)1.36  Minimum 0.0071       Minimum0.0071  Maximum 0.0135       Maximum0.0135  "},{"text":"; RickerThe simulation model used for this study, a modified version of a program developed byHampton  and Majkowski (1987a), takes these four processes Rto account, and implements a simulated sampling procedure to obtain length-frequency data.The most important characteristics of the modellare: Each cohort (i.e., all the fishes belonging to one recruitment pulse) is treated individually. The life of each recruit is traced from the time of recruitment to the time of death (due either to natural causes or to fishing); causes or to fishing); Yearly cohort strength (N,) is assumed to be a random normal variate; Yearly cohort strength (N,) is assumed to be a random normal variate; Age at recruitment (t,) is determined by genarating a gamma random variate with mean cr/P and Age at recruitment (t,) is determined by genarating a gamma random variate with mean cr/P and variance d B 2 (see Equations 3.3,3.7 and 3.8); variance d B 2 (see Equations 3.3,3.7 and 3.8); Recruitment can be uni-or bimodal, i.e., fishes may recruit at two different ages; in the case of Recruitment can be uni-or bimodal, i.e., fishes may recruit at two different ages; in the case of bimodal recruitment, the proportion of recruib corresponding to each pulse can be determined; bimodal recruitment, the proportion of recruib corresponding to each pulse can be determined; Individual growth follows a von Bertalanffy equation, modified for seasonal oscillations (Pauly and Individual growth follows a von Bertalanffy equation, modified for seasonal oscillations (Pauly and Gaschijtz 1979; see Equation 2.1); Gaschijtz 1979; see Equation 2.1); The von Bertalanffy growth parameter L , vaties between individual fishes according to a normal The von Bertalanffy growth parameter L , vaties between individual fishes according to a normal 1975): 1975): recruitment; recruitment; growth; growth; natural mortality; and natural mortality; and fishing mortality. fishing mortality.  "},{"text":"Table 3 .  3. Average parameters and percentage of bias obbined with ELEFAN in Series II experiments, with increasing 3. Average parameters and percentage of bias obbined with ELEFAN in Series II experiments, with increasing variability in L , andlor K. variability in L , andlor K.  "},{"text":"Table 3 .4. Average parameters and percentage of bias obtained with the SLCA method in Series II experiments, with increasing variability in parameters  Simulated Estimated Bias (%) SimulatedEstimatedBias (%)  "},{"text":"Table 3 .5. Average parameters and percentage of bias &in& with the P-W method in Series II experiments, with increasing variability in parameters L, andlor K.  Simulated Estimated Bias (%) SimulatedEstimatedBias (%)  "},{"text":"Table 3 ,6. Average parameters obtained for the experiments of Series Ill, in which growth was assumed to oscillate seasonally and the coefficient of variation of    Simulated   Estimated    BIAS (%)  SimulatedEstimatedBIAS (%) Method Type CVL, CVK L , K C ,WP 9' ZIK L, K 9' C UK MethodType CVL,CVKL ,KC,WP9'ZIKL,K9'CUK   (%) (\"/ . I (cm) (year')  1   (cm) (year1)   (%)(\"/ . I(cm)(year')1(cm) (year1) ELEFAN (c*o) 1 2 0 20 0 20 52.24 55.09 0.460 0.69 810.0 0.432 0.60 0.0 3.099 3.115 -- 4.49 10.18 -13.52 -7.92 0.05 0.59 -8.31 6.46 -- ELEFAN (c*o)1 20 200 2052.24 55.090.460 0.69 810.0 0.432 0.60 0.03.099 3.115--4.49 10.18 -13.52 -7.920.05 0.59 -8.31 6.46-- ELEFAN (GO) 1 2 0 20 0 20 53.22 53.69 0.449 0.00 '0.0 0.444 0.00 10.0 3.104 3.105 -- 6.44 -10.16 7.38 -11.16 0.23 0.27 -- ELEFAN (GO)1 20 200 2053.22 53.690.449 0.00 '0.0 0.444 0.00 10.03.104 3.105--6.44 -10.16 7.38 -11.160.23 0.27-- SLCA 1 2 0 20 0 20 48.76 62.62 0.547 0.512 -- I 3.1 14 3.303 -- -2.48 25.24 9.44 2.44 0.56 6.42 -- SLCA1 20 200 2048.76 62.620.547 0.512--I3.1 14 3.303---2.48 25.249.44 2.440.56 6.42-- P-W 1 2 0 20 0 20 46.53 68.82 -- - 1 1 -  2.7 4.7 -6.94 37.64 --  --15.0419 -48.1140 P-W1 20 200 2046.53 68.82---1 1 -2.7 4.7-6.94 37.64----15.0419 -48.1140  "},{"text":"Table 3 . 8 . Average parameters and percentage of bias obtained with each method for samples with variable size-dependent selection effects (parameter a,  "},{"text":"Table 3 . 12. Results obtained with a set of length data and the ELEWN method, when including length-at-age data in the adjusting procedure. Bias is always calculated in relation to the parameters 4 e d for the simulation of the length-frequency data. - -  "},{"text":"Table 4  .1. Means and coefficients of variation of the parameters L, and K in experimental .1. Means and coefficients of variation of the parameters L, and K in experimental guppy and tilapia populations.     guppy and tilapia populations. Species N L , CVL, K CVK SpeciesNL ,CVL,KCVK   (cm) (%I  (%) (cm)(%I(%) L. reticulatus (females) 7 4.91 12 0.035 22 L. reticulatus (females)74.91120.03522 L. reticulatus (males) 4 2.38 5 0.12 11 L. reticulatus (males)42.3850.1211 Tilapia 70 9.25 26 0.035 30 Tilapia709.25260.03530  "},{"text":"Table 4 .2. Coefficients of variation of L, and K. Method CVof L , C V O ~ K Notes MethodCVof L ,C V O ~ KNotes  (\"A) (%)  (\"A)(%) Allen (1966) 25.6 30.0 all points included Allen (1966)25.630.0all points included Allen (1966) 20.0 30.1 outlying points Allen (1966)20.030.1outlying points    not included not included Soriano et al. 15.5 44.4 outlying points Soriano et al.15.544.4outlying points ( 1 990)   not included ( 1 990)not included  "},{"text":"Table 5 . 1. Sources of length-frequency data used in the present study. Names without year denote personal comrnunicalions by the researcherlinstitution indicated. C = commercial catch. S = research survey.      --    -- - -----  Code    sampling Sarrpiing Sampling Uass Nuher Nu* CodesamplingSarrpiingSamplingUassNuher Nu*  name  species Source period area method inleru dassss samples Observations namespeciesSourceperiodareamethodinlerudassss samplesObservations 1 CAs1 U m b r i n a m a l M. HahwbcWRGBrazil    1.0 26 4 pooled 1CAs1U m b r i n a m a lM. HahwbcWRGBrazil1.0264pooled 2 C A B Umbrina -ai J. KotaslCEPSUL-Brazii    1.0 26 4 2C A BUmbrina -aiJ. KotaslCEPSUL-Brazii1.0264 3 CJAMA C p a s c i o n ~i s H. ValenbinYlPBrazil    0.5 39 12 pooled 3CJAMAC p a s c i o n ~i sH. ValenbinYlPBrazil0.53912pooled 4 CJAMt C. @r&ans!s (mafes) Santos. 1968    1 . O 15 4 pooled 4CJAMtC. @r&ans!s (mafes)Santos. 19681 . O154pooled 5 ' CJAMP C. &mkmsk flernalm) Santos, 19BB    1 .O 15 4 pooled 5 'CJAMPC. &mkmsk flernalm)Santos, 19BB1 .O154pooled 6 CJAMB Cprosoirmpmicsnsb Vazzder 8 Baaga. 1983    1 .O 23 4 6CJAMBCprosoirmpmicsnsbVazzder 8 Baaga. 19831 .O234 7 CSTAl cplcsdrn samrus HaimPvld 8 Maceira. 1981    2.0 25 4 7CSTAlcplcsdrn samrusHaimPvld 8 Maceira. 19812.0254 8 CSTRIl cytlasdrnrnug J. KotksK:EPSUL-Brazil    1. 0 41 4 8CSTRIlcytlasdrnrnugJ. KotksK:EPSUL-Brazil1. 0414 9 10 11 CORV MIFUR MlFURl Micfcpqolhfumlen' Micropogonias funlien M w f u m i e n M. AeyJlNAPE-Uruguay Vazzder a a].. 1973 J. KotaarCEPSUL-Brazil    1 . O 2 0 1 . o 41 22 55 8 4 4 P O O M 9 10 11CORV MIFUR MlFURlMicfcpqolhfumlen' Micropogonias funlien M w f u m i e nM. AeyJlNAPE-Uruguay Vazzder a a].. 1973 J. KotaarCEPSUL-Brazil1 . O 2 0 1 . o41 22 558 4 4P O O M 12 M l M W M - hrrniefi J. KOtWCEPStlL-Emu    1.0 46 3 12M l M WM -hrrniefiJ. KOtWCEPStlL-Emu1.0463 13 COW1 Micropogon~as h i e n Lowe-Mdhnneli, 1966    1 .O 24 10 13COW1Micropogon~as h i e nLowe-Mdhnneli, 19661 .O2410 14 CREGA Qlmsmr€@is Massmaim. 1963    0.5 43 6 14CREGAQlmsmr€@isMassmaim. 19630.5436 15 CYNOB c y n o s m m Thomas, 1960    5.0 15 1 15CYNOBc y n o s m mThomas, 19605.0151 16 JVOG John- voglerf Mulhiah, 1982    2.0 15 12 16JVOGJohn-voglerfMulhiah, 19822.01512 17 LXANT L e ~v s xanlhurvs Pacheco, 1962    0.5 50 7 17LXANTL e ~v s xanlhurvsPacheco, 19620.5507 10 PESC Macrodon ancylWorlWor Manin Juraa, 1980    1 .O 28 12 10PESCMacrodon ancylWorlWorManin Juraa, 19801 .O2812 19 PESC1 MawdonaxyMm J. KoIaslCEPSUL-Brazil    1 . O 36 4 19PESC1MawdonaxyMmJ. KoIaslCEPSUL-Brazil1 . O364 20 PCOlB Pset&schena c a i h Rajan. 1967    2.5 31 12 20PCOlBPset&schena c a i hRajan. 19672.53112 21 22 23 PDlAC PSEME UCANA Pmrmbea diemthus Pseudcwms BB-i s Umbinacwariensis Rao, 1% Poinsard h Tmadec, 1966 Dardigm, 1961    5.0 1 .o 1 .o 18 47 34 12 12 5 P l e d 21 22 23PDlAC PSEME UCANAPmrmbea diemthus Pseudcwms BB-i s UmbinacwariensisRao, 1% Poinsard h Tmadec, 1966 Dardigm, 19615.0 1 .o 1 .o18 47 3412 12 5P l e d  "},{"text":"Table 5 . 2. Growth parameters estimated for 23 sets of length data on Sciaenidae with ELEFAN, ELEFAN-C, SLCA and P-W methods. Code    ELEFAN     ELEFANC     SLCA    P-W CodeELEFANELEFANCSLCAP-W name Spedes L , K C Sta. Poim ESPIASP y L , K C WP Sta.Poim ESPIASP -$ C, K b Smre  CI. C, UK 12 nameSpedesL ,KC Sta. Poim ESPIASP yL ,KCWPSta.Poim ESPIASP-$C,KbSmreCI.C,UK12 CASl Un-brina m a i 52.53 0.355 0 3 124.0 0.373 2.991 49.60 0.624 1.00 0.00 1 H9.00 0.465 3186 43.30 0.320 0.5 61.W 2.778 13 37.50 2.337 0.9% CASlUn-brina m a i52.530.3550 3 124.0 0.373 2.991 49.60 0.624 1.00 0.001 H9.000.465318643.30 0.320 0.561.W2.7781337.502.3370.9% CAS2 Umbrina c m a i 47.90 0.302 0 1 126.50 0 . W 2.841 43.45 0.630 0.75 0.30 3 137.50 0.51 1 1075 39.30 0.470 0.E 6380 2.861 10 38.51 2.m0 0.984 CAS2Umbrina c m a i47.900.3020 1 126.50 0 . W 2.841 43.45 0.630 0.75 0.303 137.500.51 1107539.30 0.470 0.E63802.8611038.512.m00.984 CJAMA CynoscionjamaiaKsb 33.56 0.315 0 4 H9.00 0.261 2.550 35.00 0.250 0.50 0.86 5 128.00 0.323 2.486 31.00 0.435 0.9 24.60 2.621 19 35.32 5722 0.894 CJAMACynoscionjamaiaKsb33.560.3150 4 H9.000.261 2.550 35.00 0.250 0.50 0.865 128.000.3232.48631.00 0.435 0.924.602.6211935.3257220.894 CJAMl C.jmaicsnsa(males) 40.12 0.361 0 1 125.25 0.527 2.7M 40.34 0.222 1.00 0.00 2 120.25 0.581 2558 33.50 0.552 0.6 33.90 2.7W 7 31.88 3.867 0.985 CJAMlC.jmaicsnsa(males)40.120.3610 1 125.250.527 2.7M 40.34 0.222 1.00 0.002 120.250.581255833.50 0.552 0.633.902.7W731.883.8670.985 CJAM2 C. juMicsnsirpernales) 3260 0.380 0 2125.25 0.571 2.606 32.80 0.376 0.38 0.00 2125.25 0.581 2607 29.90 0.400 0.9 10.20 2.553 7 34.03 4.526 0.958 CJAM2C. juMicsnsirpernales)32600.3800 2125.250.571 2.606 32.80 0.376 0.38 0.002125.250.581260729.90 0.400 0.910.202.553734.034.5260.958 CJAM3 Cynascionjamaicensis 35.70 0.262 O 2 1 7.50 0.520 2.524 28.78 0.371 0.86 0.00 3 120.50 0.564 2.487 44.50 0.355 0.8 61.70 2.647 15 31.44 4.706 0.921 CJAM3Cynascionjamaicensis35.700.262O 2 1 7.50 0.520 2.524 28.78 0.371 0.86 0.003 120.500.5642.48744.50 0.355 0.861.702.6471531.444.7060.921 CSTRl CpaScion shiafu6 M-75 0.525 0 4 1 6.00 0.499 3.315 58.06 0.5BD 0.37 0.70 2 110.00 0.%7 3.291 53.80 0.460 0.9 68.90 3.124 19 5 3 3 2.731 0.978 CSTRlCpaScion shiafu6M-750.5250 4 1 6.00 0.499 3.315 58.06 0.5BD 0.37 0.702 110.000.%73.29153.80 0.460 0.968.903.124195 3 32.7310.978 CSTRll Cyntschshiahrs 60.62 0.495 0 1 137.50 0.275 3.260 64.06 0.318 0.89 0.22 2 127.50 0.369 3.116 56.30 0.490 0.7 73.513 3.191 28 51.50 1.550 0.995 CSTRllCyntschshiahrs60.620.4950 1 137.500.275 3.260 64.06 0.318 0.89 0.222 127.500.3693.11656.30 0.490 0.773.5133.1912851.501.5500.995 C O W Y w l u m i e n ' 68.- 0.150 0 1 149.50 0.429 2.846 63.72 0.232 1.00 0.00 7 124.50 0.498 2.974 56.00 0.510 0.8 162.50 3204 77 66.09 3.750 0.932 C O WY w l u m i e n '68.-0.1500 1 149.500.429 2.846 63.72 0.232 1.00 0.007 124.500.4982.97456.00 0.510 0.8162.5032047766.093.7500.932 MIFUR MicropogwrEasbmien' 71.90 0.160 0 4 136M) 0.399 2.918 62.43 0.322 0.98 0.50 1 132.D3 0.608 3.099 62.75 0235 0.1 35.75 2.%6 12 68.33 4.795 0.913 MIFURMicropogwrEasbmien'71.900.1600 4 136M)0.399 2.918 62.43 0.322 0.98 0.501 132.D30.608 3.09962.75 0235 0.135.752.%61268.334.7950.913 MIFUR1 MbqmgoyIvmien' 77.11 0.177 0 2131.50 0.407 3.022 74.00 0.232 1.00 0.62 1147.50 0.542 3.113-78.00 0370 0. 7 203.40 3.361 19 67.90 1.960 0.920 MIFUR1MbqmgoyIvmien'77.110.1770 2131.500.407 3.022 74.00 0.232 1.00 0.621147.500.5423.113-78.00 0370 0. 7203.403.3611967.901.9600.920 MIFURZ MicropogonrasIvmien' 70.57 0.153 0 113250 0.481 2.882 72.M 0.177 0.91 0.43 113750 0.606 2.972 71.90 0.260 0.2 46.60 3.128 19 67.53 2.954 0.992 MIFURZMicropogonrasIvmien'70.570.1530 113250 0.481 2.882 72.M 0.177 0.91 0.431137500.6062.97271.90 0.260 0.246.603.1281967.532.9540.992 CORVl CREGA cwoa M -a ? i a s W C y ~s c f u n ~ Qnusdon~brk 51.80 43.10 181.15 0.220 0241 0.308 o 1 1~7.50 1 . m 4.005 0 4 138.50 0.402 2.771 48.13 0.435 0.48 0.60 0 3 125.50 0.307 2.651 39.75 0.204 1.M) 0.00 ---. 1 133.50 5 130.CC1 . . 0.467 0.372 3 . m 2.508 - 39.50 0.510 0.9 39.10 0.140 0.9 144.8 0.520 0.2 28.M 22.70 . 11.00 2.901 2 . a 4.038 11 18 8 47.72 39.84 154.24 4.854 4.568 2.908 0.789 0.987 0.951 CORVl CREGA cwoaM -a ? i a s W C y ~s c f u n ~ Qnusdon~brk51.80 43.10 181.150.220 0241 0.308 o 1 1~7.50 1 . m 4.005 0 4 138.50 0.402 2.771 48.13 0.435 0.48 0.60 0 3 125.50 0.307 2.651 39.75 0.204 1.M) 0.00 ---.1 133.50 5 130.CC1 . .0.467 0.3723 . m 2.508 -39.50 0.510 0.9 39.10 0.140 0.9 144.8 0.520 0.228.M 22.70 . 11.002.901 2 . a 4.03811 18 847.72 39.84 154.244.854 4.568 2.9080.789 0.987 0.951 JVOG Jallnkww 34.40 0.590 0 1 111.10 0.714 2.844 34.88 0.570 0.10 0.50 3 113.50 0.722 2.841 42.50 0.440 0.5 144.90 2.WO 3 29.65 0.658 0.989 JVOGJallnkww34.400.5900 1 111.10 0.714 2.844 34.88 0.570 0.10 0.503 113.500.7222.84142.50 0.440 0.5144.902.WO329.650.6580.989 UANT L & I ~x a f h w u s 28.23 0.202 0 4 1 7.00 0.436 2.207 27.98 0.284 0.80 0.00 6 H 1.50 0.427 2.347 28.40 0.160 0.9 60.80 2.11 1 34 26.45 2.451 0.995 UANTL & I ~x a f h w u s28.230.2020 4 1 7.000.436 2.207 27.98 0.284 0.80 0.006 H 1.500.4272.34728.40 0.160 0.960.802.11 1 3426.452.4510.995 K S C Maaodon WCyk?dUll 45.90 0.240 0 9 129.50 0.463 2.704 45.80 0.244 0.30 0.80 6 123.50 0,474 2.709 34.70 0.700 0.4 413.10 2.926 18 43.53 2269 0.951 K S CMaaodon WCyk?dUll45.900.2400 9 129.500.463 2.704 45.80 0.244 0.30 0.806 123.500,4742.70934.70 0.700 0.4413.102.9261843.5322690.951 I'€Scl Macrodonan@&n 49.68 0.388 0 3 126.9 0.290 2.981 48.70 0.253 0.40 0.74 1 128.50 0.340 2.778 42.50 0.210 0.2 25.00 2.579 15 43.16 5.159 0.932 I'€SclMacrodonan@&n49.680.3880 3 126.90.290 2.981 48.70 0.253 0.40 0.741 128.500.3402.77842.50 0.210 0.225.002.5791543.165.1590.932 PC010 Pseudasckma caibor 89.W 0.250 0 9 178.80 0.387 3.297 89.14 0.255 0.44 0.17 6 138.80 0.430 3.307 87.30 0.400 0.6 25a50 3.464 25 91.16 4379 0.860 PC010Pseudasckma caibor89.W0.2500 9 178.80 0.387 3.297 89.14 0.255 0.44 0.176 138.800.4303.30787.30 0.400 0.625a503.4642591.1643790.860 POlAC P m t o n i b e a ~h v s 135.79 0.228 0 8 192.50 0.437 3.624 125.62 0.279 0.65 0.70 1 197.50 0.518 3.644 108.30 0.400 0.4 82.80 3.671 15 11 1.37 3.241 0.802 POlACP m t o n i b e a ~h v s135.790.2280 8 192.50 0.437 3.624 125.62 0.279 0.65 0.701 197.500.5183.644108.30 0.400 0.482.803.6711511 1.373.2410.802 PSENE Pseudoldi[hrrs-b 57.32 0.361 0 t 114.10 0.276 3.074 57.52 0.354 0.10 0.80 1 114.00 0.318 3.069 59.60 0.380 0.3 2M.80 3.130 20 57.21 3.476 0.994 PSENEPseudoldi[hrrs-b57.320.3610 t 114.10 0.276 3.074 57.52 0.354 0.10 0.801 114.000.3183.06959.60 0.380 0.32M.803.130 2057.213.4760.994 UCANA UmtrhacmrisMis 43.20 0.238 0 5 114.00 0.619 2.648 43.28 0.242 0.85 0.00 2 121.00 0.661 2.656 38.40 0.120 0.4 32.90 2248 13 40.13 1.422 0.934 UCANAUmtrhacmrisMis43.200.2380 5 114.00 0.619 2.648 43.28 0.242 0.85 0.002 121.000.6612.65638.40 0.120 0.432.9022481340.131.4220.934  "},{"text":"Table 5 .3. Mean, median, mode, standard deviation, coefficient ofbariation, minimum and maximum of the estimates of l-, K and $' obtained with ELEFAN, ELEFAN-C, SLCA and P-W method on 2llength data sets of sciaenid fishes.  Parameter Method Mean Median W e SD CV (010) min max ParameterMethodMeanMedianW eSDCV (010)minmax  ELEFAN 56.01 50.74 4F.90 24.01 42.87 28.23 135.79 ELEFAN56.0150.744F.9024.0142.8728.23135.79 L, ELEFAN-C 53.94 48.42 43180 22.57 41.84 27.98 125.62 L,ELEFAN-C53.9448.424318022.5741.8427.98125.62  SLCA 50.97 42.90 4250 20.50 40.22 28.40 108.30 SLCA50.9742.90425020.5040.2228.40108.30  P-W 50.62 43.35 4413 21.20 41.88 26.45 111.37 P-W50.6243.35441321.2041.8826.45111.37  ELEFAN 0.300 0.256  0.122 40.67 0.150 0.590 ELEFAN0.3000.2560.12240.670.1500.590 K ELEFAN-c 0.339 0.282  0.141 41 5 9 0.177 0.630 KELEFAN-c0.3390.2820.14141 5 90.1770.630  SLCA 0.378 0.400 a400 0.145 38.36 0.1 20 0.700 SLCA0.3780.400a4000.14538.360.1 200.700  ELEFAN 2.878 2.845 a841 0.314 10.91 2.207 3.624 ELEFAN2.8782.845a8410.31410.912.2073.624 @' ELEFAN-C 2.901 2.973 3841 0.330 1 1.38 2.347 3.644 @'ELEFAN-C2.9012.97338410.3301 1.382.3473.644  SLC A 2.896 2.901 a861 0.390 13.47 2.111 3.671 SLC A2.8962.901a8610.39013.472.1113.671  "},{"text":"Table 5 . 4. Friedman test (by ranks) to compare the growth paramaers.    Parameter K   Table of ranks  Parameter KTable of ranks File Code ELEFAN ELEFAN-C SLCA ELEFAN ELEFAN-C SLCA FileCodeELEFANELEFAN-CSLCAELEFANELEFAN-CSLCA  CAS 1   0.320    CAS 10.320  CAS2   0.470    CAS20.470  CJAMA   0.435    CJAMA0.435  CJAMl   0.552    CJAMl0.552  CJAM2   0.400    CJAM20.400  CJAM3   0.355    CJAM30.355  CSTRI   0.460    CSTRI0.460  CSTRll   0.490    CSTRll0.490  CORV   0.510    CORV0.510  MlFUR   0.235    MlFUR0.235  MIFUR1   0.370    MIFUR10.370  MIFUR2   0.260    MIFUR20.260  CORVl   0.510    CORVl0.510  CREGA   0.140    CREGA0.140  JVOG   0.440    JVOG0.440  LXANT   0.160    LXANT0.160  PESC   0.700    PESC0.700  PESC1   0.210    PESC10.210  PCOlB   0.400    PCOlB0.400  PDlAC   0.400    PDlAC0.400  PSENE   0.380    PSENE0.380  UCANA   0.120    UCANA0.120     SUM =    SUM =   Calculated statistic = 2.528     Calculated statistic = 2.528   F = 3.22; 0.05; 2; 42 ===> No significant diffwences    F = 3.22; 0.05; 2; 42 ===> No significant diffwences    Parameter I$'     Parameter I$'  "},{"text":"Table 5 . 5. Friedman rank test comparing the estimates of I (top) beween methods and results of the test for multiple comparison(Conover 1980).   Asymptotic Length (LJ  Table of Ranks Asymptotic Length (LJTable of Ranks  Code     Code  ELEFAN ELEFAN-C SLCA P-W ELEFAN ELEFAN-C SLCA P-W ELEFANELEFAN-CSLCAP-WELEFAN ELEFAN-C SLCAP-W  CASl   37.50  CASl37.50  GAS2   38.51  GAS238.51  CJAMA   35.32  CJAMA35.32  CJAMl   3 1.88  CJAMl3 1.88  CJAM2   34.03  CJAM234.03  CJAM3   31.44  CJAM331.44  CSTRl   53.32  CSTRl53.32  CSTRI1   51.50  CSTRI151.50  CORV   66.09  CORV66.09  MIFUR   68.33  MIFUR68.33  MIFUR1   67.90  MIFUR167.90  MIFUR2   67.53  MIFUR267.53  CORVl   47.72  CORVl47.72  CREGA   39.84  CREGA39.84  JVOG   29.65  JVOG29.65  LXANT   26.45  LXANT26.45  PESC   43.53  PESC43.53  PEW1   43.16  PEW143.16  PCOIB   91.16  PCOIB91.16  PDIAC   11 1.37  PDIAC11 1.37  PSENE   57.21  PSENE57.21  UCANA   40.13  UCANA40.13     SUM =  SUM =  Calculated statistic = 5.719    Calculated statistic = 5.719  F = 2.76; 0.05; 3; 60 ===r Significant differences  F = 2.76; 0.05; 3; 60 ===r Significant differences  Multiple comparison ===> Calculated statistic = 15.533 Multiple comparison===>Calculated statistic =15.533   Rank   Rank  Methods difference Signifimt  MethodsdifferenceSignifimt -- ELEFAN -ELEFAN-C ELEFAN -SLCA ELEFAN -P-W ELEFAN-C -SLCA ELEFAN-C -P-W SLCA -P-W 8 25 27 17 19 2 -no Yes Yes Yes Yes no  --ELEFAN -ELEFAN-C ELEFAN -SLCA ELEFAN -P-W ELEFAN-C -SLCA ELEFAN-C -P-W SLCA -P-W8 25 27 17 19 2-no Yes Yes Yes Yes no  "},{"text":"Table 6 . 1. Estimates of Z obtained with different combinations of points of the catch curve. Parameters: L, = 50, K = 0.5, Z = 1.6. Number of classes: 24 Points included Z Bias (%) Points includedZBias (%) 1 -24 1 -10 1.660 1.576 3.75 -1,50 1 -24 1 -101.660 1.5763.75 -1,50 1 -2 0 1.590 -0.63 1 -2 01.590-0.63 10-20 20 -24 1.758 1.813 9.87 13.31 10-20 20 -241.758 1.8139.87 13.31  "},{"text":"Table 6 . 2. Estimates of Z obtained from catch curves of populations with different growth strategies, calculated with the original growth parameters and with the growth parameters estimated by ELEFAN I.   Parameters    No. Last   ParametersNo.Last True   Estimated Parameters of class  Bias TrueEstimatedParametersofclassBias L , K Z L, K used classes included Z (%) L ,KZL,KusedclassesincludedZ(%) 30 1.8 5.0 30.46 1.449 me 30 30 5.376 7.52 301.85.030.461.449me30305.3767.52      estimated 30 30 4.197 -16.06 estimated30304.197-16.06      estimated 30 29 4.471 -10.58 estimated30294.471-10.58 50 0.6 1.9 55.40 0.489 me 26 26 1.872 -1 -47 500.61.955.400.489me26261.872-1 -47      estimated 26 26 1.957 3.00 estimated26261.9573.00      estimated 26 25 1.965 3.42 estimated26251.9653.42 50 0.2 0.6 91.90 0.1 79 true 30 30 0.598 -0.33 500.20.691.900.1 79true30300.598-0.33      estimated 30 30 0.643 7.17 estimated30300.6437.17      estimated 30 29 0.672 12.00 estimated30290.67212.00 110 0.1 0.3 139.80 0.088 true 3 1 31 0.291 -3.00 1100.10.3139.800.088true3 1310.291-3.00      estimated 3 1 3 1 0.408 36.00 estimated3 13 10.40836.00      estimated 31 30 0.416 38.67 estimated31300.41638.67  "},{"text":"Table 6 .4. Estimates of Z obtained from samples with sizedependent selection effects, without and with 10% individual variability of the growth parameters.    Parameters  No. Classes  ParametersNo.Classes GV (%)  used  of used Bias GV (%)usedofusedBias L , K L, K classes Z (\"' 4 L ,KL,KclassesZ(\"' 4  "},{"text":" manth 1 monh 1 monh hornorurn reared for 25 weeks in experimental tanks (calculated after original data from Doyle, pers. cornrn.).            TIME (in wmkl   TIME (in wmkl . -  --  - -.r  - - - - . -----.r----    Slmulaled      Estimakd    has (%] SlmulaledEstimakdhas (%] P a 1 SO 1 1r2 $0 2  Range L ' K SI.Polnt ESPIASP 0'  L, K q Pa 1SO 11r2$0 2RangeL ' KSI.Polnt ESPIASP0'L,Kq   0             0   0             0   0             0   0             0   0             0   1 month            1 month   1 month            1 month   1 mmth            1 mmth   1 month            1 month   1 month            1 month   0             40.m 42.m 040.m42.m                4 2 M 45.50 4 a d 48.54 4 2 M45.504 a d48.54   0             3 7 m 40.50 44.5 46.50 03 7 m40.5044.546.50                41.00 44.50 47.C 47.50 41.0044.5047.C47.50   0             37.50 3 8 m 41.C 40.50 037.503 8 m41.C40.50                42.00 4450 465 48.00 42.00445046548.00   0             3850 39.00 U M I 42.00 435 455 46.00 03850 39.00U M I 42.00435 45546.00   0             42.00 37.50 44.00 10.50 48C 43i 46.50 44.50 042.00 37.5044.00 10.5048C 43i46.50 44.50                4250 46.50 411: 48.00 425046.50411:48.00 0.5 0.5 0 5 0.5 0 5 0.5 lmanm 0.8 1 monlh 0.8 I mmth 0.8 imonth 1 month 1 manlh 7.47 9.47 7.47 54.80 52.85 52.67 0 500 2113.20 0.445 211320 0.432 2nD.W) 50.85 '0.466 4121.w 0.758 0.663 0753 0.616 53.30 0.485 1/11 80 0.506 50.40 0.475 4121.W 0.737 3.128 3.081 3 142 3.080 3.138 3.082  41.50 9.60 -11.00 094 3 7 m 5.68 -1380 -0.50 5 34 41.00 0.00 1.48 i m 3550 -7.00 -0.54 6 80 -3.00 1.37 0.80 .5 00 -0 50 a 0 0 r r r . 41.50 43.50 3 8 M 43.50 38.50 40.50 ma& 45130 4ac 43.5 46C 42C Y 491 49.50 4500 49.00 4150 wm 41150 0.5 0.5 0 50.5 0 5 0.5lmanm 0.8 1 monlh 0.8 I mmth 0.8imonth 1 month 1 manlh7.47 9.47 7.4754.80 52.85 52.67 0 500 2113.20 0.445 211320 0.432 2nD.W) 50.85 '0.466 4121.w 0.758 0.663 0753 0.616 53.30 0.485 1/11 80 0.506 50.40 0.475 4121.W 0.7373.128 3.081 3 142 3.080 3.138 3.08241.50 9.60 -11.00 094 3 7 m 5.68 -1380 -0.50 5 34 41.00 0.00 1.48 i m 3550 -7.00 -0.54 6 80 -3.00 1.37 0.80 .5 00 -0 50 a 0 0 r r r . 41.5043.50 3 8 M 43.50 38.50 40.50 ma& 451304ac 43.5 46C 42C Y 49149.50 4500 49.00 4150 wm 41150 0.5 0.5 0.5 0.5 1 month 0 8 1 monlh 0.0 1 monm 1 month 7-47 7-47 55.23 ~0.434 1111.60 51.90 0.430 2119.00 5505 0.441 2dl4.20 51.70 0.432 7126.40 0.6% 0.737 0.616 0.750 3.122 3.064 3.126 3.064  10.45 -13.20 0.80 41.50 40.00 3.80 -14.00 -1.07 39W 10.10 -11 80 0.94 30.50 3.58 -13.60 .1.07 41.00 48.00 4550 4 5 . m 43-00 43.50 a t 5aC 48C 415 4LC 48.50 saw 47.m 44.50 0.5 0.50.5 0.51 month 0 8 1 monlh 0.01 monm 1 month7-47 7-4755.23 ~0.434 1111.60 51.90 0.430 2119.00 5505 0.441 2dl4.20 51.70 0.432 7126.400.6% 0.737 0.616 0.7503.122 3.064 3.126 3.06410.45 -13.20 0.80 41.50 40.00 3.80 -14.00 -1.07 39W 10.10 -11 80 0.94 30.50 3.58 -13.60 .1.07 41.0048.00 4550 4 5 . m 43-00 43.50a t 5aC 48C 415 4LC48.50 saw 47.m 44.50                41.50 47.00 52 t 5203 41.5047.0052 t5203                41.M 44.00 45. : 4 5 m 41.M44.0045. :4 5 m        SLCA       41.00 27.50 4hW 4.00 4 a t 42: 48.00 42.W SLCA41.00 27.504hW 4.004 a t 42:48.00 42.W                41.00 45.50 46: 46.m 41.0045.5046:46.m                43.m 38.50 48.53 42.00 47.C U.! 47.m u.m 43.m 38.5048.53 42.0047.C U.!47.m u.m   -            3850 38 50 42.50 42 50 43.c U.! 13.W u.m -3850 38 5042.50 42 5043.c U.!13.W u.m   0             42.50 45.00 46.C 46.50 042.5045.0046.C46.50   0 0             38.50 u.m 42.00 4 8 , s a t 52: 4.50 5bm 0 038.50 u.m42.00 4 8 , sa t 52:4.50 5bm   0 0             41.50 43.m 30.m 45.50 46 50 41.00 50. t 48.C 41.1 51.00 41.00 0 041.50 43.m 30.m45.50 46 50 41.0050. t 48.C 41.151.00 41.00   month monlh mmth month month            4450 4150 3850 4150 4250 urn 48.50 u . 5 0 43.50 42.00 45m 47.00 52: 4&c 45! 4 a sa! 53.50 46.00 46.00 51.50 month monlh mmth month month4450 4150 3850 4150 4250 urn48.50 u . 5 0 43.50 42.00 45m 47.0052: 4&c 45! 4 a sa!53.50 46.00 46.00 51.50   0 0             4150 4 l m 44% a.58 45.00 47.50 YU &! sa! 50.50 46.50 49.50 0 04150 4 l m 44%a.58 45.00 47.50YU &! sa!50.50 46.50 49.50   0             4 4 m 48.00 48( 47.50 04 4 m48.0048(47.50   0             4 2 m 45.50 45( 45.00 04 2 m45.5045(45.00   0             4 1 m 41DO 4550 4100 49( 45! a 5 0 44.50 04 1 m 41DO4550 410049( 45!a 5 0 44.50 0.5 0.5 0.5 0.5 0.5 0 5 0.5 0.5 0.5 0.5 1 monh 0.8 1 monlh 0.8 1 month 0.8 1 month 0 8 lmonth 0.8 1 month 1 month 1 mmth I month lmonlh 7-47 k 4 7 7-47 7-47 7-47 47.10 45.40 46.00 45.50 45.40 M.670 ,(0.737 YI.872 ,b.721 p733 0.6 0.8 0.6 0.8 0.6 320 5 302.2 310.0 313.1 288.3 3.172 3.182 3.158 3.174 3.179  -5.80 34.W -9.20 47.40 8.40 34.40 -9.00 44.20 -9.20 46.60 2.43 42a0 4 3 s 2.73 2.29 43% 2.49 40.621 266 2065 44.00 U1. m 48.00 43.957 2 . m 4&! 52! 5a( 46. 2.: 46.m 5 x 0 50. m 47.115 3.083 0.5 0.5 0.5 0.5 0.50 5 0.5 0.5 0.5 0.51 monh 0.8 1 monlh 0.8 1 month 0.8 1 month 0 8 lmonth 0.81 month 1 month 1 mmth I month lmonlh7-47 k 4 7 7-47 7-47 7-4747.10 45.40 46.00 45.50 45.40M.670 ,(0.737 YI.872 ,b.721 p7330.6 0.8 0.6 0.8 0.6320 5 302.2 310.0 313.1 288.33.172 3.182 3.158 3.174 3.179-5.80 34.W -9.20 47.40 8.40 34.40 -9.00 44.20 -9.20 46.602.43 42a0 4 3 s 2.73 2.29 43% 2.49 40.621 266 206544.00 U1. m 48.00 43.957 2 . m4&! 52! 5a( 46. 2.:46.m 5 x 0 50. m 47.115 3.083                0.051 0.057 0.1 0.065 0.0510.0570.10.065                1 . 5 5 6 41.457 43. 4 4 . m 1 . 5 5 641.45743.4 4 . m        P-W m l o d      62.686 46.450 49. 50.197 P-W m l o d62.68646.45049.50.197     Simulated    Eslimatsd  Bias (%I SimulatedEslimatsdBias (%I   P wl SO 1 lr2 $0 2 Rangti L, UK r2 L, UK PwlSO 1lr2$0 2RangtiL,UKr2L,UK    0 9 0.5 0 0 -- -- 1 1 4 f 50.27 11-47 , 48.21 3.224 2.838 0 . 9 s 0.998 0.54 -3 58 -11.31 0 75 0 9 0.50 0----1 1 4 f 50.27 11-47 , 48.213.224 2.8380 . 9 s 0.9980.54 -3 58 -11.31 0 75  I  0.5 0.5 0.5 0 0 0 --- --- 11-48 ' 54.00 11.49 50.03 il\"47 47.86 3.W 3.409 2.770 0.997 0.998 0.886 8 W 1.M -4.28 -13.44 18.75 6 53 I0.5 0.5 0.50 0 0------11-48 ' 54.00 11.49 50.03 il\"47 47.863.W 3.409 2.7700.997 0.998 0.8868 W 1.M -4.28 -13.44 18.75 6 53    1 month           1 month    1 month           1 month    1 monh           1 monh    1 manh           1 manh    1 month           1 month     0           0     0           0     0           0     0           0     0           0    1 month  1 month        1 month1 month    1 monh           1 monh    1 month           1 month    1 monh           1 monh     1 monh           1 monh  "}],"sieverID":"b6895153-1ebe-40c5-ba21-057136430d67","abstract":"SH 207 TR4 =127 c.1 q LCJC' ' \"' 11 \"(:0 :2C'O1 ~' \" .,1..; ~A~l OO -(j\" ,."}
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