Simulation of tail weight distributions in biological year 1986-2006 landings of brown shrimp, Farfantepenaeus aztecus, from the Northern Gulf of Mexico fishery.
Abstract: Size distribution within reported landings is an important aspect of northern Gulf of Mexico penaeid shrimp stock assessments. It reflects shrimp population characteristics such as numerical abundance of various sizes, age structure, and vital rates (e.g. recruitment, growth, and mortality), as well as effects of fishing, fishing power, fishing practices, sampling, size-grading, etc.

The usual measure of shrimp size in archived landings data is count (C) the number of shrimp tails (abdomen or edible portion) per pound (0.4536 kg). Shrimp are marketed and landings reported in pounds within tail count categories. Statistically, these count categories are count class intervals or bins with upper and lower limits expressed in C. Count categories vary in width, overlap, and frequency of occurrence within the landings. The upper and lower limits of most count class intervals can be transformed to lower and upper limits (respectively) of class intervals expressed in pounds per shrimp tail, w, the reciprocal of C (i.e. w = 1/C).

Age based stock assessments have relied on various algorithms to estimate numbers of shrimp from pounds landed within count categories. These algorithms required underlying explicit or implicit assumptions about the distribution of C or w. However, no attempts were made to assess the actual distribution of C or w. Therefore, validity of the algorithms and assumptions could not be determined. When different algorithms were applied to landings within the same size categories, they produced different estimates of numbers of shrimp.

This paper demonstrates a method of simulating the distribution of w in reported biological year landings of shrimp. We used, as examples, landings of brown shrimp, Farfantepenaeus aztecus, from the northern Gulf of Mexico fishery in biological years 1986-2006. Brown shrimp biological year, Ti, is defined as beginning on 1 May of the same calendar year as [T.sub.i] and ending on 30 April of the next calendar year, where subscript i is the place marker for biological year. Biological year landings encompass most if not all of the brown shrimp life cycle and life span. Simulated distributions of w reflect all factors influencing sizes of brown shrimp in the landings within a given biological year. Our method does not require a priori assumptions about the parent distributions of w or C, and it takes into account the variability in width, overlap, and frequency of occurrence of count categories within the landings. Simulated biological year distributions of w can be transformed to equivalent distributions of C.

Our method may be useful in future testing of previously applied algorithms and development of new estimators based on statistical estimation theory and the underlying distribution of w or C. We also examine some applications of biological year distributions of w, and additional variables derived from them.
Subject: Fish industry (Influence)
Fisheries (Influence)
Shrimps (Research)
Authors: Caillouet, Charles W., Jr.
Hart, Rick A.
Nance, James M.
Pub Date: 03/22/2011
Publication: Name: Marine Fisheries Review Publisher: Superintendent of Documents Audience: Academic Format: Magazine/Journal Subject: Agricultural industry; Business Copyright: COPYRIGHT 2011 U.S. Department of Commerce ISSN: 0090-1830
Issue: Date: Spring, 2011 Source Volume: 73 Source Issue: 2
Topic: Event Code: 310 Science & research
Product: Product Code: 0900000 Fishing, Hunting & Trapping; 0913080 Shrimp NAICS Code: 114 Fishing, Hunting and Trapping; 114112 Shellfish Fishing SIC Code: 0912 Finfish; 0913 Shellfish; 0919 Miscellaneous marine products; 0921 Fish hatcheries and preserves
Geographic: Geographic Scope: United States Geographic Code: 1USA United States
Accession Number: 272740311
Full Text: Introduction

Size distribution within reported landings is an important aspect of northern Gulf of Mexico penaeid shrimp stock assessments. It reflects population characteristics such as numerical abundance of various sizes, age structure, and vital rates (e.g. recruitment, growth, and mortality), as well as effects of fishing, fishing power, fishing practices, sampling, size-grading, etc. (Kutkuhn, 1962; Neal, 1967; Rothschild and Brunenmeister, 1984; Nance et al., 1994; Diop et al., 2007; Caillouet et al., 2008; Nance et al., 2010; Parrack (1); Nichols (2)). Age of shrimp cannot be determined directly (Parrack, 1979; Rothschild and Brunenmeister, 1984; Neal and Maris, 1985). Therefore, age structure of shrimp in reported landings has been determined indirectly by estimating numbers of shrimp from pounds allocated to marketing size categories, and transforming size into age using growth curves (Neal, 1967; Rothschild and Brunenmeister, 1984; Nance et al., 1994; Parrack (1); Nichols (2)).

Most but not all reported landings from northern Gulf of Mexico shrimp fisheries are size-graded. The usual measure of shrimp size in archived landings data is count (C), the number of shrimp tails (abdomen or edible portion) per pound (0.4536 kg). Shrimp are marketed and landings reported in pounds within tail count categories. Statistically, these count categories are count class intervals or bins with upper and lower limits expressed in C. The upper and lower limits of most count class intervals can be transformed to lower and upper limits (respectively) of class intervals expressed in pounds per shrimp tail, w, the reciprocal of C (i.e. w = 1/C)

Age based stock assessments have relied on various algorithms to estimate numbers of shrimp from pounds landed within count categories (e.g. Neal, 1967; Rothschild and Brunenmeister, 1984; Nance et al., 1994; Diop et al., 2007; Parrack (1); Nichols (2)). These algorithms required underlying explicit or implicit assumptions about the distribution of C or w. However, no attempts were made to assess the actual distributions of C and w. Therefore, validity of the algorithms and assumptions could not be determined. When different algorithms were applied to landings within the same size categories (e.g. Parrack (1) vs. Nichols (2)), they produced different estimates of numbers of shrimp (Cail louet, 2003).

Estimating numbers of shrimp from pounds landed within size categories is statistically challenging for additional reasons. Some count categories representing the largest shrimp have an implied lower limit of zero (e.g. < 15 count), and some representing the smallest shrimp have an implied upper limit of [infinity] (e.g. > 67 count). Neither zero nor [infinity] can be transformed to real values of w. Count categories also exhibit considerable variability in width, overlap, and frequency of occurrence within the landings. Certain count categories dominate the landings, reflecting what are referred to as standard count categories: <15, 15-20, 21-25, 26-30, 31-10, 41-50, 51-67, and > 67 count (Caillouet et al., 2008).

This paper demonstrates a method of simulating the distribution of w in reported biological year landings of shrimp, as a basis for further investigation and evaluation of previously used algorithms and development of new ones. We used, as examples, landings of brown shrimp, Farfantepenaeus aztecus, from the northern Gulf of Mexico fishery in biological years 1986-2006. Neal (1967) defined brown shrimp biological year, [T.sub.i], as beginning 1 May of the same calendar year as [T.sub.i] and ending 30 April of the next calendar year, where subscript i is the place marker for biological year (Table 1). Most landings in the ith biological year are assumed to be produced from cohorts recruited to the fishery within that same biological year. In other words, a biological year encompasses most of the cycle and life span of brown shrimp within this intensive fishery.

Our approach does not require a priori assumptions about the parent distributions of w or C, and it takes into account the variability in width, overlap, and frequency of occurrence of count categories within the landings. Simulated biological year distributions of w can easily be transformed to equivalent distributions of C. Our method may be useful in future testing of previously applied algorithms and development of new estimators based on statistical estimation theory and the underlying distribution of w or C. We also examine some applications of biological year distributions of w and additional variables derived from them.

Materials and Methods

Fishery

The brown shrimp fishery of the northern Gulf of Mexico is bounded by statistical subareas 10-21, and comprises inshore (estuarine) and offshore (Gulf of Mexico) territorial waters of Texas, Louisiana, Mississippi, Alabama, and a portion of Northwestern Florida, as well as adjoining Federal waters landward of the 50 fm depth contour within the U.S. Exclusive Economic Zone (EEZ) (Fig. 1). Brown shrimp produce annual crops (Neal and Maris, 1985), with recruitment to the fishery occurring in May-July (Rothschild and Brunenmeister, 1984). Although life span is 20-27 mo (Baxter, 1971), most brown shrimp are harvested within 6 mo of age. (3) Neal (1967) conducted virtual population analyses of brown shrimp in statistical subareas 18 and 19 (Fig. 1), and found that estimated numbers of brown shrimp in reported landings during biological year 1964 represented 97.7% of the total virtual population over a 17-mo period. This finding indicated that only 2.3% (by number) of the shrimp recruited as new cohorts in biological year 1964 contributed to the landings in biological year 1965. If shrimp landed in a given biological year within our time series (1986-2006) included survivors from cohorts recruited in preceding biological years, this could have affected our biological year simulations of w and other variables derived from them. However, such a carryover would be small, because it would involve only the larger sizes of shrimp which are lowest in pounds and fewest in numbers within the landings.

[FIGURE 1 OMITTED]

Landings Data

Brown shrimp landings data are archived by the National Marine Fisheries Service (NMFS) Galveston Laboratory, Texas. Statistically, reported landings are fishery-dependent samples taken without replacement from the brown shrimp population. They are multitudinous but have limitations (Kutkuhn, 1962; Snow, 1969; Prytherch, 1980; Parrack (1); Nichols (2); Poffenberger (4)) which may bias not only our simulated distributions of w and additional variables derived from them, but also may have biased previous estimates of numbers of shrimp from pounds landed within count categories. Not all brown shrimp that are caught are landed, and not all that are landed are reported (Kutkuhn, 1962; Berry and Benton, 1969; Baxter, 1973; Snow, 1969; Prytherch, 1980; Nance et al., 1991; Caillouet et al., 2008; Poffenberger (4)). Nonreported catch includes shrimp marketed directly to consumers, marketed as fishing bait (not all, but some), discarded for various reasons, kept for personal use by shrimpers, or otherwise not reported. Thus, reported landings are less than the actual catches, and also represent incomplete samples of the actual landings (Caillouet et al., 2008).

Reported shrimp landings data are recorded by calendar year, month, statistical subarea (Fig. 1), depth zone, shrimping trip, and count category or unknown size category, along with other information (Kutkuhn, 1962; Snow, 1969; Prytherch, 1980; Poffenberger4). We treated the unknown size category as a catch-all category. In selecting records for a working file of size-graded landings data for our simulations, we excluded all landings originally reported in the unknown size category, as well as landings added to the unknown size category after we judged their count categories to be outliers (see Data Selection and Preparation below). The resultant unknown size category contained landings that were:

1) not size-graded,

2) size-graded incorrectly or size limits not recorded,

3) not assigned to a count category for other reasons (e.g. pieces of shrimp tails), or

4) size-graded but reported in count categories we judged to be outliers.

Previous investigators (e.g. Rothschild and Brunenmeister, 1984; Parrack (1); Nichols (2)) also excluded certain landings from their analyses for various reasons. Two methods of grading shrimp, box-grading and machine grading, were described by Kutkuhn (1962), Snow (1969), Prytherch (1980), and Poffenberger (4). Differences between these grading methods and variations in their relative contributions to size-graded landings over time may have biased our simulated distributions of w and variables derived from them, but they may also have biased previous estimates of numbers of shrimp within count categories.

Data Selection and Preparation

Our final working file contained archived landings records selected from biological years 1986-2006, but only those we considered to have legitimate count class limits. We initially consulted NMFS port agents (who collect landings data) to obtain their opinions about the true range in size of brown shrimp tails in the landings. It was agreed that the maximum C (smallest shrimp) for brown shrimp in the landings was around 250 tails per pound (equivalent to w = 0.004 lb, or 1.8 g), and minimum C (largest shrimp) around 9 tails per pound (equivalent to w [approximately equal to] 0.111 lb, or [approximately equal to] 50.3 g).

Preparation of the working file involved filtering and editing a copy of archived data from biological years 1986-2006 as follows:

1) If a record was originally coded as belonging to the unknown category, it was excluded.

2) If an upper or lower limit of a count category was not recorded (i.e. left blank), the record was excluded.

3) If a recorded lower limit exceeded the recorded upper limit of a count category, the limits were assumed to have been inadvertently transposed at data entry, and the record was retained in the working file after being recoded by interchanging its count category limits.

4) If recorded upper and lower limits of a count category were both C = 0, the record was excluded.

5) If the recorded upper limit of a count category was 0 < C < 9, both the lower and upper limits were recoded as C = 9, and the record was retained in the working file.

6) If only the recorded lower limit of a count category fell within C < 9, but the recorded upper limit was [greater than or equal to] 9, the lower limit was recoded as C = 9, and the record was retained in the working file.

7) If the recorded lower and upper limits of a count category were C > 250, the record was excluded.

8) If the recorded upper limit of a count category was C > 250, but the recorded lower limit was C [less than or equal to] 250, the upper limit was recoded as C = 250, and the record was retained in the working file.

9) All other archived records were retained in the working file.

We then performed statistical analyses of the working file to identify and remove records having count class limits we judged to be outliers. For each biological year, we used SYSTAT (5) to fit preliminary weighted linear regressions of upper limits on lower limits of the count categories, where the weighting factor was the number of observations (i.e. shrimping trips) associated with each unique count category (i.e. unique combination of upper and lower limits). Figure 2A is an example of a preliminary regression and data plot for biological year 2006. Statistical weighting by number of shrimping trips was our way of dealing with variability in frequency of occurrence of count categories in the working file. Records removed from the working file by filtering, editing, and identification of residual outlier count categories represented a higher percentage of observations than percentage of pounds landed (Table 2); i.e. they contained relatively low pounds per observation.

[FIGURE 2 OMITTED]

We fitted final weighted linear regressions of upper limits on lower limits within the final working file for each biological year (Table 3). The weighting factor for these regressions was the number of observations (i.e. shrimping trips) associated with each unique count category remaining in the final working file. These final regressions characterized the relationship between legitimate count category upper and lower limits for each biological year. Figure 2B is an example final regression and data plot for biological year 2006. Slopes and intercepts of the final linear regressions (Table 3) for each biological year were examined for trends, using polynomial regression. Coded biological year ([T.sub.i] --1996) was substituted for [T.sub.i] in these polynomial regressions, to avoid problems that otherwise might have been caused by correlations among powers of [T.sub.i] (Sokal and Rohlf, 2000).

Aggregation and Cumulation of Landings

Landings from the final working file were aggregated (summed) by biological year and count category lower limits, [C.sub.ij], where j is the place marker for the [C.sub.ij] within a biological year; j = 0, ..., [m.sub.i], where [m.sub.i] is the total number of [C.sub.ij] in each biological year (Table 1). Upper limits of count categories (equivalent to lower limits of class intervals of w) were ignored. Because the number of unique count categories in the final working file varied among biological years, the number of [C.sub.ij] also varied among biological years, as did [m.sub.i]. Summing the landings by biological year and [C.sub.ij] produced a subset of data with much lower spatial-temporal resolution than that of more detailed data sets used in previous, bottom-up approaches to estimating numbers of shrimp within count categories (Neal, 1967; Rothschild and Brunenmeister, 1984; Nance et al., 1994; Diop et al., 2007; Parrack (1); Nichols (2)).

Biological year summations of landings combined all spatial-temporal influences (statistical subarea, depth zone, and month) on size of brown shrimp in the landings. These influences included sex ratio, recruitment, growth, mortality, fishing effort, fishing power of shrimp trawlers, experience of captains and crews, gear selectivity, discarding, data collection procedures, grading methods, and possibly other factors that affect count category landings within a biological year. Spatial influences were collapsed to the level of the entire fishery, and temporal influences to the level of biological years. Summation of landings by [C.sub.ij] combined landings within count categories having [C.sub.ij] as their lower limit. The simple hypothetical example below depicts this process:

The sum of observations over all count categories having [C.sub.ij] as their lower limit became the weighting factor, [q.sub.ij], for each [C.sub.ij] and the sum of pounds associated with it. In the hypothetical example above, [C.sub.ij] = 9, [q.sub.ij] = 6, and both are associated with 1,740 lb landed. Examples of variation in [C.sub.ij] and [q.sub.ij] for biological years 1986, 1996, and 2006 are shown in Figure 3. Dominant [C.sub.ij] were conspicuous as indicated by their [q.sub.ij], and many were identical or close to the [C.sub.ij] of standard count categories, as expected.

Within each biological year, the pounds associated with [C.sub.ij] were cumulated over the observed range of [C.sub.ij], from the highest to the lowest [C.sub.ij] (i.e. from the smallest to largest shrimp tails). These cumulative pounds were then converted to proportions of cumulative pounds landed, [P.sub.ij] (Table 1), from the highest to the lowest [C.sub.ij]. Figure 4A is an example of the stair-stepped relationship between [P.sub.ij] and [C.sub.ij] for biological year 2006, and Figure 4B is the equivalent stair-stepped relationship between [P.sub.ij] and [w.sub.ij], where [w.sub.ij] = 1/[C.sub.ij].

Modified Richards Function

We searched for an asymptotic, asymmetrical sigmoid regression model to convert the stair-stepped relationship between [P.sub.ij] and [w.sub.ij] to a smooth curve for each biological year. The regression model we chose was a simplified form of the Richards function (Richards, 1959):

[FIGURE 3 OMITTED]

P = Pmax [(l - [e.sup.a-bw]).sup.c] (1)

where, P is the cumulative proportion of pounds landed at w,

w is shrimp tail weight in pounds, over the observed range from minimum to maximum w,

Pmax is the upper asymptote,

a is the parameter which allows w at which P = Pmax/2 to vary,

b is the parameter which represents the maximum intrinsic rate of increase in P per unit w, which occurs at the inflection point on the curve,

c is the parameter that allows the sigmoid shape of the curve to vary (symmetrical or asymmetrical), and

e is the base of natural logarithms.

Because we constrained Pmax to equal 1 in fitting all the regressions, Eq. (1) was simplified into the following regression model:

P = [(l - [e.sup.a-bw]).sup.c]. (2)

For each biological year, we used GraphPad Prism (version 5.02) to fit Eq. (2) to [P.sub.ij] on [w.sub.ij] by weighted nonlinear regression, where the weighting factor was [q.sub.ij]. In this way, parameters [a.sub.i], [b.sub.i], and [c.sub.i] (Table 1) were estimated for each biological year (Table 4). The lower case parameter [c.sub.i] should not be confused with the upper case count [C.sub.ij]. We tried fitting a number of other asymmetrical sigmoid functions available in GraphPad Prism, but Eq. (2) was the best fitting of those we examined. While we recognize that additional curve fitting methods and models could have been tested, Eq. (2) was adequate for purposes of demonstrating our simulation approach. By fitting Eq. (2), we smoothed the relationship between [P.sub.ij] on [w.sub.ij], and obtained an equation representing this relationship for each biological year (Table 4). We also calculated the adjusted [r.sup.2] as an approximation of how well Eq. (2) fit the data points for each biological year (Table 4), but recognize it is not strictly applicable to nonlinear regression.

[FIGURE 4 OMITTED]

Simulating Biological Year Distribution of Tail Weight

The next step toward simulating the distribution of w was to generate a new set of data pairs for each biological year, using the fitted equations in Table 4. First, we generated equally spaced values of [w'.sub.k] (Table 1), from a minimum, [w'.sub.0] (= 0.005155 lb), to a maximum, [w'.sub.999] (=0.111111 lb), where the kth place marker for the [w'.sub.k] was k = 0, ..., 999 (Table 1). The increment, g, between the [w'.sub.k] was then calculated as

g = ([w'.sub.999] - [w'.sub.0])/999

= 0.000106 lb.

The [w'.sub.k] were generated by

[w'.sub.k] = k(g) + [w'.sub.0].

We then generated values of [P'.sub.ik] for each [w'.sub.k] for each biological year, using the following equation and estimates of parameters [a.sub.i], [b.sub.i], and [c.sub.i] from Table 4:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Three reasons for applying [w'.sub.0] = 0.005155 lb (derived from 1/194) as the minimum shrimp tail weight for all biological year simulations were:

1) The lowest maximum [C.sub.ij] observed (in the working file) among all biological years was 194 count, the reciprocal of the highest minimum [w.sub.ij].

2) Imaginary numbers were generated by Eq. (3) for the minimum [P'.sub.ik] in some biological years when the actual minimum [w.sub.ij] observed in those years was applied (this probably was due in part to the fact that Eq. (3) did not fit the data points representing very small shrimp tails closely in those years).

3) It was consistent to constrain [w'.sub.k] to be the same for all biological years.

The first derivative of Eq. (3), [delta][P'.sub.ik]/[[delta]'.sub.wk], was

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

For each biological year, we used Eq. (4) to generate first derivatives for each [w'.sub.k]. To transform these first derivatives (Eq. (4)) into [p'.sub.ik] (Table 1), which was the proportion of pounds landed at [w'.sub.k] for each biological year, we divided them by the sum of all first derivatives over the range in [w'.sub.k], for each biological year. This sum was calculated as

[999.summation over (k=0)]([delta][P'.sub.ik]/[delta][w'.sub.k]).

In other words, for each biological year, [p'.sub.ik] at each [w'.sub.k] was calculated as

[P'.sub.ik] = ([delta][P'.sub.ik]/[delta][w'.sub.k])/[999.summation over (k=0)]([delta][P'.sub.ik]/[delta][w'.sub.k]).

Biological year yield, [Y.sub.i], encompassed all landings within a biological year, including those retained in our final working file as well as those that had been excluded from it. For each biological year, number of shrimp tails, [f'.sub.ik] (Table 1), at each [w'.sub.k] was calculated by

[f'.sub.ik] = [Y.sub.i] ([p'.sub.ik])/[w'.sub.k]. (5)

Equation 5, describing the relationship between [f'.sub.ik] and [w'.sub.k], is the simulated distribution of w for the ith biological year.

We would have been able to exclude some steps in our simulation sequence had the final working file represented total reported landings from each biological year (i.e. [Y.sub.i]). However, the final working file was a subset of size-graded landings selected from the archived landings, and it did not contain landings we excluded (i.e. those relegated to the unknown category), whereas [Y.sub.i] contained all landings for each biological year. Therefore, Eq. (5) applied the subset of proportions [p'.sub.ik] to the total yield Yi to estimate [f'.sub.ik] for each biological year.

We recognize that relative distributions of w for each biological year, and their corresponding cumulative relative distributions, also could have been derived from our simulated distributions of w. They might be of interest in some applications of our approach, but they were not essential to the purpose of our paper. They can easily be calculated from the information provided in this paper. However, the concept of cumulative relative distribution of w in biological year landings of brown shrimp is important in that it would estimate the probability of occurrence of tail weight [less than or equal to] w; i.e. it would be an approximation of the cumulative distribution function (CDF) for w. This is the major part of the explanation of why we chose lower limits, [C.sub.ij] (equivalent to upper limits of [w.sub.ij]), for aggregating and cumulating landings, and then transformed [C.sub.ij] to [w.sub.ij] in preparation for fitting Eq. (2). Because a simulated distribution of w can be used to calculate the relative distribution of w and cumulative relative distribution of w, it is relevant to future testing of past algorithms and development of new ones to estimate numbers of shrimp from pounds landed within class intervals of w or C in the landings. Although we excluded certain landings (unknown size category) and ignored upper limits of legitimate count categories in simulating biological year [p'.sub.ik], our simulations of [f'.sub.ik] included all biological year landings ([Y.sub.i]); i.e. all biological year landings contributed to simulation of biological year distributions of w.

Biological Year Total Number of Shrimp Tails ([N.sub.i]), Mean [C.sub.i], and Mean [w.sub.i]

The total number of shrimp tails, [N.sub.i], in the landings from a biological year [T.sub.i] was simulated by

[N.sub.i] = [999.summation over (k=0)][f'.sub.ik]. (6)

Crude estimates of biological year mean count ([N.sub.i]/[Y.sub.i]) and its equivalent mean tail weight ([Y.sub.i]/[N.sub.i]) were calculated. We examined trends in both of these means via polynomial regression, where coded years ([T.sub.i] - 1996) were substituted for [T.sub.i].

Tail Weight at Half of [Y.sub.i]

Given that a fitted equation representing the relationship between [P.sub.ij] and [w.sub.ij] was available for each biological year (Table 4), we estimated tail weight, [w50.sub.i], at which half of the annual yield, [Y.sub.i]/2, was harvested in each biological year (note that when Pmax is constrained to equal 1, [w50.sub.i] = Pmax/2 = 0.5). Each equation (Table 4) was solved for [w50.sub.i] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This statistic is similar in concept to LD50, the estimated lethal dose (concentration) of a toxic substance at which 50% mortality occurs in exposed subjects. In our application, it is a potentially useful index of the relationship between brown shrimp size and yield (see Caillouet et al., 2008). We examined w50i via polynomial regression, where coded years ([T.sub.i] - 1996) were substituted for [T.sub.i].

Results

Polynomial Regressions

We recognize that polynomial regression is an empirical approach to fitting a curve to a time series of data, and that the resulting polynomial terms have no structural meaning (Sokal and Rohlf, 2000). We applied it only to detect possible trends in the variables we simulated, and to demonstrate possible applications of our simulated distributions of w. Obviously, many other curve fitting approaches could have been used to examine the time series for each variable. Causes and effects within this brown shrimp fishery could have influenced the detected polynomial trends, despite variability (deviations from regression) caused by fluctuations in annual recruitment and other factors which are typical in shrimp populations (Caillouet et al., 2008).

Weighted Linear Regressions of Upper vs. Lower Limits of Count Categories

Data plots and preliminary weighted linear regressions of upper on lower limits of unique count categories in each biological year (see the example for the year 2006 in Fig. 2A) showed that count category outliers remained in the data after filtering and editing. In the year 2006 example, the outliers were concentrated near the minimum lower limit count of 9 (largest shrimp), which elevated the intercept of the fitted line in Figure 2A as compared to the intercept of the fitted line in Figure 2B, in which residual outliers had been removed. The unusually wide class intervals of outlier count categories could lead to serious biases in estimating numbers of shrimp within such count categories. Also, we emphasize that each data pair (upper and lower limits) was weighted, so the actual numbers of residual outliers are much higher than the number of data points representing outliers in Figure 2A (Table 2).

As expected, final weighted linear regressions of upper limits on lower limits of count categories were close fitting in all biological years as shown by high adjusted [r.sup.2] (Table 3, Fig. 2B). These final regressions characterized the relationship between upper and lower limits of what we considered to be legitimate count categories in each biological year. All slopes of these final regressions were slightly greater than 1 (Table 3), indicating that count class intervals in the working file widened as their lower limits increased. Trends in slopes and intercepts of these regressions are shown in Figures 5A and 5B, respectively.

Biological Year [m.sub.i] and Weighted Regressions of [P.sub.ij] on [w.sub.ij]

The biological year total number, [m.sub.i], of [C.sub.ij] exhibited a concave quadratic (parabolic) trend (Fig. 6); [m.sub.i] dropped from 77 in 1986 to 35 in 1995, then increased but not to its earlier highest level. This trend in [m.sub.i] reflected changes in the total number of legitimate count categories over the biological years. However, the total number of count categories in biological year [T.sub.i] exceeds [m.sub.i], because upper limits of count categories were ignored in our simulations; i.e. landings at the count category level were combined at the count category lower limit level, [C.sub.ij] (see Aggregation and Cumulation of Landings). Wide variation in biological year numbers of count categories and the consequential quadratic trend in mi (Fig. 6) are interesting and worthy of further investigation. They could reflect changes in size-related marketing strategies, recruitment, and perhaps other influences on choices of count categories in the landings.

[FIGURE 5 OMITTED]

Weighted nonlinear regressions of [P.sub.ij] on [w.sub.ij] for all biological years were close fitting, as indicated by very high adjusted [r.sup.2.sub.i] (Table 4). Over all biological years, adjusted [r.sup.2.sub.i] equaled or exceeded 0.981. Examples of plotted data points [P.sub.ij] vs. [w.sub.ij] and fitted curves for 1986, 1996, and 2006 are shown in Figures 7A-C, respectively. Inflection points of the regressions were far to the lower left in such plots (Fig. 7A-C), suggesting that brown shrimp were fully recruited to the landings at very small sizes, which is a very important finding.

[FIGURE 6 OMITTED]

The total sample size,

[[m.sub.i].summation over (j=0)][q.sub.ij]

(Fig. 8), for each biological year regression (Table 4), and the adjusted [r.sub.i.sup.2] (Fig. 9) for these regressions, declined over biological years. In other words, adjusted [r.sub.i.sup.2] and total sample size were dependent, as expected (Fig. 10); i.e. the larger the sample size the higher the adjusted [r.sub.i.sup.2]. We emphasize that the total sample size (Fig. 8) used in fitting the regressions of [P.sub.ij] on [w.sub.ij] for each biological year was less than the actual number of shrimping trips in the archived data for each biological year, because landings from some trips were initially in the unknown category or later placed there by filtering, editing, and outlier removal from the working file. Therefore, the data points and trend in Figure 8 should not be taken to represent total shrimping trips in the biological years.

As is common in fitting models containing more than one parameter, the parameter estimates often are not independent (i.e. orthogonal). Graph-Pad Prism provided estimates of dependency of estimated parameters [a.sub.i], [b.sub.i], and [c.sub.i] within each biological year regression (dependency = 1 represents complete dependency, and dependency = 0 indicates orthogonality). Over biological years, dependency was 0.865-0.985 for parameter [a.sub.i], 0.968-0.981 for parameter [b.sub.i], and 0.978-0.994 for parameter [c.sub.i]. Not only did all these parameters show strong dependency within each biological year regression, but they also appeared related to each other over biological years (Fig. 11A-C).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Simulated Distributions of w

Example distributions of [w'.sub.k] for biological years 1986, 1996, and 2006 are shown in Figures 12A-C. All were strongly skewed to the right. Their most striking feature was their likeness to negative exponential curves. Therefore, we plotted them in the form of ln([f'.sub.ik]) vs. [w'.sub.k] for all biological years (Fig. 13). Straight lines for ln([f'.sub.ik]) vs. [w'.sub.k] would have indicated that these simulated distributions of w followed a negative exponential pattern, once full recruitment to the landings was reached at very small sizes (Fig. 13). Only slight concavity was evident in all the curves.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Biological Year Total Number of Shrimp Tails and Yield

Interestingly, although the biological year total number of shrimp tails, [N.sub.i] (Fig. 14), and yield, [Y.sub.i] (Fig. 15), showed hints of declines, they exhibited no significant trends over biological years, because of wide year to year variation. A close linear relationship between [N.sub.i] and [Y.sub.i] (Fig. 16) was expected; i.e. the more pounds landed the greater the number of shrimp tails in the landings, and vice versa. However, biological year mean count, [N.sub.i]/[Y.sub.i] (Fig. 17A) was not constant, because simulated distributions of w and [Y.sub.i] were not constant over biological years (Fig. 12A-C, Fig. 13). Biological year mean tail weight ([Y.sub.i]/[N.sub.i]) also was not constant (Fig. 17B). We emphasize that [N.sub.i]/[Y.sub.i] and [Y.sub.i]/[N.sub.i] are crude estimates of mean count and mean tail weight, respectively, and do not represent biological year central tendency of C and w in the landings very well. Trends in [N.sub.i]/ [Y.sub.i] (Fig. 17A) and [Y.sub.i]/[N.sub.i] (Fig. 17B) were cubic (sigmoid), mirroring each other as expected.

[FIGURE 11 OMITTED]

Tail Weight at Which Half of the Biological Year Yield was Harvested

The cubic trend in [w50.sub.i] is shown in Figure 18. As expected, it is similar in shape to that of [Y.sub.i]/[N.sub.i] (Fig. 17B). However, the two trends (Fig. 17B, Fig. 18) were not parallel, because the slope of the regression of [w50.sub.i] on [Y.sub.i]/[N.sub.i] did not equal 1 (Fig. 19). Instead, [w50.sub.i] was 1.459 times [Y.sub.i]/[N.sub.i]. Although significantly different from zero, the intercept of the regression of [w50.sub.i] on [Y.sub.i]/[N.sub.i] was very small (i.e. near the origin).

Discussion

It is clear that brown shrimp landings data should be filtered, edited, and residual records representing outlier count categories removed before distributions of shrimp tail weight are simulated. The same should be (and in most cases have been) done before numbers of shrimp are estimated from landings within count categories, regardless of the algorithm used to estimate numbers of shrimp within count categories, unless the algorithms are based on actual sampling of size distributions within count categories (Ehrhardt and Legault, 1996). The problem of unreported landings and other limitations of reported landings data affect not only our simulations, but all other uses of reported landings to estimate numbers of shrimp within count categories. These data problems cannot be rectified retroactively, but should be addressed in the future.

Our simulated biological year distributions of brown shrimp tail weight could be biased to unknown degrees by many factors. This is true of all estimates of numbers of shrimp derived from landings within count categories, whether at the highest possible level of data resolution (i.e. an individual shrimping trip within a statistical subarea, depth zone, and month), or at lower levels of data resolution represented by various spatial-temporal aggregations of landings data, including ours. Our simulated distributions of shrimp tail weight should not be taken as equivalent to distributions of brown shrimp tail weight in the population of the northern Gulf of Mexico. However, our simulated distributions of w in biological year landings no doubt have some yet undetermined relationship to actual distributions of shrimp tail weight in the brown shrimp population in biological years. This relationship cannot be determined retroactively due to lack of or paucity of required data. Unreported landings are much less than reported landings, but our simulated distributions of shrimp tail weight only represent landings that were reported and archived.

Despite landings data deficiencies, our simulated distributions of w, and other fishery-dependent statistics derived from them, can be useful in examining changes in the brown shrimp fishery over biological years. Their relationships to other important fishery-dependent and fishery-independent variables could be examined in attempts to explain causes and effects.

Our method could be applicable to fisheries of other penaeid shrimp species for which landings are recorded within size categories expressed in C or w. It might also be applicable to finfish fisheries in which landings are reported within size categories expressed in number of fish per unit weight or in weight per fish. The method may also be applicable to shrimp landings aggregated at spatial-temporal levels lower (i.e. higher resolution) than that of an entire fishery and biological year.

Our results suggest that brown shrimp were fully recruited to the fishery at small sizes in each biological year, then declined in number with w in a pattern similar but not identical to that of a negative exponential curve. In a study of distributions of growth rates of shrimp in captivity, Banks et al. (2009) examined effects of bin width, sample size, and sampling frequency on distributions of weight per shrimp. Interestingly, the shapes of their distributions of weight per shrimp were similar to those of our simulated distributions of w. Although we did not simulate relative distributions of w or corresponding cumulative relative distributions of w, we noted that they could be simulated from our approach, and they too might be of interest and use in shrimp stock assessments.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

Simulated biological year distributions of w could be used to estimate numbers of parents and recruits, for purposes of determining parent-recruit relationships (Rothschild and Brunenmeister, 1984; Gracia, 1991; Ehrhardt and Legault, 1996; Parrack (1); Nichols (2)). Numbers of parents or recruits could be extracted from curves representing distributions of w by integrating them over the size ranges of parents and recruits. However, estimates or assumptions about size at maturity and growth patterns of males and females would be required, as well as estimates of size-specific sex ratios in the landings (Gracia, 1991; Ehrhardt and Legault, 1996 Parrack (1), Nichols (2)).

[FIGURE 17 OMITTED]

It may be possible to estimate instantaneous total mortality rate (Z) from simulated biological year distributions of w by transforming them to bounded length distributions and applying length-based models similar to those of Ehrhardt and Ault (1992) (Ehrhardt (6)). Alternatively, the length-based models used by Ehrhardt and Ault (1992) might be reformulated for direct application to biological year distributions of w for purposes of estimating Z (Ehrhardt (6)). Biological year distributions of w could also be transformed to age-frequencies for age-structured stock assessments. This would require conversion of tail weight to age using sex-specific growth curves and knowledge of size-specific sex ratios in the landings (Parrack, 1979; Rothschild and Brunenmeister, 1984; Gracia, 1991; Ehrhardt and Legault, 1996; Parrack (1); Nichols (2)).

[FIGURE 18 OMITTED]

[FIGURE 19 OMITTED]

Simulated distributions of w of brown shrimp in biological year reported landings are linked, by definition and calculations, to biological year yield. Fishing effort influences size-composition of the landings and therefore influences yield, although environmental variables affecting recruitment also affect yield (Caillouet et al., 2008; Nance et al., 2010). Numbers of shrimp estimated from landings within count categories have been used in evaluating the influence of environmental factors on abundance, growth, and survival (Diop et al., 2007).

Our method provides an alternate way to estimate abundance of shrimp in reported annual landings, as compared to algorithms used by previous investigators. However, the relationship between abundance of shrimp in the landings and in the population remains undetermined. Our simulated distributions of w provide examples for comparison with explicit or implicit assumptions made by previous investigators about the distributions of C and w. They also provide information of potential use in developing new estimators of number of shrimp from landings data, based on statistical estimation theory and the underlying distribution of w or C. Finally, there may be other useful applications of our approach and results that we have not realized or anticipated.

Acknowledgments

Special recognition goes to Charles H. Lyles, Jr., for his pioneering development of the system used to collect and report shrimp fishery statistics, and to all shrimp industry participants whose cooperation made it possible over the years. We are especially grateful to those who perpetuated and improved this system, and to those who collected, processed, and archived shrimp fishery statistics, making them available for analyses such as ours. Joseph H. Kutkuhn's comprehensive statistical examination and evaluation provided an early and important understanding of the usefulness and limitations of landings data in shrimp stock assessment. Succeeding investigators expanded and improved this understanding, for which we are grateful. We greatly appreciated reviews of our manuscript by Nelson M. Ehrhardt, and three anonymous reviewers. We thank Jo Anne Williams for assistance in drafting the figures; James Primrose and John Cole for assistance in landings data compilation; and Brian Linton, James A. Bailey, and Stephen A. Bailey for assistance with derivatives of the modified Richards function, and in solving this function for w50. This paper is dedicated to the memory of the senior author's parents, Charles W. Caillouet, Sr. (1908-1971) and Elida P. Millet Caillouet (1906-2004).

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--, R. A. Hart, and J. M. Nance. 2008. Growth overfishing in the brown shrimp fishery of Texas, Louisiana, and adjoining Gulf of Mexico EEZ. Fish. Res. 92:289-302.

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--and J. S. Ault. 1992. Analysis of two length-based mortality models applied to bounded catch length frequencies. Trans. Am. Fish. Soc. 121:115-122.

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--, E. X. Martinez, and E. F. Klima. 1994. Feasibility of improving the economic return from the Gulf of Mexico brown shrimp fishery. N. Am. J. Fish. Manage. 14:522-536.

--, C. W. Caillouet, Jr., and R. A. Hart. 2010. Size-composition of annual landings in the white shrimp fishery of the northern Gulf of Mexico, 1960-2006: its trend and relationships with other fishery-dependent variables. Mar. Fish. Rev. 72(2):1-13.

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--and R. C. Maris. 1985. Fisheries biology of shrimps and shrimplike animals. Chapter 1: Fisheries biology of shrimps and shrimplike animals. In A. J. Provenzano, Jr. (Editor), Economic aspects: fisheries and culture, vol. 10, The biology of crustacea, p. 1-110. Acad. Press, Inc., N.Y.

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(1) Parrack, M. L. 1981. Some aspects of brown shrimp exploitation in the northern Gulf of Mexico. Presented at the Workshop on the Scientific Basis for the Management of Penaeid Shrimp, Key West, Fla., Southeast Fisheries Science Center, National Marine Fisheries Service, NOAA, Miami, Fla. Unpubl. rep., 50 p.

(2) Nichols, S. 1984. Updated assessments of brown, white and pink shrimp in the U.S. Gulf of Mexico. Presented at the Workshop on Stock Assessment, Miami, Fla., Southeast Fisheries Science Center, National Marine Fisheries Service, NOAA, Miami, Fla. Upubl. rep., 54 p.

(3) Fishery Management Plan for the Shrimp Fishery of the Gulf of Mexico, United States Waters. Gulf of Mexico Fishery Management Council, Tampa, Fla., Nov. 1981 (online at http://www. gulfcouncil.org).

(4) Poffenberger, J. R. 1991. An overview of the data collection procedures for the shrimp fisheries in the Gulf of Mexico, National Marine Fisheries Service, Southeast Fisheries Science Center, Miami, Fla. (online at http://www.sefsc. noaa.gov/gssprogram.jsp).

(5) Mention of trade names or commercial firms does not imply endorsement by the National Marine Fisheries Service, NOAA.

(6) Ehrhardt, N. M. Rosentiel School of Marine and Atmospheric Science, Miami, FL. Personal commun., August 2010.

Charles W. Caillouet, Jr., conservation volunteer, retired in 1998 from the Galveston Laboratory, National Marine Fisheries Service, NOAA, 4700 Avenue U, Galveston, TX 77551. Rick A. Hart and James M. Nance are with the Galveston Laboratory, National Marine Fisheries Service, NOAA, 4700 Avenue U, Galveston, TX 77551 (corresponding author: rick.hart@noaa.gov).
Table 1.--Symbols and descriptions of variables used in analyses
of biological year reported landings of brown shrimp from the
northern Gulf of Mexico fishery. These apply only to size-graded
landings in legitimate count categories; i.e. data selected by
filtering, editing, and removing residual outliers from archived
landings data.

Symbols               Descriptions of variables

[T.sub.i]             biological year, from 1 May of a given calendar
                      year through 30 April of the next calendar year,
                      where i = 0, ..., 20 is the place marker for
                      biological years 1986-2006

[C.sub.ij]            the jth lower limit of a legitimate count
                      (number per pound) category in landings data
                      from the ith biological year, where y = 0, ...,
                      [m.sub.j]

[m.sub.i]             the total number of [C.sub.ij] in landings data
                      from the ith biological year

[w.sub.ij]            the jth upper limit of a pounds per shrimp tail
                      category, where [w.sub.ij] = 1/[C.sub.ij], in
                      landings data from the ith biological year

[P.sub.ij]            the jth cumulative proportion of pounds landed
                      at [w.sub.ij] in ith biological year

[q.sub.ij]            the jth weighting factor for the [P.sub.ij] and
                      [w.sub.ij] data pairs in the ith biological
                      year. This weighting factor, [q.sub.ij] is the
                      sum of observations over all count categories
                      having [C.sub.ij] as their lower limit (or
                      [w.sub.ij] as their upper limit), regardless of
                      the recorded upper limits of these count
                      categories

[w'.sub.k]            the kth simulated value of weight per shrimp
                      tail, where 0.005155 lb [less than or equal to]
                      [w'.sub.k] [less than or equal to] 0.111111 lb,
                      k = 0, ..., 999, and the interval between the
                      [w'.sub.k] is 0.000106

[P'.sub.ik]           the kth cumulative proportion of pounds landed
                      at [w'.sub.k] in the ith biological year, which
                      is simulated from the modified Richards function
                      fitted to [P.sub.ij] on [w.sub.ij] in the ith
                      biological year

[a.sub.i]             the parameter, estimated from the modified
                      Richard's function fitted to [P.sub.ij] on
                      [w.sub.ij] in the ith biological year, which
                      allows the [w'.sub.k] at which [P'.sub.ik] =
                      [Pmax.sub.i]/2 to vary among biological years

[b.sub.i]             the parameter, estimated from the modified
                      Richard's function fitted to [P.sub.ij] on
                      [w.sub.ij] in the ith biological year, which
                      represents the maximum intrinsic rate of
                      increase in [P'.sub.ik] per unit [w'.sub.k] at
                      the inflection point of the curve

[c.sub.i]             the parameter, estimated from the modified
                      Richard's function fitted to [P.sub.ij] on
                      [w.sub.ij] in the ith biological year, which
                      allows the sigmoid shape of the curve to vary
                      (symmetrical or asymmetrical) among biological
                      years

[p'.sub.ik]           the kth simulated proportion of pounds landed at
                      [w'.sub.k] in the ith biological year

[Y.sub.i]             the ith biological year yield, which includes
                      pounds of brown shrimp tails landed in
                      legitimate count categories and in the unknown
                      size category combined

[f'.sub.k]            the kth simulated number of shrimp tails at
                      [w'.sub.k], where 0.005155 lb [less than or
                      equal to] [w'.sub.k] [less than or equal to]
                      0.111111 lb, in the ith biological year

[N.sub.i]             the simulated total number of shrimp tails
                      landed in the ith biological year

[w50.sub.i]           the simulated pounds per shrimp tail at which
                      half of [Y.sub.i] is harvested in the ith
                      biological year

[N.sub.i]/[Y.sub.i]   the simulated mean count of brown shrimp in the
                      landings from the ith biological year

[Y.sub.i]/[N.sub.i]   the simulated mean pounds per shrimp tail of
                      brown shrimp in the landings from the ith
                      biological year

Table 2.--Number of observations (shrimping trips)
and pounds landed in the NMFS-archived records,
compared to those remaining after filtering, editing,
and removal of residual outlier count categories, for
brown shrimp landings in the northern Gulf of Mexico
fishery in biological years 1986-2006.

                           Observations
Records                  (shrimping trips)   Pounds (tails)

Archived                     2,425,373       1,682,806,769
                              100.0%             100.0%
After filtering and          2,319,554       1,668,305,100
  editing                     95.64%             99.14%
After residual outlier       2,308,674       1,664,449,467
  removal                     95.19%             98.91%

Count                     Pounds
Category   Observations   landed

9-12            2           500
9-15            3         1,200
9-20            1            40
Total           6         1,740

Table 3.--Final weighted linear regressions of upper
(U on lower (L) limits of count categories in brown
shrimp landings data selected by filtering, editing, and
removal of residual outliers from the NMFS-archived
landings data. The weighting factor was the number
of shrimping trips associated with each unique count
category (i.e. unique U and L data pair) in the landings
data selected from each biological year. Sample size
was the sum of these weighting factors for each bio-
logical year (see Fig. 2).

Biological
year,        [Intercept                     Sample        Adjusted
[T.sub.i]      .sub.i]     [Slope.sub.i]     size      [r.sub.i.sup.2]

1986         -0.0609420      1.172878       141,523         0.988
1987         -0.7467180      1.189633       159,010         0.988
1988          0.4795405      1.160103       158,733         0.992
1989          0.0479237      1.171595       147,315         0.992
1990          0.6609263      1.157006       137,647         0.993
1991         -0.1564869      1.180132       122,065         0.992
1992          0.1879885      1.169404       117,633         0.991
1993         -0.5079871      1.187639       105,907         0.989
1994          0.0533226      1.173414       111,968         0.992
1995         -0.1264397      1.179191       102,643         0.993
1996         -0.7873293      1.195364        97,111         0.989
1997         -1.4422680      1.210573        98,415         0.987
1998         -1.0609130      1.199557        91,378         0.988
1999         -1.2453040      1.204808        92,638         0.985
2000         -0.3466840      1.179789        95,775         0.990
2001          0.1584804      1.169094        89,022         0.992
2002         -0.2696291      1.187891       122,160         0.992
2003         -1.0236740      1.206445       103,013         0.993
2004         -1.1202140      1.208740        82,006         0.993
2005         -0.3479283      1.192750        69,662         0.994
2006         -0.9036610      1.208740        63,050         0.995

Table 4.--Biological year yield ([Y.sub.i]), parameter estimates,
and other statistics for weighted nonlinear regressions (modified
Richards function, Eq. (2)) of cumulative proportions of pounds
landed, [P.sub.ij], on pounds per shrimp tail, [w.sub.ij], in
brown shrimp landings data selected by filtering, editing, and
removal of residual outliers from the NMFS-archived landings
data. The weighting factor was the number of shrimping trips,
[q.sub.ij], associated with each data pair, [P.sub.ij] and
[w.sub.ij], in the selected landings data. For a given biological
year, the number of data points analyzed (total sample size) was
the sum of these weighting factors, [[m.sub.i].summation over
(j=0)] [q.sub.ij].

Biological      Yield               Estimated parameters
year,         [Y.sub.i]
[T.sub.i]       pounds      [a.sub.i]   [b.sub.i]   [c.sub.i]

1986          94,738,424    0.2770842   53.75477    1.016937
1987          89,394,421    0.3177634   61.64658    1.074417
1988          79,859,436    0.2713897   57.15293    1.286208
1989          94,170,525    0.2802385   58.89570    1.243767
1990         105,121,282    0.2865627   55.59358    0.968642
1991          85,602,708    0.1627544   47.08183    1.095961
1992          68,425,417    0.2294646   55.76027    1.150789
1993          66,431,237    0.2427682   55.10823    0.865503
1994          67,049,354    0.2126820   51.46945    1.107677
1995          75,859,021    0.2123855   48.21137    0.829590
1996          73,500,416    0.2459783   55.83692    0.888528
1997          65,389,618    0.2837078   55.32308    0.761172
1998          80,514,861    0.2723718   61.82822    0.975136
1999          81,035,496    0.2308989   56.10879    0.836558
2000          94,463,851    0.3038908   59.25881    1.084649
2001          87,660,251    0.3287329   74.62214    1.352838
2002          73,180,653    0.3917993   81.88587    1.447248
2003          82,309,001    0.3194503   79.86258    1.376817
2004          74,233,767    0.2973424   57.98183    0.884165
2005          58,819,403    0.2349768   56.86499    1.169126
2006          85,047,627    0.0818478   51.68214    1.640370

             Total sample
                 size
Biological    [[m.sub.i]    Adjusted
year,         .summation    [r.sup.2
[T.sub.i]    over (j=0)]    .sub.i]

1986             141,523     0.994
1987             159,010     0.997
1988             158,733     0.997
1989             147,315     0.996
1990             137,647     0.992
1991             122,065     0.993
1992             117,633     0.995
1993             105,907     0.989
1994             111,968     0.996
1995             102,643     0.991
1996              97,111     0.991
1997              98,415     0.994
1998              91,378     0.992
1999              92,638     0.987
2000              95,775     0.995
2001              89,022     0.987
2002             122,160     0.988
2003             103,013     0.986
2004              82,006     0.981
2005              69,662     0.981
2006              63,050     0.991
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