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CONTENTS
Introduction
Population Dynamics Model
Statistical Estimation Approach
Southern Demersal Working Group Recommendations
Model Diagnostic
Model Results
Conclusions
Acknowledgments
Literature Cited
Appendix (PDF only)

Northeast Fisheries Science Center Reference Document 02-03

An Age-Structured Assessment Model for Georges Bank Winter Flounder

Jon K.T. Brodziak
National Marine Fisheries, NEFSC, 166 Water St., Woods Hole MA 02543

Web version posted March 27, 2002

Citation: Brodziak, J.K.T. 2002. An age-structured assessment model for Georges Bank winter flounder. Northeast Fish. Sci. Cent. Ref. Doc. 02-03; 54 p.

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Abstract

An age-structured assessment model for Georges Bank winter flounder (Pseudopleuronectes americanus) stock during 1964-2000 is developed to provide an alternative to VPA-based analyses of stock status. Age-structured population dynamics of winter flounder are modeled using standard forward-projection methods for statistical catch-at-age analyses.

Trends in the relative abundance of population biomass are measured by research survey indices for Georges Bank winter flounder. Three surveys were available: the NEFSC autumn groundfish survey (1963-2000), the NEFSC spring groundfish survey (1968-2000), and the Canadian spring groundfish survey (1987-2000). Two alternative models were examined in detail: (1) a model that used all three research survey time series (WINC, WINter flounder model including Canadian survey) and (2) a model that used the two NEFSC research survey time series (WIN, WINter flounder model). Both the WINC and WIN models provided similar trends in population biomass and fishing mortality, indicating that results were robust to the inclusion of the Canadian research survey time series. Based on model diagnostics, the WIN model that used the two NEFSC research survey time series provided the best fit to the data. Conditioned on the accuracy of the model and the assessment data, results of the best fit model indicate that: (i) Spawning biomass exceeded 20,000 mt in 1964 but declined to less than 3,000 mt in the early-1990s. Spawning biomass in year 2000 was roughly 9,900 mt; (ii) Fishing mortality (fully-recruited, age-4 estimate) increased steadily from less than 0.2 in the early-1960s to over 1.0 in the late-1980s and early-1990s, but has declined since then to roughly 0.32 in 2000; (iii) Stock-recruitment data show that the stock produced large year classes (>15 million recruits) in the 1960s and 1970s when spawning biomass was near or above 10,000 mt; (iv) Surplus production data show that the stock was most productive during the 1970s and early-1980s, with annual surplus production of roughly 3,000 mt. Since the mid-1980s annual surplus production has decreased to roughly 2,000 mt.


Introduction

An age-structured assessment model for Georges Bank winter flounder (Pseudopleuronectes americanus) stock during 1964-2000 is developed to provide an alternative to VPA-based analyses of stock status. Age-structured population dynamics of winter flounder are modeled using standard forward-projection methods for statistical catch-at-age analyses (Fournier and Archibald 1982, Methot 1990, Ianelli and Fournier 1998, Quinn and Deriso 1999). We describe the underlying population dynamics model, statistical estimation approach, Southern Demersal Working Group recommendations, model diagnostics, and model results below.


POPULATION DYNAMICS MODEL

The age-structured model is based on forward projection of population numbers at age. This modeling approach is based on the principle that population numbers through time are determined by recruitment and total mortality at age through time. That is, if one knew the time series of inputs and outputs to the population numbers and the initial population size at age, then one would have complete information on the population size, spawning biomass, and total mortality through time. In practice, one uses available sampling data and a statistical model of how the data were observed to estimate parameters to determine the time series of population sizes.

Population numbers at age through time are key variables in the age-structured model and the population numbers at age matrix N=(Ny,a)YxA contains this information. This matrix has dimensions Y by A, where Y is the number of years in the assessment time horizon and A is the number of age classes modeled. The oldest age (A) comprises a plus-group consisting of all fish age-A and older. The time horizon for winter flounder is 1964-2000 (Y=37). The choice of time horizon was determined by the availability of landings data which are first available in 1964 and a relative abundance index, the NEFSC autumn groundfish survey. The number of age classes in the model is 7, representing ages 1 through 7+.

Recruitment (numbers of age-1 fish) in year y (Ry) is modeled as a lognormal deviation from an average recruitment parameter (µR), where the Vy are independent and identically distributed (iid) normal random variables with zero mean and constant variance.

For all years, y, from 1965-2000, Ry = Ny1 is estimated from the recruitment deviation and average recruitment parameter. The recruitment deviations are constrained to sum to zero over all years.

Initial population abundance at age in 1964 is based on recruitment deviations from average recruitment for 1959-1964 and natural mortality. For all ages a < A, the numbers at age in the first year (ystart=1) are estimated as a lognormal deviation from average recruitment as reduced by natural mortality (M)

For the plus group, the initial numbers at age is the sum of numbers at ages 7 and older based on average recruitment and recruitment deviations for ages 7 and older in 1964 along with the natural mortality rate

Total mortality rates at age through time are also key variables in the population dynamics model. The total instantaneous mortality at age matrix Z=(Zy,a)YxA and the instantaneous fishing mortality at age matrix F=(Fy,a)YxA both have dimensions Y by A. Instantaneous natural mortality at age is assumed to be constant with M equal to 0.2. Thus, for all years y, and age classes a, total mortality at age is the sum of fishing and natural mortality

To determine total mortality, fishing mortalities will be estimated. While natural mortality might be estimable in some rare data-rich situations, M is often highly correlated with other parameters and is not estimable in practice (see for example, Schnute and Richards 1995).

Population numbers at age through time are computed from the initial population numbers at age, recruitment through time, and total mortality at age through time. For each age class, indexed by "a", that is younger than the plus group (a < A), the number at age is sequentially determined using a standard survival model

For the plus group, numbers at age are the sum of survivors of age A-1 and survivors from the plus group in the preceding year

Estimation of fishing mortality at age is facilitated by making the simplifying assumption that fishing mortality can be modeled as a separable process. This assumption implies that Fy,a is determined from the average selectivity pattern of age-a fish (Sa) and fully-recruited fishing mortality in year y (Fy)

While more complicated models of time-varying selectivity may be useful, this approximation is likely to be satisfactory if observation errors in the catch-at-age data are substantial.

Fully-recruited fishing mortality in each year is modeled as a lognormal deviation from average fishing mortality (µF), where the Uy are iid normal random variables with zero mean and constant variance

The fishing mortality deviations (Uy) are constrained to sum to zero over all years.

Fishery selectivity at age is modeled as being time-invariant throughout the assessment time horizon. This approach was chosen for parsimony and because there was believed to be substantial errors in the observed fishery age composition, especially in recent years. In particular, winter flounder catch-at-age data to estimate fishery selectivity are limited to 1982-2000, a period when the fishery was prosecuted primarily by domestic trawl fishing vessels. Since 1993, fishery sampling intensity of Georges Bank winter flounder catches has been relatively low. As a result, temporal changes in fishery selectivity would likely be difficult to detect given relatively high measurement errors in the fishery age composition data.

The average fishery selectivity at age is estimated for ages 1 through 6. For ages 7 and older, fishery selectivity is assumed to be equal to the age-6 selectivity value. This approach was chosen to reflect the fact that age-7 fish were not likely to differ much from age-6 fish in their fishery selectivity. Two constraints are applied to the estimated selectivity at age coefficients. First, the selectivities are constrained to average 1 for estimated ages. This forces the scale of each coefficient to be near unity. Second, a constraint is applied to ensure that estimated selectivities change smoothly between adjacent ages. Details of the implementation of both constraints are described in the section on statistical estimation approach. Last, for each year, the selectivity at age values are rescaled so that the maximum selectivity at age value is unity. This rescaling ensures that estimated fully-recruited fishing mortality rates are directly comparable to biological reference points such as F0.1.

Fishery removals from the population are accounted for through the fishery catch numbers at age matrix C=(Cy,a)YxA and the fishery catch biomass at age (yield) matrix Y=(Yy,a)YxA. Both C and Y have dimensions Y by A. Fishery catch at age in each year is computed in a standard manner from Baranov's catch equation using population numbers, fishing mortality, and total mortality at age

Catch biomass at age in each year (Yy,a) is approximated by the product of catch numbers at age and the long-term mean weights at age, where Wa is the mean weight at age computed as the average of mean Georges Bank weights at age from fishery sampling during 1982-2000

Use of the long-term mean weights at age is likely to be a useful approximation unless mean weights at age have varied substantially through time. Since fishery sampling has been relatively poor in recent years, the use of a long-term average was considered to be adequate given the likely errors in the observed annual mean weights at age computed from fishery samples.

Total fishery catch biomass in year y (Yy) is the sum of yields by age class

The calculated total fishery catch biomass time series is compared to observed values using a lognormal probability model. This model feature was included because it was expected that observed catches were not accurately reported in some years and that discards were not estimated for inclusion in the catch-at-age data.

Similarly, the proportion of fishery catch at age a in year y (Py,a) is computed from estimated catch numbers

The time series of fishery proportions at age are fitted to observed fishery values using a multinomial probability model (see for example, Fournier and Archibald 1982, Quinn and Deriso 1999). This model feature accounts for the possibility that the fishery catch-at-age data are measured with error.

Trends in the relative abundance of population biomass are measured by research survey indices for Georges Bank winter flounder. Three surveys were available: the NEFSC autumn groundfish survey (1963-2000), the NEFSC spring groundfish survey (1968-2000), and the Canadian spring groundfish survey (1987-2000). The survey biomass index in year y (Iy) for any of the surveys is modeled as a catchability coefficient (QSURVEY) times the population biomass that is vulnerable to the survey, where SSURVEY,a is survey selectivity at age a and pSURVEY is the fraction of annual total mortality that occurs prior to the survey

The survey biomass index time series are fitted to observed values using a lognormal probability model. This model feature accounts for the possibility that the survey relative abundance indices are measured with error.

Survey selectivity accounts for differential vulnerability of winter flounder age classes to the survey fishing gear and also for differential vulnerability due to differences in the behavior and distribution of juvenile and adult fish. For each of the three surveys, selectivity at age is modeled using Thompson's exponential-logistic model (Thompson 1994), where , ß, and are parameters and survey selectivity for winter flounder is assumed to be time invariant

This model has the useful property that the maximum selectivity value is unity. For values of >0 survey selectivity is dome-shaped, and survey selectivity is flat-topped (i.e., constant at older ages) when =0.

Survey age composition data provide information on the relative abundance of winter flounder age classes captured with the survey gear. Survey catch proportion at age a in year y (PSURVEY, y, a) is computed from survey selectivity, the fraction of mortality occurring prior to the survey, and population numbers at age

The time series of survey proportions at age are fitted to observed fishery values using a multinomial probability model. This model feature accounts for the possibility that the survey age composition data are measured with error.


STATISTICAL ESTIMATION APPROACH

The population dynamics model is fit to observed data using an iterative maximum likelihood estimation approach. The statistical model consists of ten likelihood components (Lj) and two penalty terms (Pk). The model objective function () is the weighted sum of the likelihood components and penalties where each summand is multiplied by an emphasis coefficient (j) that reflects the relative importance of the data.

Each likelihood component is written as a negative log-likelihood so that the maximum likelihood estimates of model parameters are obtained by minimizing the objective function. The Automatic Differentiation Model Builder software is used to estimate a total of roughly 95 parameters depending upon the model configuration. The likelihood components and penalty terms are described below.

1. Recruitment

Recruitment strength is modeled by lognormal deviations from average recruitment for the period 1959-2000. A total of 42 recruitment deviation parameters (Vy) and one average recruitment parameter (µR) are estimated based on the objective function minimization. The recruitment likelihood component (L1) is

where

(18)

and the Vy are iid normal random variables with zero mean and constant variance and n1 is the number of recruitment deviations.

2. Fishery age composition

Fishery age composition is modeled as a multinomial distribution for sampling catch numbers at age. The constant NE ,Fishery, y denotes the effective sample size for the multinomial distribution for year y and is assumed to be 200 fish per year during 1982-1993, 100 fish per year during 1994-1997 and 2000, and 50 fish per year during 1998-1999. These different sample sizes were chosen to reflect the relative intensity of fishery sampling of Georges Bank winter flounder. The observed number of fish at age in the fishery samples is computed as the effective sample size times the observed proportion at age, denoted with a superscript "OBS" for all variables.

The negative log-likelihood of the multinomial sampling model for the fishery ages (L2) is

The second term in summation over ages indexed by "a" is a constant that scales L2 to be zero if the observed and predicted proportions were identical. Six fishery selectivity coefficients (S1 through S6) are estimated based on the objective function minimization.

3. NEFSC Fall survey age composition

Fall survey age composition is also modeled as a multinomial distribution for sampling survey catch numbers at age. The constant NE ,Fall, y denotes the effective sample size for the multinomial distribution for year y and is assumed to be constant across time for the years 1982-2000 when winter flounder autumn survey catch-at-age data are available. The observed number of fish at age in the survey samples is computed as the effective sample size times the observed proportion at age. The effective sample size was assumed to be 100 fish in each year. The negative log-likelihood of the multinomial sampling model for the autumn survey ages (L3) is

As with the fishery age composition, the second term in the summation over the age index "a" is a constant that scales L3 to be zero if the observed and predicted proportions were identical. Three fall survey selectivity coefficients (Fall, ßFall, Fall) are estimated based on the objective function minimization using the survey selectivity model (Eqn. 14).

4.NEFSC Fall survey biomass index

The fall survey biomass index is modeled by lognormal deviations of predicted values from observed values during 1964-2000, where the log-transformed deviations DFall, y are iid normal random variables with zero mean and constant variance

The fall survey biomass likelihood component (L4) is

where n4 is the number of observed fall survey index values. One fall survey catchability coefficient (QFall) is estimated based on the objective function minimization.

5. NEFSC Spring survey age composition

Spring survey age composition is also modeled as a multinomial distribution for sampling survey catch numbers at age. The constant NE ,Spr, y denotes the effective sample size for the multinomial distribution for year y and is assumed to be constant for the years 1982-2000 when winter flounder spring survey catch-at-age data are available. The observed number of fish at age in the survey samples is computed as the effective sample size times the observed proportion at age. The effective sample size was assumed to be 100 fish in each year. The negative log-likelihood of the multinomial sampling model for the spring survey ages (L5) is

Three spring survey selectivity coefficients (Spr, ßSpr, Spr) are estimated based on the objective function minimization using the survey selectivity submodel (Eqn. 14).

6. NEFSC Spring survey biomass index

The spring survey biomass index is modeled by lognormal deviations of predicted values from observed values during 1968-2000, where the log-transformed deviations DSpr, y are iid normal random variables with zero mean and constant variance

The spring survey biomass likelihood component (L6) is

where n6 is the number of observed spring survey index values. One spring survey catchability coefficient (QSpr) is estimated based on the objective function minimization.

7. Canadian Spring survey age composition

Canadian spring survey age composition is also modeled as a multinomial distribution for sampling survey catch numbers at age. The constant NE ,CANSpr, y denotes the effective sample size for the multinomial distribution for year y and is assumed to be constant for the years 1987-2000 when winter flounder Canadian spring survey catch-at-age data are available. The observed number of fish at age in the survey samples is computed as the effective sample size times the observed proportion at age. The effective sample size was assumed to be 200 fish in each year. The negative log-likelihood of the multinomial sampling model for the Canadian spring survey ages (L7) is

Three Canadian spring survey selectivity coefficients (CANSpr, ßCANSpr, CANSpr) are estimated based on the objective function minimization using the survey selectivity model (Eqn. 14).

8. Canadian Spring survey biomass index

The Canadian spring survey biomass index is modeled by lognormal deviations of predicted values from observed values during 1987-2000, where the log-transformed deviations DCANSpr, y are iid normal random variables with zero mean and constant variance

The Canadian spring survey biomass likelihood component (L8) is

where n8 is the number of observed Canadian spring survey index values. One Canadian spring survey catchability coefficient (QCANSpr) is estimated based on the objective function minimization.

9. Catch biomass

Catch biomass is modeled by lognormal deviations of predicted values from observed values during 1934-1999, where T y are iid normal random variables with zero mean and constant variance

The catch biomass likelihood component (L9) is

where n9 is the number of observed catch biomass values.

10. Fishing mortality

Annual values of fully-recruited fishing mortality are modeled as lognormal deviations from average fishing mortality during the period 1934-2000. A total of 37 fishing mortality deviation parameters (Uy) and one average fishing mortality parameter (µF) are estimated based on the objective function minimization. The fishing mortality likelihood component (L10) is

where

and n10 is the number of observed catch values.

11. Fishery selectivity

Two constraints on fishery selectivity are included in a penalty function. The fishery selectivity penalty function (P1) is

The first term constrains the fishery selectivity coefficients to scale to an average of 1. The second term constrains the fishery selectivity coefficient of age a+1 to be near to the linear prediction of this value interpolated from age a and age a+2 selectivities over the range of estimated selectivity coefficients.

12. Fishing mortality penalty

One constraint on fishing mortality is imposed to ensure that during the early phases of the iterative estimation process the observed catch could not be generated by an extremely small F on an extremely large population size. The fishing mortality penalty function (P2) is

The constraint is weighted with a value of 10 for the initial estimation phases and is weighted with a value of 0.001 for all later estimation phases. The value of 0.1 was used because this value is sufficient to ensure that the estimated mean F will be on the order of the value of natural mortality for Georges Bank winter flounder.

Initial values are input for all parameters before the estimation phases are conducted. A total of nine estimation phases were used for the iterative minimization of the objective function. Any parameters first estimated in a given phase, say N, are estimated in all subsequent phases, N+1, N+2, etc. The first phase estimates average recruitment. The second phase estimates average fishing mortality and fishing mortality deviations. The third phase estimates recruitment deviations. The fourth phase estimates fishery and NEFSC spring survey selectivity coefficients. The fifth phase estimates the spring survey catchability coefficient. The sixth phase estimates the NEFSC fall survey selectivity coefficients. The seventh phase estimates the fall survey catchability coefficient. The eighth phase estimates the Canadian spring survey selectivity coefficients. The ninth phase estimates the Canadian spring survey catchability coefficient.

The twelve emphasis values (s) used for the baseline model were:

  1. Recruitment 1=10
  2. Fishery age composition 2=1
  3. NEFSC Fall survey age composition 3=1
  4. NEFSC Fall survey biomass index 4=10
  5. NEFSC Spring survey age composition 5=1
  6. NEFSC Spring survey biomass index 6=10
  7. Canadian Spring survey age composition 7=1
  8. Canadian Spring survey biomass index 8=10
  9. Catch biomass 9=100
  10. Fishing mortality 10=1
  11. Fishery selectivity penalty 11=10
  12. Fishing mortality penalty 12=1

SOUTHERN DEMERSAL WORKING GROUP RECOMMENDATIONS

After making some adjustments to the initial model configuration to better reflect the timing of the surveys and the emphasis factors for the fishery and survey age composition likelihood components, the Southern Demersal Working Group recommended that two final models be examined: (1) a model that used all three research survey time series (WINC, WINter flounder model including Canadian survey) and (2) a model that used the two NEFSC research survey time series (WIN, WINter flounder model). Both the WINC and WIN models provided similar trends in population biomass and fishing mortality, indicating that results were robust to the inclusion of the Canadian research survey time series.


MODEL DIAGNOSTICS

Model diagnostics showed that the WIN model provided a better fit to the observed catch biomass series (RMSE=0.137) than the WINC model (RMSE=0.149). The WIN model also provided a better fit to the NEFSC fall biomass series (RMSE=0.356) than the WINC model (RMSE=0.373). The fits to the NEFSC spring biomass series were nearly identical for the two models (WIN, RMSE=0.472 vs WINC, RMSE=0.473). In addition, the trend in the observed Canadian spring biomass series was lower than the WINC model predictions during 1998-2000, suggesting that the Canadian survey was not tracking relative abundance in recent years. Overall, the WIN model that used the two NEFSC research survey time series provided the best fit to the catch biomass and NEFSC survey biomass series while the WINC model provided a poor fit to the Canadian survey biomass series in recent years (see Figure 4 below). The condition numbers of the hessian matrices of the two models were also different with the WINC model having a much higher condition number (=6.83•1012) than the WIN model ( =2.83•107). This indicated that the numercial solution of the WINC model was not well-determined relative to the WIN model. Based on model diagnostics, the WIN model that used the two NEFSC research survey time series was considered to be the best model among the statistical catch-at-age models examined for winter flounder. Computer code to fit the WIN model, the input data file, and the standard deviation parameter file are listed in the Appendix.

Plots of diagnostics for the two models include the discrepancies between observed data and predicted values for the catch biomass series (Figure 1), the fall survey biomass series (Figure 2), the spring survey biomass series (Figure 3), and the Canadian spring survey biomass series (Figure 4, shown for the WINC model only). For the best fit WIN model, diagnostic plots include the fishery age composition series (Figure 5), the fall survey age composition series (Figure 6), and the spring survey age composition series (Figure 7). For the WINC model, a diagnostic plot of the Canadian spring survey age composition series is also shown (Figure 8).


MODEL RESULTS

Model estimates of spawning biomass, fishing mortality, recruitment, and population biomass for the WIN model during the period 1963-2000 are listed in Table 1. Fishery and survey selectivity estimates at age are shown in Figure 9. Recruitment estimates are shown in Figure 10 (see also Table 1). Population biomass estimates are shown in Figure 11 (see also Table 1). Spawning biomass estimates (at start of the spawning season) are shown in Figure 12 (see also Table 1). Fishing mortality estimates are shown in Figure 13 (see also Table 1). Stock-recruitment data are shown in Figure 14. Surplus production implied by the age-structured estimates of exploitable biomass and observed catches are shown in Figure 15.

Other model outputs included depletion ratios for year 2000, relative to 1964, for spawning biomass (46%) and population biomass (53%). Similarly, depletion ratios for year 2000, relative to 1982, were computed for spawning biomass (88%) and population biomass (81%). Long-term average recruitment was estimated to be 5.550 million age-1 fish during 1959-1999.

Sensitivity to the assumed value of M was investigated by systematically varying this parameter using the likelihood profile feature of the AD Model Builder software. This analysis showed that the model was not stable for moderate departures from M=0.2. In particular, running the baseline model under alternative assumptions about M showed that the model did not converge in its final configuration for M=0.195, 0.196, 0.2005, 0.201, 0.2015, 0.2025, 0.203, while it did converge for M=0.197 (-lnL=3445.2), M=0.198 (-lnL=3444.8), M=0.199 (-lnL=3444.2), and M=0.202 (-lnL=3443.0). The objective function value for the assumed value of M=0.2 was -lnL=3443.8. This suggested that the objective function surface was a complicated function of natural mortality and that the model was sensitive to the assumed value.

Based on the Southern Demersal Working Group's recommendations, a sensitivity analysis was conducted on the value of the effective sample size time series for the fishery age composition likelihood. This was done to see how the model results might change if different effective sample sizes were used. The SDWG suggested multiplying the effective sample size time series by and 2. The use of multipliers of less than 0.8 did not lead to model convergence, presumably because there was insufficient information assigned to the fishery age composition in these cases. Nonetheless, estimated spawning biomass, a key model output, was insensitive to using effective sample sizes that were 80% and 200% of the baseline values, which ranged from 50 to 200 fish (Figure 16). Overall, this suggested that the model solution would not be well-determined if effective sample sizes for the fishery age composition were below 40 fish per year, but, for values above this, the results appeared to be robust.


CONCLUSIONS

Conditioned on the accuracy of the model and the assessment data, results of the best fit model indicate that:


ACKNOWLEDGMENTS

I thank the members of the Southern Demersal Working Group for their helpful comments and suggestions. I also thank the members of the 34th Northeast Regional Stock Assessment Review Committee for their helpful comments and thoughtful review.


LITERATURE CITED

Fournier, D. A., and C. P. Archibald. 1982. A general theory for analyzing catch at age data. Can. J. Fish. Aquat. Sci. 39:1195-1207.

Ianelli, J. N., and D. A. Fournier. 1998. Alternative age-structured analyses of NRC simulated stock assessment data. NOAA Tech. Memo. NMFS-F/SPO-30. pp. 81-96.

Methot, R. D. 1990. Synthesis model: an adaptive framework for analysis of diverse stock assessment data. Int. North Pac. Fish. Comm. Bull. 50:259-277.

Quinn, T. P., II, and R. B. Deriso.1999. Quantitative fish dynamics. Oxford University Press, New York, 542 pp.

Schnute, J. T., and L. J. Richards. 1995. The influence of error on population estimates from catch-age models. Can. J. Fish. Aquat. Sci. 52:2063-2077.

Thompson, G. G. 1994. Confounding of gear selectivity and the natural mortality rate in cases where the former is a nonmonotone function of age. Can. J. Fish. Aquat. Sci. 51:2654-2664.


Table 1. Baseline model results for Georges Bank winter flounder.

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