Fisheries Research 74 (2005) 96–115
Assessing a stock assessment framework for the green sea urchin Strongylocentrotus drobachiensis fishery in Maine, USA Minoru Kanaiwa a,∗ , Yong Chen a , Margaret Hunter b a
218 Libby Hall, School of Marine Sciences, University of Maine, Orono, ME 04469, USA b Maine Department of Marine Resources, West Boothbay Harbor, ME 04575, USA
Received 6 October 2004; received in revised form 17 March 2005; accepted 29 March 2005
Abstract A Bayesian stock assessment framework with a size-structured population dynamics model used to assess the green sea urchin, Strongylocentrotus drobachiensis, fishery in Maine, USA was evaluated, using a simulation approach, for its performance in describing sea urchin population dynamics under different recruitment dynamics and data quality. This study suggests that the current stock assessment model performs well in estimating key sea urchin fishery parameters such as exploitable stock biomass, total stock biomass, natural mortality, and fishing mortality under different simulation scenarios, and can capture the dynamics of the Maine sea urchin population. The recruitment dynamics of the sea urchin are likely to vary greatly with large changes occurring in its ecosystem. The finding that the current assessment framework is able to capture different patterns of recruitment dynamics implies that the current assessment framework will remain effective in future stock assessments of the Maine sea urchin fishery. © 2005 Elsevier B.V. All rights reserved. Keywords: Green sea urchin; Strongylocentrotus drobachiensis; Bayesian size-structured stock assessment framework; Fisheries management; Maine; Simulation
1. Introduction The green sea urchin, Strongylocentrotus drobachiensis, fishery is of great importance to Maine’s economy. The sea urchin fishery developed and grew in the late 1980s as a result of expanding export markets, and landings peaked in 1992 (Fig. 1). Since 1992, ∗ Corresponding author. Tel.: +1 207 581 4405; fax: +1 207 581 4990. E-mail address:
[email protected] (M. Kanaiwa).
0165-7836/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fishres.2005.03.006
the fishery has experienced substantial declines in landings, mainly resulting from a large decrease in sea urchin stock abundance, although the change in catch per unit effort (CPUE) is relatively small (Fig. 1). A size-structured population dynamics model was developed and used to assess the sea urchin stock because sea urchins are difficult to age and have large variations in growth among individuals (Quinn and Deriso, 1999; Russell and Meredith, 2000). Bayesian inference was used in fitting a population dynamics model to data because the use of prior distributions enables the
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Fig. 1. Landing and catch per unit of effort (CPUE) for the sea urchin fishery in Maine.
incorporation of prior knowledge of the fishery into parameter estimation and the use of likelihood functions in Bayesian inference makes it easy to incorporate data of various sources and uncertainties associated with the data (Taylor et al., 1996). Data available for this study were mainly from the fishery and outliers are evident (Chen et al., 2000). Thus a robust Bayesian method, which is less sensitive to outliers, is used in the parameter estimation (Chen et al., 1994; Fournier et al., 1990; Chen and Fournier, 1999; Chen and Hunter, 2003). The current sea urchin stock assessment framework is modified from an early version of the model reported in Chen and Hunter (2003). The early version of the model assumed that the stock recruitment relationship follows the Beverton–Holt model, which may not be realistic for the sea urchin due to possible depensation processes in low densities (Wahle and Peckham, 1999; Steneck et al., 2004). The modified model, however, estimates annual recruitment in the assessment, independent of spawning stock biomass. Thus there is no built-in assumption between spawning stock biomass and recruitment. As a result, we can evaluate the estimated recruitment and spawning stock biomass to
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determine if there is a relationship between spawning stock biomass and subsequent recruitment. Like many stock assessment frameworks with complex population dynamics models, although the current stock assessment framework had been used in assessing the Maine sea urchin stock, its performance and ability in modeling the sea urchin population dynamics had not been evaluated. Thus, we had limited information on its performance under different recruitment dynamics and data quality. Understanding the model performance under different recruitment dynamics is especially important to the Maine sea urchin because its ecosystem is experiencing large changes (Steneck et al., 2004) which are likely to result in recruitment dynamics different from ones observed before. Only through an extensive simulation study can we understand the ability of the current stock assessment framework to capture different patterns of population dynamics resulting from changes in recruitment dynamics. The quality of parameter estimation (and thus stock assessment) can be influenced by the choice of the models, quality of input data, degree of realism about error assumptions, and appropriateness of the selected estimators (Chen and Paloheimo, 1998; Chen and Fournier, 1999). Inappropriate assessment models, together with low quality of data, unrealistic error assumptions, and inappropriate estimators tend to result in large errors in parameter estimation, leading to sub-optimal management of fisheries. For a given stock assessment framework, performance may also vary with the patterns of population dynamics. For example, some stock assessment frameworks can capture the population dynamics for a fishery with large fluctuations in recruitment, but not for a fishery with a one-way trend of recruitment (e.g., continuing increase or decrease). This calls for the careful evaluation of the stock assessment framework including how the model and statistical method may respond to errors in data and to changes in the temporal patterns of recruitment dynamics. Because the true population dynamics of a fishery are unknown, we cannot directly use field data collected from a fishery to assess a stock assessment framework. A commonly used and probably only viable approach is to simulate a fishery with known population dynamics and statistical properties of data, and then apply the proposed stock assessment framework to the simulated fishery (Chen et al., 2000). It is, however, important to develop an approach that can provide an unbiased
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assessment of the performance of the stock assessment framework in the simulation study. This calls for the avoidance of using the stock assessment model to be tested in generating the simulated fishery, because such an approach implicitly assumes we know the population dynamics models. The model estimation should not be limited to the data simulated based on the actual field data. Stock assessment models are often used to make projections of future population dynamics which may differ greatly from the past patterns. Because their performance may also depend upon patterns of population dynamics, we need to evaluate model performance for different patterns of population dynamics (e.g., constant, continue increasing or decreasing, or fluctuating over time) that may appear in the future. In this study, using random Bernoulli trials, we simulated a fishery based on data from the Maine sea urchin fishery. We then used the simulated fishery to assess the Bayesian size-structured stock assessment framework that was currently used in the assessment of the Maine sea urchin fishery (Chen and Hunter, 2003). The performance of the stock assessment framework in estimating key population and fishery parameters was evaluated under different simulation scenarios regarding data quality and recruitment dynamics. We then applied the tested stock assessment framework to the actual data collected in the Maine sea urchin fishery and from a fishery-independent survey program.
2. Description of the size-structured sea urchin model The size-structured stock assessment model for the Maine sea urchin stock consists of five submodels: (1) a growth model; (2) a recruitment model; (3) a
population-at-size model; (4) a predictive model; (5) four observational models that are established to relate observed catch per unit of effort (CPUE) with CPUE data predicted from the models, the observed catch size compositions with predicted catch size compositions, the observed survey abundance index with the abundance index predicted from the models, and observed survey size composition data with predicted stock size compositions (see Appendix A for detailed descriptions). The first four submodels describe fishing processes and the processes determining the dynamics of a fish population, and they were used to generate a model fishery. The dynamics of the model fishery were driven by reported catch. Various fisheries statistics such as catch size composition and stock biomass could be predicted from the simulated fishery. These submodels were then fine-tuned for their parameters using the four observational models, which are used for establishing objective functions for parameter estimation. We set bin sizes between 40 and 100 mm with an interval of 1 mm. The mathematical functions of these submodels and likelihood functions are described in Appendix A.
3. Bayesian parameter estimation Following Chen et al. (2000), three forms of priors were allowed for each parameter, Cauchy distribution functions, normal or log-normal functions, and uniform distributions. In this study, to reduce the impacts of priors in estimating parameters we assumed that all parameters have non-informative priors described by uniform distributions (Table 1). All the boundary values used are derived from our understanding of Maine sea urchin biology. Posterior distributions of the model parameters were estimated using the Markov Chain Monte
Table 1 Priors used for some key parameters in the Bayesian estimation Parameters
M Eps Rho R0
Boundaries Lower
Upper
0.01 −4 0.00001 5
0.5 4 0.99 50
Distribution
Mean
CV
Initial values
Uniform Normal Uniform Uniform
na 0 na na
na 0.4 na na
0.05 0 0.5 18
na = not applicable. The lower and upper boundaries were determined based on our understanding of Maine sea urchin biology and include all possible values for a particular parameter.
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Carlo (MCMC) simulation approach. The HastingsMetropolis algorithm was used for this calculation. The simulation was started from the parameters at the mode of the posterior distribution, which was identified by minimizing the total objective function, including the negative log likelihood components and the prior probability contributions. The lag between samples was 200. The model was implemented in AD Model Builder (Fournier, 1996). Detailed descriptions on estimating posterior distributions can be found in Fournier (1996). Half a million simulations were run
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to estimate the posterior distribution of model parameters (Chen et al., 2000).
4. Evaluating the developed stock assessment model To determine whether the developed model can realistically reproduce the model parameters and sea urchin population dynamics, a simulation study was conducted (Fig. 2). The simulation included the
Fig. 2. Simulation flowchart for evaluation of the Maine sea urchin stock assessment model. Built-in “true” parameters (R, M, F, BT and BL ) ˆ M, ˆ Fˆ , B ˆ T and B ˆ L ) estimated using the proposed model. R is the recruitment, M the natural mortality, F were compared with the parameters (R, the exploitation rate, G the growth transition matrix (see Chen et al., 2003), N0 the initial condition of population, BT the total biomass, BL the legal biomass, C the catch in the fishery, I the catch per unit effort, Cp the size frequency of the fishery catch, A the survey abundance index, and Sp the survey size frequency data. Letters with hat are estimated, and letters with prime are “observed” data with errors.
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following steps: (1) using catch and CPUE data collected in the sea urchin fishery and biological information (e.g. growth and maturation) to simulate a sea urchin fishery with known population dynamics; (2) applying the proposed stock assessment framework to the simulated sea urchin fishery to estimate fishery parameters and sea urchin stock dynamics such as natural mortality, fishing mortality, recruitment, spawning stock biomass, and exploitable stock biomass; (3) comparing the parameters and population dynamics used to create the simulated sea urchin fishery (i.e., step 1) with the parameters and population dynamics estimated by applying the proposed assessment framework to the simulated sea urchin fishery (Fig. 2). Small differences in the comparisons indicate that the proposed stock assessment framework is likely to capture the true patterns of sea urchin population dynamics, and thus is reliable. The natural mortality used in simulating the fishery had the value of 0.1615 (Chen and Hunter, 2003). For simplicity we set fishery selectivity (Sk ) as 1 for each size in the simulation. Thus, only a legal size switch is used as fishery selectivity and the simulated fishery has a knife-edge selectivity determined by the minimum and maximum legal sizes.
A Bernoulli trials approach was used to simulate the sea urchin fishery (Fig. 3). This approach is a probabilistic approach including simulating the lives of individual sea urchins and is accomplished by expressing various components of the model equations as random Bernoulli trials. For example, rather than calculating the number of sea urchins that survive natural mortality by Nt+1 = Nt e−M where M is the instantaneous rate of natural mortality, we simulated natural mortality acting on Nt individual sea urchins for 1 to Nt : if U(0, 1) ≤ 1 − e−M then Nt+1 = Nt − 1 where U(0, 1) is a uniform distributed random number between 0 and 1. We refer to this simulation process as a simulator. The simulator simulates the life of each sea urchin in a population subject to the fishery. All the parameters used in the simulator in generating the simulated sea urchin fishery were from the sea urchin stock assessment (Chen and Hunter, 2003).
Fig. 3. Flowchart for the individual-based simulator for simulating the sea urchin fishery.
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An initial population of N individuals was generated randomly with size-frequency chosen in proportion to the initial size structure. Each year, a certain number of recruits was added to the population. Four scenarios were considered for temporal patterns of recruitment in the simulation: fluctuating (Chen and Hunter, 2003), constant, declining, and increasing over time. Bookkeeping was done by examining each individual and adding it to the legal biomass and spawning stock biomass where appropriate. The exploitation rate was calculated as the proportion of the catch of the total biomass. In each time step (i.e., year), each sea urchin had a probability of being caught in the fishery, dying of natural mortality, growing, or maturing. When a sea urchin was caught in the simulated fishery, its size was recorded to generate catch size frequency data. CPUE for the fishery was generated as a constant proportion of the total weight of legal sea urchins in the population. Illegal catch was assumed to be zero in the simulation study. The output from the simulation model was then used as input data for the stock assessment framework to be tested. We then fitted the stock assessment model to the simulated data using the robust Bayesian approach to estimate parameters in the assessment model and key fisheries statistics such as recruitment and stock biomass. The estimation results were then compared with the true values initially used in simulating the data (Fig. 2). Because temporal patterns of population dynamics are likely to influence the performance of a stock assessment model in describing the population dynamics (Hilborn and Walters, 1992), we need to evaluate the performance of the assessment framework under different temporal patterns of recruitment, which is the driving force of population dynamics. Four different temporal patterns were considered for recruitment in the simulation: fluctuating, decreasing, increasing, and constant. For the fluctuating recruitment, recruits fluctuated over time, being low in the beginning of the fishery, increasing with landings, and then decreasing quickly after reaching the peak. This mimics the temporal variations in recruits of the Maine sea urchin fishery over time (Chen and Hunter, 2003). Recruits were low in the early stage of the fishery due to high density, but increased with the reduction of population density due to high landings in the sea urchin fishery and/or unfa-
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vorable environmental conditions, or perhaps because of an expansion of fishing into previously unexploited areas, followed by a large decrease in recruits probably due to the reduced spawning stock biomass and/or loss of habitats and increased predator crab population abundance (Steneck et al., 2004). The other three patterns involve one-way change in recruitment over time, and such a pattern is often difficult to quantify by a population dynamics model (Hilborn and Walters, 1992). They have, however, important management implications for possible stock recovery (increasing), remaining the same (constant), and deteriorating (decreasing) in the future. Errors that arise in the simulation are a result of the random sampling associated with the random Bernoulli trials for all life history and fishery processes (Fig. 2). The scenario without any other sources of errors was considered as the base scenario. Aside from the base scenario, we also considered other simulation scenarios with the combinations of different errors (Table 2). The observation errors were considered for catch, CPUE, size frequency of fishery catch, survey abundance index and survey size frequency. For CPUE and survey index data, errors were assumed to follow a log-normal distribution with its standard deviation being 0.1 (considered as low errors) and 0.25 (considered as high errors). Errors associated with size frequency data were assumed to follow a multinomial distribution and the magnitudes of the errors were decided by sample sizes (Chen, 1996). Two levels of errors were considered, sub-sample sizes of 500 (considered as high errors) and 4000 (considered as low errors). These sample sizes are consistent with the number of sea urchins measured by the Maine dock sampling program over the last decade. For 61 size classes, this is equivalent to 8 and 66 individuals per size class, respectively. The effective sample size used in the estimation (i.e., ωk,t in Eq. (A.18)), which determines the actual level of variation in size composition data, is the square root of the subsample size used in the simulation (Chen et al., 2000; Breen and Kim, 2003). Thus, the effective sample size is 22 (high errors) and 63 (low errors), respectively. The data simulated were then analyzed using the developed models. This test would evaluate the validity of the models. If a model yielded large errors in estimating parameters, we would conclude the performance of the proposed assessment framework was not
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Table 2 Design of the simulation study Scenario
Description
I
Base case without observation errors and with fluctuating recruitment Without observation errors and with continuous decrease in recruitment Without observation errors and with continuous increase in recruitment Without observation errors and with constant recruitment Small observation errors in all input data Large observation errors in all input data Large observation errors in catch, but small errors in all other input data Large observation errors in CPUE, but small errors in all other input data Large observation errors in survey abundance index, but small errors in others Large observation errors in fishery size frequency, but small errors in others Large observation errors in survey size frequency, but small errors in others No survey data and small errors in fishery data No survey data and large errors in fishery data Large errors in survey index and size frequency data, but small errors in fishery data Large errors in catch, CPUE, and catch size frequency, small errors in survey data Large errors in fishery and survey size frequency data
II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI
All input data include fishery data (i.e., catch, CPUE, catch size frequency data) and survey data (i.e., survey abundance index and survey size frequency). The recruitment for Scenarios V–XVI is the same as that for Scenario I (i.e., fluctuating recruitment).
ble with the estimation of posterior distributions in a Bayesian analysis (Berger, 1994; Chen et al., 2000). A potential problem with such an approach is lack of considering variations in process and measurement errors among simulation runs. Such variations are, however, partially incorporated by the prior distributions of parameters (Table 1) into the posterior distribution (Chen et al., 2000). Thus we should have a good evaluation of impacts of random errors on parameter estimation by evaluating the resultant posterior distributions that combine random errors in data and uncertainty in parameters.
5. Application to the Maine sea urchin data The tested stock assessment framework was applied to the actual data collected in the sea urchin fishery and from a fishery-independent survey program for Management Zone 1. The observed CPUE, survey abundance index, catch size frequency, and survey size frequency were compared with those predicted from the tested stock assessment framework. The posterior distributions of stock biomass, recruitment, and exploitation rate were evaluated. The estimated recruitment was evaluated with the corresponding stock spawning biomass to identify how the recruitment varies with spawning stock biomass.
6. Results good. Otherwise, we would conclude that the framework performed well in capturing the dynamics of the sea urchin population. The average percentage of bias for an estimated parameter was calculated as: |Ptrue,i −P¯ est,i | percentage of bias =
i
P
true,i × 100 n where Ptrue is the true value of the parameter and P¯ est the average of the n sets of posterior parameter estimates. In this study we calculated the percentage bias for natural mortality (M), total biomass (BT ), legal biomass (BL ), and exploitation rate in each year (F = C/BL ; see Fig. 2). Instead of running a large number of simulation runs in this study, we only implemented one simulation run because multiple simulation runs are not compara-
6.1. Simulation study For the first simulation scenario (base case with no observation errors and fluctuating recruits; Table 2), the recruits estimated using the proposed model were rather consistent with the recruits used in simulating the fishery (Fig. 4). Other key fishery statistics, including exploitable stock biomass, total stock biomass, exploitation rate and natural mortality, estimated using the proposed stock assessment model, were also consistent with the “true” values used in simulating the sea urchin fishery with small estimation errors (Fig. 4). For the second simulation scenario without observation errors but with a decreasing recruitment (Table 2), the recruitment estimated using the proposed model virtually mimicked the true trend of changes in
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Fig. 4. Posterior distributions of key fishery statistics built-in in the simulated fishery (i.e. true) and predicted by the proposed stock assessment framework when recruits fluctuate among years.
recruitment (Fig. 5). The true values for recruitment and all other fishery parameters were all within 95% credibility intervals of the posterior distributions. Thus, the proposed model could capture the dynamics of the sea urchin population with decreasing recruitment. For the third scenario without observation errors but with an increasing recruitment (Table 2), the recruitment estimated using the proposed model captured the trend of increase in the first several years, but tended to under-estimate the true recruitment in more recent years, resulting in large estimation errors for recruitment in recent years (Fig. 6). The under-estimation of recruitment in recent years then resulted in the underestimation of stock biomass and over-estimation of exploitation rates in recent years (Fig. 6). The true values
for recruitment, biomass, and exploitation rate were all within the 95% credibility intervals of the posterior distributions of these parameter estimates for the first several years, but beyond the 95% credibility intervals in recent years (Fig. 6). The posterior distribution of natural mortality estimates did not overlap with the true natural mortality (Fig. 6). For the fourth scenario without observation errors but with a constant recruitment (Scenario IV; Table 2), the recruitment estimated using the proposed model captured the temporal trend of the true recruitment, but the difference between the mean of the posterior distribution and true recruitment increased with time (Fig. 7). Thus estimation errors in recent years tended to be larger than errors in early years. This pattern was
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Fig. 5. Posterior distributions of key fishery statistics built-in in the simulated fishery (i.e. true) and predicted by the proposed stock assessment framework when recruits continue decreasing over time.
also applicable to exploitation rate and stock biomass (Fig. 7). When the low observation errors were added to all the input data used in Scenario I (i.e., Scenario V; Table 2), the differences between model predictions and true values for estimated recruitment, stock biomass, and exploitation rate became larger compared with those for Scenario I (Fig. 8). Such a difference increased with the increased observation errors (Scenario VI, Fig. 9; Table 2). Thus large observation errors tended to result in large estimation errors. The “observed” size composition data of landings were compared with the size composition of landings predicted using the proposed population model. For
the four simulation scenarios without observation errors but with different recruitment patterns (i.e., Scenarios I–IV), the largest differences in the “observed” and “predicted” occurred in the small size classes (Figs. 10 and 11). This was especially true for the size classes that were close to the minimum legal size. The predicted size composition data of landings were almost identical to the “observed” size composition of landings for the large size classes (Figs. 10 and 11). This suggests that the model can predict size compositions of landings well for large sea urchins, and less well for sea urchins with sizes close to the minimum legal size. However, the magnitudes of differences in Figs. 10 and 11 were small, suggesting that the
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Fig. 6. Posterior distributions of key fishery statistics built-in in the simulated fishery (i.e. true) and predicted by the proposed stock assessment framework when recruits continue increasing over time.
proposed population model predicted size composition data in the fishery and landings well in the absence of observation errors (Scenarios I–IV; Figs. 10 and 11). The difference between the “observed” and predicted size frequency data increased with the observation errors (e.g., Scenarios V and VI; Figs. 10 and 11). The difference tended to be larger for small size classes compared with that for the large size classes because the frequency was higher for small size classes.
Different combinations of errors were applied to different types of input data to identify the key input data that might have large impacts on parameter estimation (see Scenarios VII–XVI; Table 2). For Scenarios VII–XI where large observation errors were applied to each of the input variables (Table 2), Scenarios VIII–XI had similar average percentage biases in the parameter estimation, but the average percentage bias was much higher for Scenario VII,
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Fig. 7. Posterior distributions of key fishery statistics built-in in the simulated fishery (i.e. true) and predicted by the proposed stock assessment framework when recruits are constant over time.
which had large observation errors in catch (Fig. 12). This suggests that observation errors in catch had the largest impact on the quality of parameter estimation with the proposed population model. This result was expected because the catch was implicitly assumed to be error free (see Eq. (A.4)), while other variables including CPUE, survey abundance index, landing size compositions, and survey size compositions were assumed to be subject to observation errors. Thus, the quality of landings data is important in determining the quality of parameter estimation with the proposed model. The Scenarios XII and XIII had no survey data with small and large observation errors for all other input
data, respectively (Table 2). The average percentage biases for Scenarios XII and XIII were comparable to those for Scenarios V and VI (same as XII and XIII, respectively, except they included survey data), respectively. Thus the proposed population model could yield good estimates in the absence of survey data as long as the fishery data were well defined. Large observation errors were only applied to the survey data, but not to the fishery data in Scenario XIV. The average percentage biases of this scenario in the parameter estimation were smaller than those for Scenario XV which had large observation errors in the fishery data. Thus, the magnitude of observation errors associated with the fishery data was likely
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Fig. 8. Posterior distributions of key fishery statistics built-in in the simulated fishery (i.e. true) and predicted by the proposed stock assessment framework when there are small observation errors in catch, fishery abundance index, fishery size frequency, survey abundance index and survey size frequency.
to be more important than errors in the survey data in determining the quality of the parameter estimation. For Scenario XVI, which had large errors in both the survey and fishery size frequency data, the average percentage bias was higher than that for Scenarios X and XI which had large observation errors only in landings size-frequency or survey size-frequency data. 6.2. Application The model-predicted CPUE, survey abundance index, catch size composition, and survey size composi-
tion fitted those observed well, respectively (Fig. 13). Although only the most recent year of size composition data were shown, the patterns of estimated and observed size composition data in other years were similar to those in 2003 (Fig. 13). The stock assessment framework appears to perform well in describing the observed data. The estimated recruitment initially increased with the changes in spawning stock biomass. The recruitment reached its highest value in 1992. However, after the spawning stock biomass became lower than 20,000 metric tons, the recruitment decreased with decreasing spawning stock biomass (Fig. 14).
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Fig. 9. Posterior distributions of key fishery statistics built-in in the simulated fishery (i.e. true) and predicted by the proposed stock assessment framework when there are large observation errors in catch, fishery abundance index, fishery size frequency, survey abundance index and survey size frequency.
The posterior distributions of total biomass and legal biomass were similar (Fig. 15). Initially, the stock biomass increased and reached the highest level in 1992, followed by a large decrease. The average stock biomass of the most recent year is only 10% of the highest stock biomass in 1992. Stock biomass estimates had relatively large uncertainty in early years and recent years (Fig. 15). Exploitation rate calculated as the ratio of catch and legal stock biomass increased initially, and then became relatively constant at high levels. For most recent years, the exploitation rates seemed to decrease, corresponding to the decrease in landings. However, exploitation rates of most recent years had large
uncertainty, making the interpretation difficult. For recruitment, the uncertainty was small in recent years (Fig. 15).
7. Discussion Overall, this simulation study suggests that the proposed population model performs well under the simulation scenarios considered in this study. The population model can capture various temporal trends in recruitment. For Scenario III, which had a temporal pattern of increasing recruitment, the true recruitment in
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ˆ catch size frequency data for Scenarios I–VI. The Y-axis shows the Fig. 10. Summary of differences in “observed” (Cp) and predicted (Cp) ˆ and the X-axis shows bin size (mm). values of differences calculated as (Cp − Cp)
recent years tended to be under-estimated, resulting in over-estimating exploitation rate and under-estimating stock biomass. In practice such a negative bias is not risk prone, and tends to lead to the development and implementation of precautionary management plans. For a fast developing fishery associated with continuous increasing recruitment (i.e., Scenario III), such an error in estimating recruitment is not necessarily bad. The relatively larger estimation errors in more recent years compared with those for early years in the assessment are consistent with the performance of other models.
For all the scenarios, the 95% credibility intervals of posterior distributions for the key fishery parameters are tight. This may result from the use of bounded uniform distributions as prior information and the availability of unbiased estimates of size frequency information on the fishery catch and on the population (i.e., survey size frequency data). The population model calculates the exploitation rate using the landings data (i.e., Eq. (A.4), Appendix A). This implicitly assumes that the landings data are error free. Landings data are also driving the dynamics
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ˆ survey size frequency data for Scenarios I–VI. The Y-axis shows the Fig. 11. Summary of differences in “observed” (Sp) and “predicted” (Sp) ˆ and the X-axis shows bin size (mm). values of differences calculated as (Sp − Sp)
of the population in the model. This study suggests that the parameter estimation is most sensitive to errors in catch. Thus, among the input data, the quality of landings data is perhaps most important in determining the performance of the tested model. The average percentage bias in the parameter estimation for Scenario XIV was larger than that for Scenario XII and the estimation bias for Scenario VI was larger than that for Scenario XIII. This suggests that if
the survey data contained large observation errors their inclusion in the parameter estimation would not help improve the parameter estimation; rather it would reduce the quality of the parameter estimation. However, the smaller biases for Scenarios V and XV compared, respectively, with those for Scenario XII and XIII suggest that the survey data with small observation errors would improve the parameter estimation. Thus the inclusion of survey data may not necessarily improve the
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Fig. 12. The summary percentage of estimation bias for all the scenarios.
stock assessment, and only those survey data of high quality are helpful in improving the stock assessment. This statement is, however, made based on the assumption that the data collected from the fishery (e.g., catch, CPUE, and catch size composition data) are of high quality in representing the fishery. In practice data from a fishery are likely to be low quality with large observation errors or sometimes bias, and survey data from a well defined survey are likely to be of good quality.
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Fig. 14. Changes of estimated recruitment with the estimated spawning stock biomass.
In this case, the inclusion of the survey data is important in improving the stock assessment (Hilborn and Walters, 1992). By comparing the average estimation bias among Scenarios V, X, XI, and XVI, we could conclude that with the inclusion of at least one set of size frequency data with small observation errors, the bias in the parameter estimation would become smaller compared with that for the scenarios with large errors in both sets of size frequency data collected for landings and
Fig. 13. Observed and predicted CPUE: survey abundance (a), catch size composition (b), and survey size composition (c).
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Fig. 15. Posterior distributions of key fisheries statistics for Management Zone 1 sea urchin in Maine.
survey. The impacts of having large errors in both sets of size frequency data on the parameter estimation were comparable to that for large errors in catch (Scenarios IV and XIII; Fig. 12), suggesting the importance of having at least one set of good estimates for size frequency data in a stock assessment. In this study we only implemented one simulation run because multiple runs are not comparable with the estimation of posterior distributions in a Bayesian analysis. Thus, although observation errors were simulated randomly, errors in the single run realized in the simulation might not reflect such randomness and might even be biased. The levels of errors for various input data, although generated arbitrarily, have been commonly used in simulating fisheries data (Walters, 1998). Scenario VI, the worst in this study, with large errors in all input data, has the highest average percentage bias in parameter estimation among all the scenarios considered in this study. Even for this scenario, the average percentage bias is smaller than 20%, indicating the pro-
posed population model can perform well in capturing the population dynamics of sea urchins. The setting of this study can be extended for including more scenarios that might be observed in fisheries stock assessment and management. More studies can also be done to identify the relative importance of various variables in the model. The approach can also be modified for the development and evaluation of other fish stocks. The application of the tested stock assessment framework to Management Zone 1 data reveals some interesting results, in particular the decreasing exploitation rates and changes of recruitment with the spawning stock biomass. The estimated mean of exploitation rate shows a decreasing trend in recent years. However, this should be interpreted carefully because of large uncertainty associated with the exploitation rates of recent years. More information probably is needed in order to better define the current exploitation rate. The stock–recruitment relationship does not follow the
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Beverton–Holt model as we assumed in the past (Chen and Hunter, 2003), rather it is more likely to follow the Ricker model (Hilborn and Walters, 1992). The current recruitment process is likely to be governed by the depensation process because the recruits per spawning stock biomass decreased with the spawning stock biomass in recent years. More study is necessary to quantify this process and estimate the critical values that separate the compensation and depensation processes for the Maine sea urchin stock. Together with the habitat information, this will allow managers to identify key conditions for the recovery of the sea urchin stock in Maine.
Acknowledgements The financial support for this project was provided by the Maine Department of Marine Resources, Maine Sea Urchin Zone Council, Maine Sea Grant and University of Maine (Orono) to Chen. We thank the constructive and detailed comments given by the two anonymous reviewers, which greatly improved the paper.
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the dependency of recruitment in a given year with recruitment of the previous year. Thus, recruitment in year t can be defined by these parameters as 2 ) Rt = R0 exp(Rdevt − 0.5σRdev
(A.1)
where Rt is the recruitment for year t, σ Rdev is the standard deviation of the estimated recruitment deviations for all the years included in the assessment, and Rdevt is the recruitment deviation from the mean value for year t and is defined as √ √ Rdevt = Rho Rdevt−1 + 1 − Rho Epst (A.2) The Rho values of 1 and 0 suggest a perfect and no autocorrelation in recruitment between the two neighboring years, respectively (Breen and Kim, 2003). Epst describes the level of Rdevt that is determined in year t. Thus, the parameters to be estimated for recruitment are Epst , R0 , and Rho. If there are n (thus there are n Epst ’s) years of data, the number of parameters to be estimated are n + 2. If these parameters are estimated, the annual recruitment can then be estimated using Eqs. (A.1) and (A.2). A.3. Population-at-size model
Appendix A. Model descriptions A.1. Growth model To reduce the complexity of the stock assessment model, the growth model was determined outside the stock assessment modeling process. A growth transition matrix was developed to describe the probability of a sea urchin growing from one size class to another. Sixty-one size classes were established from 40 to 100 mm (test diameter midpoint values) with the width of a size class of 1 mm. The development of the growth model and the derived growth transition matrix used in the simulation is detailed in Chen et al. (2003). A.2. Modeling recruitment No functional relationship is assumed between recruitment and spawning stock biomass. Instead, we estimate an average recruitment over the modeling time period (R0 ), a parameter (Epst ) describing the variation in year t from the average recruitment R0 , and the inter-annual dependency coefficient (Rho) describing
Each year the population model calculates the target target biomass, Bt , as target Bt = (A.3) Pk,t Sk Nk,t wk k
where Pk,t is a switch that determines whether size class k is within the legal sizes and Sk is fishery selectivity, wk is the average weight of sea urchin in size class k, and Nk,t is the number of sea urchins of size k in the population in the beginning of year t. The exploitation rate, Ut , is calculated as Ut =
Ct target Bt
(A.4)
where Ct is the catch in year t observed in the fishery. The survival rate from fishing, SVk,t , can be calculated as SVk,t = 1 − Ut Sk Pk,t
(A.5)
where Sk is the selectivity of the sea urchins in size class k. Thus, the number of sea urchins in size class k
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M. Kanaiwa et al. / Fisheries Research 74 (2005) 96–115
in year t, Nk,t , can be calculated as Nt = SVt−1 Nt−1 Ge−M
A.5. Observational models (A.6)
where matrices N and SV are indexed with year (t and t − 1), and G is a growth transition matrix (see Chen et al., 2003 for its derivation and parameterization for the Maine sea urchin population), M is natural mortality, and Nt−1 is the transposed matrix. We assume that the size distribution of recruitment coming into the population is proportional to the probability of the sea urchins in the first size class growing into other size classes. Thus, the number of sea urchins in the stock in year t, after including the recruitment of the year, becomes Nt + Rt G. A.4. Predictive model Using the above population dynamics models we can generate a model fishery. The following predictions for key fishery outputs are made for the model fishery: total biomass Bttotal = (A.7) Nk,t wk k
(A.8)
catch per unit of effort in the fishery legal
= q1 Bt
where q1 = of years
n
(A.9) legal
obs t=1 (It /Bt n
)
and n is the total number
pred
stock abundance index At
= q2 Bttotal
(A.10)
size composition of the catch pred
Pk,t Sk Nk,t pred Cpk,t = pred k Pk,t Sk Nk,t where Sk =
1 1+exp(h(lk −l50% )) ,
(A.11) and
size composition of the sea urchin stock pred
Nk,t pred Spk,t = pred k Nk,t
exp(εIt )
(A.13)
and observed and predicted size compositions of the catch described as pred
Cpobs k,t = Cpk,t + εCpk,t
(A.14)
where εIt and εCpk,t follow normal and multinomial distributions, respectively. The data collected in the fishery-independent survey program are related to predicted survey data using the following two observational models: pred
Aobs = At t
exp(εAt )
(A.15)
for survey abundance index, and pred
k
pred
pred
Itobs = It
Spobs k,t = Spk,t + εSpk,t
legal-sized sea urchin biomass legal Bt = Pk,t Nk,t wk
It
Various observational models were developed to relate fishery predictions made using the above population dynamics models with fishery data measured in the sea urchin fishery. They include observed and predicted catch per unit of effort (CPUE) described as
(A.12)
(A.16)
for survey size composition data. The error terms εAt and εSpk,t follow normal and multinomial distributions, respectively. A.6. Likelihood functions The likelihood function can be derived based on the distributional assumptions on errors in the observational models (i.e., Eqs. (A.13)–(A.16)). Following Chen et al. (2000), we incorporated three forms of likelihood functions for CPUE and survey abundance index: log-normal, robust log-normal, and t-distribution; described, respectively, for CPUE (i.e., Eq. (A.13)) as pred √ 1 p(Itobs |It ) = 2πσˆ I obs t t (ln(I obs ) − ln(I pred ))2 t t × exp − 2(σˆ I obs )2 t
(A.17a)
M. Kanaiwa et al. / Fisheries Research 74 (2005) 96–115
pred
p(Itobs |It
)=
t
1
√ 2πσˆ I obs t
(ln(I obs ) − ln(I pred ))2 t t exp − + 0.01 2(σˆ I obs )2
Similar functions can be derived for the survey abundance index (i.e., Eq. (A.15)). Detailed comparative studies of the three likelihood functions were done in Chen et al. (2000). We focused this study on the tdistribution, as recommended by Chen et al. (2000). The likelihood function for size compositions of catch, following Chen et al. (2000), is described as L(Cpobs k,t ) t
k
× exp
(A.17b)
t
−2.5 pred 2 obs (ln(It ) − ln(It )) 1.329 pred p(Itobs |It ) = 1+ √ 2 4(σˆ I obs ) 4π t t
=
115
1 obs 2π Cpobs k,t (1 − Cpk,t ) + 0.1/Ω pred 2
−ωk,t (Cpobs k,t − Cpk,t )
obs 2{Cpobs k,t (1 − Cpk,t ) + 0.1/Ω}
+ 0.01 (A.18)
where Ω is the total number of size bins and ω the effective sample size. It is an adjusted multinomial function robust to outliers in the data (Fournier, 1996). A similar likelihood function can be defined for size compositions of stock estimated in the fishery-independent survey.
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(A.17c)
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