© 2006 International Council for the Exploration of the Sea
Variation in the catchability of yellow perch (Perca flavescens) in the fisheries of Lake Erie using a Bayesian error-in-variable approach
a Department of Fisheries and Wildlife Sciences, Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0321, USA
b Ontario Commercial Fisheries' Association Box 2129, 45 James Street, Blenheim, ON N0P 1A0, Canada
c Department of Integrative Biology, University of Guelph Guelph, ON N1G 2W1, Canada
*Correspondence to Y. Jiao: tel: +1 540 2315749; fax: +1 540 2317580. e-mail: yjiao{at}vt.edu.
Catch per unit effort (cpue) from fisheries, and abundance or biomass indices from fishery-independent surveys are often used to infer the dynamics of exploited populations. To do this, cpues and survey indices are usually assumed to be proportional to population size or biomass. Four sources of data on the cpue of yellow perch (Perca flavescens) in Lake Erie were available to evaluate this assumption: commercial gillnet and trapnet fisheries, an angling fishery, and a fishery-independent gillnet survey. The relationships between fisheries cpue and population biomass (estimated from an age-structured model), and between fisheries and survey cpues were analysed by error-in-variable (EIV) models because of the absence of independent estimates of population size. Cpues were not proportional to population size, estimated by biomass. Catchabilities varied widely among fisheries (gear types), time period, and areas (management units) within Lake Erie. A spatial EIV model showed that the migrations among management units were considerable. The whole-lake spatial EIV model showed that cpues were not proportional to population size.
Keywords: Bayesian, catchability, error-in-variable, Lake Erie, yellow perch
Received 15 February 2006; accepted 16 July 2006.
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Introduction |
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 Top Introduction Material and methodsResultsDiscussionReferences |
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The relationship between catch per unit effort (cpue) and population size is commonly assumed to be proportional (Quinn and Deriso, 1999), so catchability, the proportion of a population captured per unit of effort, is independent of population size under this assumption. Several studies have demonstrated non-linear relationships between cpue and population size, implying that catchability is a function of population size. Many factors may influence catchability, in fisheries and in fishery-independent surveys, including spatial and temporal aggregation of fish, non-random search behaviour of fishers, changes in fishing power, gear selectivity, and gear saturation (Paloheimo and Dickie, 1964; Pope and Garrod, 1975; MacCall, 1976; Crecco and Savoy, 1985; Crecco and Overholtz, 1990; Rose and Leggett, 1991). Assuming that catchability is independent of population size, when in fact it is not, may result in a bias in estimates of population size and incorrect inferences regarding population dynamics, which increase the potential for mismanagement of a fishery (Crecco and Overholtz, 1990; Walters and Maguire, 1996; Harley et al., 2001).
Lake Erie is the shallowest and most productive of the Great Lakes, and its fisheries input into local economies amounts to several billion dollars (Leach, 1999). Yellow perch (Perca flavescens) is one of the main fisheries in Lake Erie (Ohio Department of Natural Resources, 2005). From west to east, Lake Erie is divided into four management units (MU1–MU4) for yellow perch (Figure 1). All data on the cpue of three yellow perch fisheries (commercial gillnet, commercial trapnet, and sport angling), and partnership indices for fishery-independent gillnet surveys, are collected on the basis of these management units (Figure 2). Here, we refer to the partnership indices for fishery-independent gillnet surveys as survey cpue. The discrepancies in the cpues of the three fisheries and survey cpue over time suggest that some of these series may not be directly proportional to abundance, as assumed in the current statistical catch-at-age model (Myers and Bence, 2001).
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When analysing the catchability or the relationship between cpue and population size, one key question is how the form of the relationship between cpue and population size in the absence of independent estimates of population size can be identified. Many stock assessment biologists are well aware that indices based on cpue may not be proportional to population size (or biomass), but in the absence of a method to quantify the bias, assessment biologists fall back on starting with the simplest plausible relationship, i.e. proportionality. The biomass estimates from the age-structured model are not independent of the various cpue indices, because every index was used in fitting the model (which assumes a proportional relationship). Because both cpue and population size were estimated with substantial error, an error-in-variable (EIV) method can be used to analyse the form of the relationship between the two (Richards and Schnute, 1986).
In this study, we use a Bayesian approach to solve the EIV model in analysing the relationship between cpue and population biomass. WinBUGS (Spiegelhalter et al., 2004) was used for the purpose. The idea of the EIV model is to estimate a latent or "true" biomass, then to fit the parameters and all observation errors simultaneously when estimating the relationship between cpue and biomass. The EIV approach helps to answer the question of how to identify the form of the relationship between cpue and population size in the absence of independent estimates of population size. The EIV model was fitted separately to each management area and to each fishing method.
There is mixing of yellow perch among the four management units. A spatial correlation model with error-in-variable was therefore used to analyse the lake-wide relationship between cpue and population biomass. Errors in biomass estimates were correlated across areas, and movement of fish from one area to another produced positive errors in one and negative errors in the other, so making a case that the "stocks" were not well separated. Such correlations were also examined in a Bayesian manner, with biomass estimates from the age-structured model being used as observations, with spatial correlation errors.
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Material and methods |
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TopIntroductionMaterial and methodsResultsDiscussionReferences |
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Gillnetting is used by the commercial fishing industry in Ontario waters of Lake Erie. Gillnet cpue data extend from 1975 to 2003 in all four management units (Figure 2). Trapnetting gear was employed to a lesser degree. The Ohio trapnet cpue data in MU1 and MU2 extend from 1975 to 2003, in MU3 from 1985 to 2003 (except for 1986), and in MU4 from 1986 to 2003 (Figure 2). The cpue data for the angling fishery extend from 1975 to 2003 in MU1, MU2, and MU3; and from 1989 to 2003 in MU4 (Figure 2). Since 1989, a fishery-independent gillnet survey has been conducted in Ontario waters, with the exception of 1989 in MU1 and MU2, 1996 in MU3, and 1996 and 1997 in MU4 (Figure 2). The survey indices of relative abundance are the fishery-independent Ontario Partnership Index Fishing Survey catch data (Ontario Ministry of Natural Resources, 2000), which we refer to as survey cpue in this article. They were estimated in units of kg set–1, i.e. the total weight (kg) of fish caught divided by the total number of gillnet sets during the survey. The yellow perch stock biomass of the four MUs are from the most recent assessment done by the Lake Erie Committee (LEC), using an age-structured model (Yellow Perch Task Group, 2004). The gillnet and trapnet fisheries and angling fisheries cpues, and the survey cpue were used to tune the population abundance in the age-structured model using a proportional relationship, but with blocked catchabilities based on regulation changes of the yellow perch fisheries and ecosystem changes in Lake Erie (Table 1; Yellow Perch Task Group, 2004).
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Several models have been proposed to describe the non-proportional relationship between cpue and biomass (Peterman and Steer, 1981; Bannerot and Austin, 1983; Richards and Schnute, 1986; Arreguín-Sánchez, 1996). Commonly used models include the power model, the logistic model, and the polynomial model. Use of the power model has a long history in modelling the relationship between cpue and biomass (Ultang, 1976; Arreguín-Sánchez, 1996), and is now used widely because of its parsimony (Harley et al., 2001). We chose the power model in this study for that reason. An EIV with a Bayesian approach was used to estimate the parameters of the power model (Gustafson, 2003).
Cpue and catchability
The power model of the relationship between cpue (I) and biomass (B) was
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| (1) |
and ß are parameters used to model the relationship between I and B. Catchability (q) is the proportion of the population captured per unit of effort, i.e.
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| (2) |
We also explored the catchability of fisheries by analysing the relationship between fisheries cpues and survey cpue. The survey cpue from the fishery-independent survey is expected to be free of many of the biases caused by changes in catchability that affect the commercial fishery (Swain et al., 1994). Yellow perch gillnet surveys were made by the same methods each year; there were no obvious changes in fishing power or efficiency. The relationship between fisheries cpue and survey cpue can be written as
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| (3) |
and ß are the parameters used to model the relationship between fisheries cpue and survey cpue. The significance of the relationship between cpue or survey cpue and biomass was assessed. A ß value not significantly different from 0 implied a non-significant relationship between cpue or survey cpue and biomass (Equations (1) and (3)). We also investigated evidence for apparent changes in catchability in the gillnet fishery, the trapnet fishery, the angling fishery, and the survey. A ß value not significantly different from 1 implied a constant catchability of the fishery relative to the biomass or survey cpue (Equation (2)). A Bayesian error-in-variable approach was used to estimate parameters in the power calibration model in this study.
Error-in-variable (EIV) model
We used EIV models (Gustafson, 2003) to estimate the parameter values of and ß in Equations (1) and (3) because of the absence of an independent estimate of population biomass and measurement error of the biomass and survey cpues.
Lognormal error structure is used for both B and I; Equation (1) can be rewritten as
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| (4) |
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in Equation (4) is the true value of biomass, 
1 is independent of B, has mean 0 and variance 
, and B in Equation (4) is the observed value of biomass, 2 is independent of B, with mean 0 and variance 
. The same modelling approach is used to analyse the relationship between fishery cpue and survey cpue. The Bayesian method was used to estimate g,i, ßg,i, 
, and 
. A WinBUGS code for the Bayesian EIV model described above is available upon request.
Error-in-variable with spatial consideration in terms of catchability in the whole lake
In this part of the study, we also analyse the lake-wide cpue and population biomass relationship using a spatially correlated EIV model.
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| (5) |
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g and ßg are the same in the whole lake for a specific fishing method. The Bayesian method was used to estimate g, ßg, , , and the spatial correlation matrix W. A WinBUGS code for the spatial EIV model described above is available upon request.
Bayesian method and priors
A Bayesian method was used to estimate the parameters in the EIV models. WinBUGS software was used. WinBUGS is a numerically intensive software package that implements general Bayesian models using "Metropolis-Hasting within Gibbs sampling" (Gilks, 1996). Bayesian implementation of these models requires specification of prior distributions on all unobserved quantities. Priors for the models above are listed in Table 2. In general, non-informative priors (here, wide uniform distribution) were used for parameters g and ßg; uniform distributions were used for variances and . Uniform prior distributions work better as non-informative priors than inverse-gamma distributions for variance parameters when dealing with multi-level models (Gelman, 2005). Here, we also chose uniform distribution instead of inverse-gamma. However, we did use inverse-gamma to investigate the sensitivity of the results to inverse-gamma priors. Informative distribution was used for the true biomass, and was based on the observed biomass. Correlations between observed biomass and true biomass from the same unit were set to be between 0.5 and 1, and correlations between observed biomass and true biomass from the neighbouring units to be between –0.5 and 0.5.
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To test whether the results are sensitive to the priors, we also did analysis using inverse-gamma IG(0.001,0.001) for variances of
and in the Bayesian EIV model, which is often used as a non-informative prior (Spiegelhalter et al., 2004; Gelman, 2005). For the spatial EIV model, another two sets of priors were used to test the sensitivity to priors. Correlations between observed biomass and true biomass from the same unit were set to be between 0.6 and 1 (or 0.4 and 1), and correlations between observed biomass and true biomass from the neighbouring units to be between –0.5 and 0.6 (or –0.5 and 0.4).
Convergence diagnostics
A critical issue in using MCMC methods is how to determine when random draws have converged to the posterior distribution. Here, three methods were considered: monitor the trace, diagnose autocorrelation plot, and Gelman and Rubin statistics (Spiegelhalter et al., 2004). In this study, three chains were used. After several sets of analysis, for each chain, the first 20 000 iterations with a thinning interval of 10 were discarded, and another 10 000–50 000 iterations were used in the Bayesian EIV model. In the Bayesian spatial EIV model, for each chain, the first 40 000 iterations with a thinning interval of 10 were discarded, and another 20 000–60 000 iterations were used. The final number of the iterations that was used depended on the three diagnostic results.
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Results |
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TopIntroductionMaterial and methodsResultsDiscussionReferences |
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The gillnet, trapnet, and survey cpues showed similar trends over time in MU1 and MU2; trends diverged in MU3 and MU4 (Figure 2). The angling fishery cpue differed from the other three cpues across all four management units.
Biomass of age 2+ yellow perch as estimated by the Yellow Perch Task Group (2004) in the four management units showed different trends before 1988, but similar trends thereafter (Figure 3). Biomass decreased to a low level in the mid-1990s in all management units, then increased quickly until 2001. Trends in estimated biomass in MU2, MU4, and lake-wide were generally similar over time, declining from the 1970s to a very low level in the mid-1990s. Biomass in MU1 peaked in 1979, and biomass in MU3 decreased and remained low from the late 1970s to the mid-1980s, a trend different from that in MU2 and MU4. By 2001, the estimated biomass of yellow perch in MU2 and MU4 was greater than the greatest biomass previously recorded.
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Estimated power parameters (ß) were significantly smaller than 1 (Table 3). Estimated ß values were significantly different from 0 when the long time period (1975–2003) data were used in the gillnet fishery cpue and survey cpue, in the trapnet fishery cpue in MU3, and in the angling fishery cpue in MU1. Most of the fishery and survey cpue analyses resulted in non-significant relationships between cpue and biomass, when the blocked data were used. Obviously, the shortness of the time period is a likely reason.
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Angling fishery cpue resulted in ß values far from 1 in almost all MUs and time periods, which implied that it is not a good indicator of population biomass (Table 3). The gillnet fishery, the trapnet fishery, and the survey cpues resulted in higher ß values, although they were still significantly smaller than 1.
The relationships between fisheries cpues and survey cpue were quite different in the various fisheries (Table 4). The ß values from the gillnet fishery and trapnet fishery in MU1 and MU2 were high and were close to 1. The ß values from the angling fishery and the trapnet fishery in MU3 and MU4 were small; most of them were not significantly different from 0 (Table 4).
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When the whole-lake gillnet and trapnet and angling cpues were functions of the biomass, each resulted in ß values significantly different from 1 (Table 5). The results when fisheries cpues were used were similar to those when each unit was analysed separately. The results when survey cpue was used were quite different. The resulting ß values were smaller or even negative, the diagonal values in the correlation matrix were higher, and other b values in the correlation matrix were smaller than those when fishery cpues and biomass were analysed. The results using survey cpue in the spatial EIV model implied less migration among management units. In general, the results from the spatial EIV model suggest that migrations among MUs are considerable. When the whole-lake gillnet cpue was a function of survey cpue, the ß value was significantly larger than 0, but it was not significantly different from 1. The whole-lake trapnet and angling fishery cpues resulted in ß values not significantly different from 0 (Table 5). Surprisingly, the relationship between whole-lake survey cpue and biomass was not significant.
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The analysis indicated that estimated ß values and their confidence intervals were robust to priors of variances of
and in the Bayesian EIV model when uniform distribution and inverse-gamma distribution were used, although their left tails seemed to be influenced by the priors (Table 6). The estimated ß values were robust to the priors of variances of bs in the spatial correlation matrix W (Table 7). This was also true for most estimates of bs. The bs estimates tended to be more sensitive than the other bs in the correlation matrix W.
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Discussion |
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TopIntroductionMaterial and methodsResultsDiscussionReferences |
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For yellow perch fisheries in most management units in Lake Erie, the relationship between cpue and biomass is not proportional, and catchability is not independent of population biomass. The gillnet and trapnet fisheries cpues and the survey cpue were mostly correlated with the estimated population biomass, although in some time periods and some management units, the resulting ß values were smaller when trapnet cpue was used.
The angling fishery cpue was not a good indicator of population biomass based on the estimated ß values and their significance levels. The angling fishery can be substantially influenced by yellow perch population biomass, prey abundance, weather, and economic factors, such as fuel prices. An increase in stock size is followed by an increase in angler effort and the entry of less experienced, less efficient anglers into the fishery (M.L. Jones, Michigan State University, pers. comm.). The opposite is thought to occur when there is a decrease in stock size and the proportion of experienced, more efficient anglers increases. Selection of fish based on size by anglers may worsen the relationships between angling cpue and estimated population size or survey indices. It may be prudent to use angling harvest data only to reflect estimates of the components of the population removed to project the following year's population size, and to place less emphasis on the relationships between angling and effort to estimate population size in catch-at-age modelling. Further studies on angling catchability, fishing effort, and their relationship to population abundance, economics, and the sampling method are suggested.
The survey cpue examined here was represented exclusively by fishing within Ontario waters, in contrast to sport and trapnet data, which were exclusively from American waters. Some lack of proportionality could be attributed to perch densities that differ between waters from which the gillnet index was deduced vs. those in which sport and trapnet fisheries operated. Both the influence of environmental changes on the catchability of different fishing methods and fishing areas, and the short data series are reasons for the lack of correlation between fishery cpues and survey cpue. Here, water clarity changes because of the invasion of zebra mussel (Dreissena polymorpha) are most likely the main reason for the environmental changes during this time period.
Based on our findings, no proportionality exists between cpue and biomass. Power models are suggested for use in tuning population models. Closer examination of the estimates of variation with respect to gear-related catchability is now essential in assessing the goodness-of-fit, and in tuning the population biomass and stock assessment of yellow perch fisheries in Lake Erie.
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Acknowledgements |
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This research was supported by Virginia Polytechnic Institute's new faculty start-up funding to YJ and by the Ontario Commercial Fisheries' Association (OCFA) Grant-in-aid of Research to TN, principal investigator for the Decision Analysis and Adaptive Management (DAAM) Project for the Lake Erie walleye and yellow perch fisheries, in the Department of Integrative Biology, University of Guelph, Canada. We thank Bruce J. Morrison, John Casey, and an anonymous reviewer for their thorough reviews of the manuscript. We also thank the Yellow Perch Task Group and the Great Lakes Fisheries Commission for providing data.
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References |
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