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Perhaps the most influential, but not necessarily the best, works on shark stock assessment were those of Holden in the 1960s and 1970s. Holden (1977) was one of the first scientists to consider the problem of shark fisheries stock assessment from a general point of view. He correctly pointed out that sharks were different from bony fishes in terms of their biology but unfortunately he wrongly concluded that classic fisheries models such as stock production models could not be applied to sharks and rays. Holden dismissed these models and called for new models to be developed. He stated that the assumptions of surplus production models regarding immediate response in the rate of population growth to changes in population abundance and independence of the rate of natural increase from the age composition of the stock do not hold for sharks. These conclusions were based mainly on the time delays caused by the longer reproductive cycles of sharks and their reproductive mode, which in his view would cause a linear and direct stock-recruitment relationship.
Because of his influential paper, surplus-production models have been mostly ignored for shark stock assessment and scientists and non-scientists reading Holden's papers have sought new methods and models for dealing with shark fisheries stock assessment. For a while, Holden's thoughts influenced the works of other scientists who opted for the more detailed approach offered by age structured models (e.g. Wood et al., 1979; Walker, 1992).
The main problem of surplus production models is not that they are inadequate when applied to sharks but the way in which they were being applied. A paramount obstacle for the use of classic surplus production models in the 1960s and part of the 1970s was the equilibrium constraint (see Section 10.6 on fitting models data). At that time, due to the lack of readily available computers to perform iterative search algorithms, scientists engaged in surplus production model-fitting were forced to assume that populations were in equilibrium at all exploitation levels (i.e. that every catch observed was sustainable) to simplify the process of fitting surplus production models to data.
The dangerous consequences of this assumption are well known and explicitly warned against in fishery text books (Pitcher and Hart, 1982; Hilborn and Walters, 1992). However, the personal computer revolution has helped to overcome the equilibrium constraint through the availability of non-linear optimization routines which are accessible to virtually any fishery scientist in the world today. The diversity of approaches this offers for fitting surplus production models has translated into a new era of popularity for the utilization of what are presently known as dynamic surplus production models that have been applied to organisms as slow-growing as whales and sharks (Punt, 1991; Prager et al., 1994; Polachek et al., 1993; Babcock and Pikitch, 2001). Perhaps the most interesting outcome of all this re-appraisal of surplus production models is the view that most of the problems associated with successfully applying them are due to the quality of the fisheries data (Hilborn, 1979; see also Section 10.3), and the finding that simple surplus production fishery models can sometimes perform better than the more elaborate and biologically detailed age-structured approaches (Ludwig and Walters, 1985, 1989; Ludwig et al., 1988; Punt, 1991).
One of the reasons for the difficulty in applying these models to sharks is that the data available on shark fisheries and our knowledge about shark biological parameters may not be adequate. This is expressed clearly in the work of Anderson (1990), Anderson and Teshima (1990) and Bonfil (1996). In fisheries science, independent of the species in question, the most common problem is that lack good and sufficient data that lack contrast in the data when it is available. Another problem often overlooked is that the more ‘realistic’ age-structured models also pose problems in their application. Age-structured data are much more difficult and expensive to obtain. Further, the life cycles of most shark species, even in terms of the basic parameters of age, growth and reproduction, have just started to be unveiled during the last 20 years, and this only in the case of a handful of stocks [see Pratt and Casey, (1990) and Cortés, (2000) for reviews]. In addition, there are some relevant areas of elasmobranch population dynamics that are still largely unknown. For example: empirically derived stock-recruitment relationships have never been documented for any elasmobranch, although a strong relationship is suspected due to the reproductive strategies of the group (Holden, 1973; Hoff, 1990); the size, structure and spatial dynamics of most stocks of elasmobranchs are almost totally unknown. Inadequate knowledge of migration routes, stock delimitation and movement rates amongst them, can seriously undermine otherwise “solid”assessments and management regimes.
Hoff (1990) favored the use of dynamic surplus-production models for shark stock assessment for a variety of reasons. Punt (1988, 1991) also reported dynamic surplus production models to be the most reliable for management of slow-growing resources with limited reproductive potential such as baleen whales, when tested using a simulated fully age-structured population. Similar positive results were reported with a Schaefer model for a swordfish age-structured simulation model (Prager et al., 1994). The results of Bonfil (1996) suggest that surplus production models are good enough for shark biomass assessment but less so for management parameter estimation. He found that although generally inferior to the Deriso-Schnute model (Section 10.4.1 below), surplus production models are capable both of estimating biomass benchmarks and obtaining good biomass fits for most of the scenarios analysed.
The best advice in regard to model choice for elasmobranch stock assessment is found in Section 2.2.3. Surplus production models can, and should, be applied to elasmobranch fisheries as they are one of the easiest to implement, but their results should be taken as a first and preliminary assessment. A complete and reliable assessment should not stop there but attempt to apply delay-difference and fully-age structured models as soon as that is also possible.
10.2 SURPLUS PRODUCTION MODELS
10.2.1 Ease of application
These models are among the simplest and most widely used in stock assessment. They are easy to use because they require only two or three types of data. These models are flexible and have different variations; the Schaefer, Fox, and Pella-Tomlinson models are some of the best known.
Surplus production models (SPM) are based on the following principles:
Next biomass = last biomass + recruitment + body growth -catch -natural mortality
If there is no catch
Next biomass = last biomass + production -natural mortality
where production is the sum of recruitment and body growth, and
Surplus production = production -natural mortality
New biomass = last biomass + surplus production -catch
10.2.2 Logistic growth and the Schaefer model
Population growth has been typified in several ways, but the logistic model of population growth has been found to fit a large number of populations both in nature and in captivity. This model is expressed in the following way (differential equation or continuous model):
where B = biomass, K = carrying capacity, and r = intrinsic rate of population increase.
The carrying capacity of the system, K (or B∞), is the maximum population size that can be achieved. Mortality, age-structure, reproduction, and tissue growth are all captured by a simple parameter called the intrinsic rate of increase, or intrinsic rate of production, r. In theory, the intrinsic rate of increase is fully realized at the lowest population level while the finite rate of population growth is highest at the midpoint of K. Figure 10.1 illustrates some of these concepts and shows the trajectories of population growth for two different values of r.
Examples of population growth according to the logistic model. Two different r values are shown.
The Schaefer model is the most commonly used among SPMs (known also as Biomass Dynamic Models). This model is based exactly on the logistic population growth model. The continuous logistic model explained above can also be written in discrete form in the following way (Hilborn and Walters 1992):
When catch is included in the above equation we obtain the discrete version of the Schaefer (1954) surplus production model:
|Ct = qfBt||(10.4)|
and C is catch, q is the catchability coefficient and f is effort. In the Schaefer model above, the middle term is known as the surplus production. If the surplus production is greater than catch, population size increases; if catch equals surplus production, catch is sustainable and the population size remains constant (Bt+1= Bt); if catch is greater than surplus production, population size declines.
The Schaefer model has the following assumptions:
In practice, many of the above assumptions are not met but this does not mean that the method cannot be used. As long as it is used critically, the Schaefer model is a powerful tool for an initial assessment of a stock. The management parameters of importance from the Schaefer model are given by:
MSY = r K/4
Optimum effort (fMSY) = r/2q
10.2.3 Fox and Pella-Tomlinson models
There are other SPMs that have been proposed to more ‘realistically’ describe fisheries. Fox (1970) describes a model that is based not on the logistic population growth model but on the Gompertz growth model. The Fox model equation is:
The model is supposed to be more realistic because it assumes that the population can never be totally driven to extinction, something that sounds intuitive but may be wrong in the light of the severe depletion of fishery resources in recent years and well-documented human-caused terrestrial species extinctions. The management parameters of the Fox model are given by:
MSY = rKe-1/InK
BMSY= Ke-1 fMSY= r/q InK
Pella and Tomlinson (1969) proposed a generalized model that can take any shape, including that of the Schaefer (m =2) and Fox (m = 1) models.
However, there is a price to be paid for this ‘improvement’, one must estimate an additional parameter (m) to fit the model to the data. This model is not much more useful because despite its ‘flexibility’ the fit will probably be worse than with the Schaefer or Fox models as there is often an inverse relationship between the number of parameters to be estimated and the performance of the models (Hilborn and Walters 1992).
10.2.4 Data requirements
In its simplest form, SPMs have only two data requirements:
The abundance data can be constructed if effort data is available corresponding to the time series of catches and if we assume that CPUE is linearly related to abundance. The assessment can greatly benefit if an estimate of the virgin biomass is also available, but this is not essential for applying the model. The longer the time series and the better the quality of these data, the greater chances of having a good assessment. Modern implementation of SPMs through Bayesian approaches can incorporate additional heterogeneous information such as estimates of the intrinsic rate of increase of the stock and estimates of historical catches for which no effort or abundance index is available (McAllister and Pikitch 1998a; Apostolaki et al., 2002; Cortés et al., 2002).
10.2.5 Advantage and disadvantages of Surplus Production Models
These models offer an excellent cost/benefit ratio. Data requirements are modest compared with age-structured models yet, SPMs can yield critical information for assessment and management such as estimates of virgin and current biomass, level of population depletion, MSY and optimal effort (fopt). Most importantly, they can be used to make projections of the population under several scenarios of management (quotas or efforts) and to evaluate the outcomes of each scenario. This is possible because SPMs explicitly incorporate the time variable unlike demographic analysis and yield-per-recruit (Y/R) models. Thus, they are dynamic models that can be used to make predictions.
A further advantage (simplicity), but at the same time criticism (lack of biological reality) of SPMs is that they do not include age structure. They assume that all the processes occurring in a population can be captured by the simple processes described above while ignoring the size or age structure of the population and the dynamics of different parts of the population. Another common criticism of SPMs, especially in respect to elasmobranchs, is that they do not incorporate time delays between reproduction and recruitment. While this is true, in practice this seems to be the least of the problems for the application of SPMs to real shark fisheries. Often the shortage and bad quality of the data available for the assessment are more pressing problems. Bonfil (1996) using Monte Carlo simulation showed, that despite criticisms of these models, SPMs can be useful for certain situations when applied to elasmobranch fisheries data.
10.2.6 Examples of use of Surplus Production Models in shark stock assessment
Aasen (1964) was the first to apply the Schaefer model to a shark fishery and probably the first scientist to perform a stock assessment of an elasmobranch species. Although there was a dominant view 40 years ago that these models were not adequate for sharks due to incompatibility between the assumptions of the models and the biology of sharks, they are now widely accepted as applicable although not necessarily recommended as the best. They have been used in the multispecies shark fishery of the east coast of the USA (Otto et al 1977; Anderson 1980; McAllister and Pikitch 1998a; McAllister et al., 2001; Cortés 2002; Cortés et al., 2002), for the kitefin shark fishery in Portugal (Silva 1987), the Australian fishery for school and gummy sharks (Xiao 1995; Walker 1999) and in the multispecies skate and ray fishery of the Falkland Islands (Agnew et al., 2000).
10.3 YIELD PER RECRUIT MODEL
Beverton and Holt (1957) first developed this model, which provides a steady-state (static) view of the population that allows determination of the catch or yield relative to recruitment (catch divided by recruitment, thus the yield per recruit, or Y/R, name of the technique) that can be obtained from a stock at different levels of fishing mortality F (which is dependent on effort) and age of entry to the fishery. The method is described in detail by Pitcher and Hart (1982), Megrey and Wespestad (1988), and Quinn and Deriso (1999).
The model describes the population in terms of the biological processes of growth, recruitment and mortality and treats the exploited population as the sum of its individual members. It has more biological detail than surplus production models but is not as powerful and detailed as the fully age-structured models treated below. Also, it is inferior to SPMs in the sense that it is static, assumes that there is no dependence between stock size and recruitment, and cannot provide estimates of absolute biomass or be used for making projections of stock size according to different management strategies. Its main utility is that it indicates if the fishery is catching fish at an age that is too early or too late to obtain the maximum biomass relative to recruitment and if the level of fishing mortality is too high or could be higher.
10.3.2 Data requirements and assumptions
The calculation of yield per recruit requires the following data:
The method has the following assumptions:
This model is based on three equations:
(i) Von Bertalanffy Growth Model (in weight):
|Wt = W∞ (1 - e-k(t-t0))3||(10.7)|
(ii) Exponential survival model:
|Nt = R . e- M(tc - tr) . e - (M + F) (t - tc)||(10.8)|
where R is the number of recruits, tc is age at first capture and tr is age of recruitment to the stock.
(iii) General yield equation:
where Y represents yield (catch).
These three equations can be integrated to obtain the yield equation of Beverton and Holt (1957):
Because the level of recruitment is not known, the above equation is usually expressed in relative terms, as yield per recruit:
The model predicts the level of yield (catch) that can be obtained depending on the age of entry and maximum age in the stock and the level of natural and fishing mortality.
This model allows managers to investigate the effects of varying fishing mortality (F) or age of first entry (tc) on yield. One disadvantage of the model is that the shape of yield is completely determined by growth and mortality. If the stock has a low rate of growth and high M the yield curve is asymptotic (this wrongly suggests yield does not decrease as you fish harder and harder). Conversely, if the stock has rapid growth rate and a low M the yield curve is dome-shaped.
10.3.4 Advantages and disadvantages
The main advantages of this method it that it is relatively simple to implement and does not require historical data on catch and effort. It is a step forward from demographic methods because it informs within a relatively simple implementation procedure if fish are being exploited at the right age (or size) and also if fishing is at the right intensity. Using this method advice can be provided on the best age of entry to the fishery and an adequate level of effort, thus offering information that can potentially translate into direct management recommendations such as changing the fishing mesh size of gillnets used to catch sharks, or taking a number of boats out of the fishery to reduce fishing mortality.
The main disadvantages are that the method provides no estimate of the absolute biomass of the stock and gives only limited advice on management actions. As with life tables, a disadvantage of this method is that it is not dynamic (there is no time variable) and therefore cannot be used to make predictions. Nor does it incorporate density-dependent processes such as stock-recruitment relationships. Other disadvantages of the model are that it unrealistically assumes constant growth and mortality rates; it is more expensive to implement than SPMs as age needs to be frequently determined requiring large samples of fish; the curve shape is predetermined and inflexible; the model predicts yield even at infinite effort, which is unrealistic; and yield is not expressed in absolute terms so the real magnitude of the catch cannot be known.
Using the Y/R method alone can be misleading as pointed by Grant et al. (1979). These authors suggested that the recommended 10-fold increases in fishing mortality from their Y/R assessment was a bad advice as only a 2-fold increase could already reduce the reproductive stock of school sharks to less than half of its original abundance. Using a modified demographic method Au and Smith (1997) showed that the estimates of Y/R obtained by Smith and Abramson (1990) for the leopard shark (Triakis semifasciata) were considerably lower after adjusting for the effect of reduction in recruitment due to fishing. Also, Rago et al. (1998) found that the optimum age of entry predicted by the Y/R model would lead to recruitment failure and stock collapse in spiny dogfish (Squalus acanthias) because of the late age of maturity in this species. Another problem of the Y/R method is that a poor estimation of growth or mortality can strongly influence conclusions and lead to decisions that could put the stock in jeopardy.
The Y/R method has been used for stock assessment of school sharks (Grant et al., 1979), for little skate (Waring, 1984), for leopard shark (Smith and Abramson, 1990), for silky sharks (Bonfil, 1990), for sandbar sharks (Cortés, 1998) and for porbeagle (Campana et al., 1999, 2001). To my knowledge this method has not been used as the main basis for the management of any elasmobranch species.
10.4 DELAY-DIFFERENCE MODEL
10.4.1 Application and assumptions
The delay-difference model of Deriso (1980) is a clever simplification that allows the inclusion of biological information of the species to be taken into account in a simple way. This model belongs to an intermediate class known as partially age-structured models. They represent a step forward from the rather simple surplus-production models that ignore biological processes like recruitment and individual growth, while avoiding the demanding data requirements of the more sophisticated fully age-structured models and they consider age structure implicitly, not explicitly.
The biological realism of the delay-difference model arises from terms for recruitment, natural and fishing mortality and growth. Yet, this model can be simplified to be fitted to data on catch and effort and an index of abundance, as in the case of surplus production models. Additional requirements are knowledge of the growth in weight of the species and an estimate of natural mortality. An important advantage of the model is that it has fewer model parameters to be estimated in comparison to fully age-structured models. Thus, it can be applied to fisheries with limited amounts of data while still offering a more realistic representation of population dynamics.
The delay-difference model of Deriso (1980) was further generalized by Schnute (1985). The model incorporates four main types of biological information: body growth, recruitment, survival and a measure of age-structure. The main formula of the model links present available biomass (exploitable biomass or that recruited to the gear) to available biomass and population numbers from the previous year. The advantage of the model lies on several simplifications that allow the incorporation of important population dynamics processes into a simple equation. However, perhaps its more important characteristic is that the model allows for time-lags in the dynamics of the stock, such as are found in species with slow growth and late age of entry to the fishery. This ability to take into account time-delay is what gives the model its name of ‘delay-difference’ model. The derivation of the delay-difference model here is taken from Hilborn and Walters (1992).
The model assumes that body growth of the exploitable stock can be represented by a linear function (the Brody equation):
|wα = α + ρ wα+1||(10.12)|
where wa is body weight at age a, and α and ρare constants. This equation states that after a certain age, the typical von Bertalanffy model of growth in weight shown in Figure 10.2 can be represented by a linear equation of weight at age a against weight at age a+1.
Individual growth in weight according to the von Bertalanffy Growth Model.
The parameters α and ρof the Brody equation are determined by linear regression as shown in Figure 10.3. This figure shows several possible linear regressions, which differ in how many points are considered for the regression (i.e. different starting points). Which regression is chosen and therefore which parameters α and ρused in the model depends on the age of entry to the fishery.
The delay-difference model also assumes that all fish older than age k (in this particular model age of entry to the fishery) are vulnerable to fishing and have the same natural mortality M. Another simplification of the model is that the total survival rate S at time t is given by
|St = e-z||(10.13)|
and can be decomposed into terms for constant and variable (harvest) survival by
Ford-Walford plot of weights at age. Solid diamonds represent the original data points and each straight line is a linear regression using a different starting age (0, 2, 4, 6, and 7).
|St = Ψ(1 - ht)||(10.14)|
where Ψ is the natural survival rate and h is the harvest rate in year t. This assumes that harvest (fishing) takes place in a short time during the beginning or end of the year. Biomass at age can be represented as numbers at age times average weight at age:
|Bα = Nα wα||(10.15)|
This can be extended for the whole exploited population plus the recruitment R:
where k is the age of recruitment (to the gear or fishery). Population number N, can be expressed as survivors from last year at age a-1, and all the weights at age a can be expressed using the Brody equation, thus arriving at the following formula:
Factoring out terms that do not depend on age results in sums over age k and older for year t-1:
|Bt = St-1αNt-1 + St-1ρBt-1 + wk Rt||(10.18)|
and total numbers in the population are
|Nt = St-1Nt-1 + Rt||(10.19)|
But the term αNt-1 can be expressed as
|αNt-1 = αSt-2 + αRt-2||(10.20)|
and the term α St-2 Nt-2 can be expressed in terms of Bt-1 and Nt-2 using the equation for Bt above as
|αSt-2Nt-2 = Bt-1 - ρSt-2Bt-2 - wkRt-1||(10.21)|
Combining the last two equations and with some more algebraic manipulations gives the delay-difference equation (Schnute, 1985):
|Bt = (1 + ρ)St-1Bt-1 - ρ St-1St-2Bt-2 - ρwk-1St-1Rt-1 + wkRt||(10.22)|
This is the original form of the model and it requires 7 parameters to predict biomass dynamics and to fit the model to catch and CPUE data:
Recruitment can be expressed using either the Ricker or the Beverton and Holt model, simplified by assuming that the population was in equilibrium (i.e. virgin population) when exploitation began. For the Ricker recruitment model the equations are
|Rt+1 = St-k+1e(α' - b'St-k+1)||(10.23)|
For the Beverton and Holt recruitment model the equations are:
Other parameters needed to fit the delay-difference model can be estimated externally or internally with the following assumptions:
This leaves us with only 3 parameters to be estimated during model fitting by non-linear methods:
Thus, the delay-difference model can be simplified by fixing values for the first 3 parameters listed above and fitted to the catch and effort data by finding the values of the last 3 parameters using nonlinear iterative methods such as those included in spreadsheet software. The parameter a or a' of the recruitment model is eliminated by the assumptions above.
10.4.2 Advantages and disadvantages of the delay-difference model
The advantages of this model are:
The main disadvantages of this model are (Hilborn and Walters, 1992):
10.4.3 Use of delay-difference models in shark stock assessment
This simplification of age-structured population dynamics was initially welcomed with excitement but has been seldom used in practice due to the availability of more sophisticated models that can be easily applied thanks to the powerful computer technology now available. The delay-difference model has not been used often for the assessment of shark fisheries but Monte Carlo simulations performed by Bonfil (1996) showed that it performed better than surplus production models for estimating stock size in shark-like fishes. In addition, this model was used as part of the assessment of the school and gummy shark fisheries of Australia by Walker (1999). Cortés (2002) and Cortés et al. (2002) used a simplified version of the Deriso (1980) delay-difference model known as lagged recruitment, survival and growth model as part of the assessment of small and large coastal sharks, respectively, off the U.S. eastern seaboard.
10.5 VPA AND CATCH-AT-AGE ANALYSIS
10.5.1 VPA structures
This family of methods is based on catch-at-age data, i.e. the catch is disaggregated into age-groups. These methods are more detailed and are more realistic than the previously reviewed models. Nevertheless, age-structured models are also extremely data demanding and require much detailed information that is often expensive to obtain.
Age-structured models can be classified into two groups (Hilborn and Walters, 1992): Virtual Population Analysis or VPA and statistical catch-at-age analysis or CAGEAN. These methods are recursive algorithms that calculate stock size based on catches broken down by each age-class. Using these methods it is possible to estimate the magnitude of fishing mortality, recruitment and the numbers at age in the stock for each past year using only catch-at-age and an estimate of natural mortality, M.
VPA does the calculations without having any specific statistical underlying assumptions. In contrast, the more sophisticated CAGEAN methods depend on formal statistical models and have been developed to the degree that various types of data can be integrated in a statistical framework to be used for the assessment. Thus, data on S/R (stock-recruitment) relationships, CPUE time series, biomass time series and others can be integrated into a powerful analysis. The stock synthesis method of Methot (1989) is one of the best examples of a sophisticated CAGEAN model.
10.5.2 Cohorts as the basis of VPA and CAGEAN
A fundamental part of age-structured models is the concept of a cohort. A cohort comprises all the individuals (fish in this case) that were born in the same year. An example of a human cohort is all the persons that were born in 1960. The cohort of 1960 can be followed through time year-after-year by looking at individuals that are age 1 in 1961, age 2 in 1962, and so on. The size of the 1960 cohort in the year 2003 consists of all the individuals that were born in 1960 and have survived to that year. The cohort concept is illustrated in Figure 10.4.
Diagrammatic representation of the 1960 cohort of humans (all individuals born in 1960). N represents the numbers of age A alive each year for cohort 1960.
VPA and CAGEAN are recursive algorithms that track the history of each cohort in the exploited population back in time from the present to the time when each cohort was born or more commonly to the time it recruited to the fishery, i.e. they calculate the number of fish alive in each cohort for each past year, following each cohort through time. They are used to reconstruct the entire exploited population to estimate fishing mortality and numbers at age for each age class in each year.
10.5.3 Virtual Population Analysis
The VPA is also known as a cohort analysis because each cohort is treated separately. The method is based on the following equation:
N alive at beginning of next year = (N alive at beginning of this year) - (catch this year) - (natural mortality this year)
In this particular case recruitment is not considered because we are analysing only a single cohort. We can change the above equation to:
N alive at beginning of this year = (N alive at beginning of next year) + (catch this year) + (natural mortality this year)
Assuming that natural mortality, M, is known and that at some age x there are no more fish alive (that is, all fish in the cohort die after age x) we can iteratively calculate the number of fish alive each year, starting from the oldest age and moving backwards to the youngest.
The basis of the method is the assumption that if we know that this year we have zero fish of the oldest age left alive and we know how many of them we caught last year (in theory those were the last fish of that age left in the sea after those which died of natural causes) and if we know the instantaneous natural mortality rate then, for fisheries where the fishing period is short it can be assumed that there is no natural mortality during the short fishing period so that:
|Nt= Nt+1 + Ct + Dt||(10.27)|
|Nt- Dt= Nt+1 + C||(10.29)|
|Nt- Nt(1-S) = Nt+1 + Ct||(10.30)|
|Nt- Nt+ NtS = Nt+1 + Ct||(10.31)|
|NtS = Nt+1 + Ct||(10.32)|
|Nt= (Nt+1 + Ct) / S||(10.33)|
where N is number of fish, C is catch, D is the number of deaths, t is time (year) and S is the finite survival rate.
The last equation above is the key equation for VPA or cohort analysis, when fishing takes place in a single short period of time during which we can consider M to be negligible. This equation allows the calculation of the numbers alive last year from the numbers this year, the catch-at-age and natural mortality, but because we assume there were no more fish left of the oldest age this year (they were all caught or died) we can calculate the numbers last year with only catch and mortality as parameter values.
An illustrative example of the principles of VPA
Consider a shark species that lives only to 10 years (such as Rhizoprionodon terraenovae) when we assume that all the individuals die. Consider a situation where this species recruits to a fishery at age 3. Further consider that this fishery takes place in only a couple of weeks each year when the fish aggregate to mate. The information needed for a cohort analysis is an estimate of M, which for this stock is considered to be 0.5 (finite rate) and the total catch of fish in each age class for each year. A table with such hypothetical catch data is given in column 3 of table 10.1 and represents the total numbers in the catch for the cohort of Rhizoprionodon terraenovae born in 1980. Using these data and the following equations provides estimates of:
Hypothetical example of data required and the results of a cohort analysis for a short-lived elasmobranch, loosely based on the life history of Rhizoprionodon terraenovae. See text for methods used to calculate each column.
|Year||Age||Catch||Cohort size at start of year||Cohort size before fishery||Harvest rate||Instantaneous fishing mortality rate|
The numbers at age t at the start of the year is given by
|Nt= (Nt+1 + Ct) / s||(10.34)|
For numbers alive at the beginning of the fishery:
The harvest rate is given by:
The instantaneous fishing mortality rate is given by:
|Ft = - In (1-ht)||(10.37)|
Table 10.1 shows the results of the calculations for the cohort born in 1980; but other cohorts can be treated in the same way for a full VPA. For the last cohort in the last year of data we assume there are no fish left, they all die after age 10 in 1990. The table is constructed for this cohort using equation (1) to calculate cohort size at the beginning of each year (note that fish age 10 in 1990 were age 9 in 1989, etc.). The equations for VPA when fishing takes place during the whole year (continuous fishing) are more complicated and can be found in Hilborn and Walters (1992) and Quinn and Deriso (1999) Sparre and Venema (1992) describe a length-based VPA method.
The above example of cohort analysis includes only one cohort. For a complete VPA the same method should be applied for all cohorts that have completely ceased to exist, which is all cohorts that are no longer present in the fishery. One remaining problem after doing this is that there is no information to do the analysis for living cohorts (those still present in the fishery) and these are usually the most important for managers.
One way to solve the problem of incomplete cohorts is to estimate the fishing mortality rate of cohorts currently being fished and use this to estimate the sizes of the incomplete cohorts. Two ways used to estimate the size of current cohorts are: (a) to obtain population size estimates from surveys or mark recapture methods or (b), more commonly, to assume a value for the current F and estimate previous values from there.
This last case, known as the terminal F assumption, comes from the following equation:
There are two ways to estimate F here: (a) from tag-recapture methods or (b), from effort (f) data while assuming that q can be obtained from the relation F = fq.
The catchability coefficients (q) for each age can be obtained from the complete cohorts and assuming q is constant over time we can use that together with effort data to calculate F for each age. Another variation of this approach is known as the ‘tuned’ VPA which first uses the q's from complete cohorts and this is used to derive a new set of catchability coefficients for the incomplete cohorts.
Disadvantages of VPAs
A problem of VPA is that using the wrong M estimate can lead to severely overestimated or underestimated cohort sizes. More worryingly, when catchability increases as the stock declines in size, using the assumption that the terminal F has not changed has been found to introduce great errors, overestimating the stock size and probably recommending larger catches than can be sustained, which can lead to overfishing of the stock.
Another problem is that to obtain the necessary catch-at-age data it is essential to perform routine ageing of large samples of fish from the catch (which is costly) and if the ages are wrongly estimated this will introduce systematic biases to the assessment.
Use of VPAs for shark stock assessment
Smith and Abramson (1990) used a backward VPA in combination with Y/R to estimate replacement rates of leopard sharks off California
10.5.4 Catch-at-age analysis
10.5.4.1 Methods and assumptions
CAGEAN or statistical catch-at-age analysis is similar to VPA but differs in that it uses formal statistical methods to estimate the current abundance of incomplete cohorts. CAGEAN methods also provide a means to estimate natural mortality rate provided that the data have clearly contrasting levels of fishing effort and total mortality rate.
CAGEAN starts by using the catch curve concept (Section 8) to calculate the instantaneous total mortality rate for each age class from the catch at age data. In the same way that curves for the catches at age of one single year are calculated, the same concept is applied to the catches of all cohorts between subsequent years. The equation used for normal catch curves (one single year of data) is a linear regression of the numbers-at-age in the catch (Ca) against age (a), where the slope of the line is the estimate of Z, and the intercept of the Y axis represents the logarithm of the recruitment (R) times the vulnerability to the gear (v):
|ln(Ca) = 1n(Rv) -Za||(10.39)|
To use the catch equation to estimate mortality within a single cohort we use a modified version of the catch curve is used with the following equation:
|ln(Cai) = 1n(Rjv) - Za||(10.40)|
where j denotes a specific cohort. This allows the estimation of the total mortality and the relative recruitment ‘strength’ of each cohort. This method assumes that fishing and natural mortality are constant and that vulnerability to the fishing gear is constant above a given age. One problem is that these catch curves do not allow an estimate of the natural mortality rate or vulnerability, so their usefulness is limited. CAGEAN is a modification of these techniques. An introduction to the CAGEAN methods explained below is provided by Hilborn and Walters (1992) and is recommended for beginners: Quinn and Deriso (1999) offer an updated and mathematically more rigorous treatment of the same topics.
10.5.4.2 Paloheimo method
There are several versions of the CAGEAN method. That of Paloheimo (1980) is the simplest and the one analysed here with some detail. The Paloheimo method uses the following equations and some algebra to arrive at its key equation. It starts with the catch equation, which in this version assumes that fishing mortality is responsible for a fraction (F/Z) of the total mortality:
Second, numbers at age a can be related to recruitment times cumulative fishing and natural mortality for each previous age by
|Na = Re -ΣF-ΣM||(10.42)|
A linear relation between effort and fishing mortality is assumed
|F = fq||(10.43)|
where f is effort and q is the catchability coefficient.
These equations can be combined and manipulated to give:
This equation relates CPUE at age to recruit numbers, the catchability coefficient, total and natural mortality and fishing effort. It can be shown that this becomes:
The Paloheimo method assumes that M is constant over years and uses a well-known approximation for the last term (which is valid for values of Z that are no larger than 0.7):
The Paloheimo equation can be arranged to give:
where j = year, a = age, and k = the number of years that the cohort has been fished. Equation 10.47 is a linear multiple regression of the form:
|Y= b0+ b1X1+ b2X2||(10.48)|
Given the needed data (usually catch by age for several ages and the corresponding effort that produced the catches), this equation can be solved with standard multiple regression packages to obtain estimates of Rq, q, and M.
The following example taken from Hilborn and Walters (1992) shows an application of Paloheimo's method. Table 10.2 presents the required data on catch at age and corresponding effort for Lake Erie perch.
Data on catch at age and corresponding effort for the 1971 cohort of Lake Erie perch
(Hilborn and Walters, 1992).
The estimates of the parameters given by Paloheimo's methods are:
Ln (Rq) = 2.37
q = - 0.22
M = 4.34
The correlations between the parameters are shown in Table 10.3.
Parameter correlations for the CAGEAN analysis based on the Paloheimo method for the data of Table 10.2.
(from Hilborn and Walters, 1992).
These results are suspicious and suffer from strong parameter correlation. This occurs because of poor data contrast (see Section 10.6); q is negative, which is impossible, while M is extremely high. To perform this catch-at-age analysis, not only the catches at age for each year for this cohort are needed, but also the fishing effort used to catch them. These efforts are all of the same magnitude and almost constant (i.e. poor contrast in effort) and this is why there is a strong negative correlation between q and M.
To simultaneously analyse data for three cohorts of Lake Erie perch using this method (see Hilborn and Walters 1992 for further details), dummy variables may be used to form an experimental design table, to perform a multiple linear regression. In this case, the equation becomes:
|Y = b1X1+ b2X2+ b3X3+ b4X4+ b5X5||(10.49)|
The first three coefficients represent the recruitment level of each cohort. The dummy variables X1–3 take the values 1 or 0 depending on which cohort we are analyzing, so that the corresponding coefficient b (recruitment) is included or excluded. The last two terms are the same as before and are the fishing effort and number of years of accumulated natural mortality. An analysis the results would still not be satisfactory because there is still poor data contrast in the effort for this set of data despite the fact that there are data for 3 different cohorts and 4 different years of fishing. It is still impossible to differentiate between the effects of natural and fishing mortality from these data. However, it is possible to obtain good estimates of the recruitment levels because there is good contrast in the relative abundance data (CPUE).
10.5.4.3 Doubleday's method
A more general approach to the catch-at-age method was proposed by Doubleday (1976). This method does not assume a linear relationship between the variables and is thus more difficult to calculate, requiring non-linear estimation methods. Its advantages are that fishing mortality F is not assumed to be proportional to effort, so the method can be applied in the absence of effort data. However, this method suffers from the general problem that good contrast is needed between fishing mortalities for good parameter estimation. The main Doubleday equation is presented below and more details about this method can be found in Hilborn and Walters (1992) and Quinn and Deriso (1999).
10.5.4.4 Other methods
A more developed and powerful catch-at-age method is that developed by Fournier and Archibald (1982). Paloheimo and Doubleday derived their models assuming an underlying deterministic process but in nature variables are measured and are subject to natural variability, which may be interpreted as noise. The method of Fornier and Archibald is flexible and accounts for explicit estimation of errors in:
Their method also explicitly accounts for a stock-recruitment relationship. This method is sophisticated both mathematically and statistically and is not analysed here, but has the advantage that it can include several types of external information that can help in the estimation of parameters, such as estimates of recruitment levels, fishing mortalities from other studies and effort data. A further sophistication of this type of analysis was developed by Methot (1989) and is able to use CPUE, gear selectivity and independent survey biomass data in the estimation of parameters.
10.6 PRINCIPLES OF FITTING MODELS TO DATA
Some of the models used in fisheries stock assessment are simple but the estimation of their parameters, which implies fitting the models to the data, is not always simple. In the case of the surplus production models treated above, there are three main approaches that are commonly employed for the estimation of their parameters.
First, one might assume equilibrium conditions, that is, that all the catches observed so far in the fishery are sustainable at the corresponding level of fishing effort. This assumption is invariably wrong and must be avoided. Equilibrium methods were used to simplify the computations because of difficulties in calculating parameter values analytically. However, modern computers allow the use of other methods mentioned below or even more sophisticated ones and there is no longer any need to assume equilibrium conditions.
10.6.2 Linear regression
A better option than assuming equilibrium conditions is to use linear regression. In the case of the Schaefer model, it is shown below that this model can be expressed as a linear equation to which standard regression methods can be applied to provide the values of the parameters and fit the model to our data.
Given the Schaefer model equation for biomass dynamics in a fishery:
Thus, substituting the last equation in the first gives:
Rearranging, dividing by Ut and multiplying by q gives:
Equation 10.55 is a linear equation of the general form:
|Y = b0 + b1X1 + b2X2||(10.56)|
which can be easily solved using the multiple regression facilities available in most spreadsheet software programs.
Although regression methods are easily applied to solve fisheries models, it has been demonstrated that they can give biased answers (Uhler 1979). They can also produce obviously wrong answers, such as negative values of r or q, which are biologically impossible. The general corollary is that illogical answers only mean bad data!
10.6.3 Time-series fitting
The most recommended method for fitting fisheries models to data is time-series fitting. Hilborn and Walters (1992) note that this method was first proposed by Pella and Tomlinson (1969) and implies taking an initial estimate of the stock size at the beginning of the time series of data (catch and CPUE) and using the Schaefer model to predict each point in the entire time series of data. Initial parameter values (guesses) are iteratively adjusted to minimize the difference (εt) between the observed CPUE and the CPUE predicted by the Schaefer model:
|εt = (Ǔt - U t)2||(10.57)|
Where U (CPUE) is:
This means that r, q, K, and the initial biomass size B0 be estimated. Usually, the problem of finding the best parameter values while minimizing the difference given by equation 10.57 is solved by using nonlinear estimation procedures such as those available in spreadsheets.
10.6.4 Bayesian estimation
Bayesian estimation is a powerful method for fitting fisheries models to data because it allows the incorporation of previous knowledge about the system into the estimation process, effectively helping to find better solutions. The types of additional information that can be incorporated into Bayesian estimation are extremely varied and include, e.g. fishery CPUE, independent survey CPUE, catches, estimates of intrinsic rate of population growth from life-table analyses, biological limits, knowledge from similar stocks and mark-recapture information.
Bayesian estimation is also extremely useful because it quantifies the uncertainty of the parameter estimates. The method uses previous knowledge to determine a probability distribution for the parameters that are to be estimated. This distribution is known as the prior probability distribution or ‘the prior’. Although relatively new in fisheries stock assessments, Bayesian estimation has rapidly become a powerful and accepted method to fit models to data.
Bayes theorem is based on the conditional probability and states that the probability of a parameter or group of parameters given certain data is equal to the product of: (a) the probability of the data given the parameters and (b) the probability of the parameters themselves, all divided by the sum over all possible parameter values of the product of (a) and (b):
The left term of the equation is the posterior probability distribution or ‘posterior’. The right-most terms in the numerator and denominator, imply that previous knowledge about the shape of the distribution of the parameters is available. This is the strength of the method as it allows additional ‘external’ information, such as biological or fisheries information to be included into the estimation process.
Depending on the type of ‘external’ information that can be incorporated, different possible prior distributions can be used for the parameters such as the binomial, normal, uniform, Poisson, multinomial and others. For more details about the types of distributions for different types of data users should consult a statistical text book.
A rudimentary, but simple way, to implement Bayesian statistics is to calculate the “kernel”, which is based on the sum of squares.
where L is the likelihood of the parameters and SS, the sum of squared differences between the real data and the estimated data points derived from a given set of model parameter values for t-1 degrees of freedom.
Bayesian approaches have been applied to elasmobranch fisheries by, e.g. McAllister and Pikitch (1998a,b), Punt and Walker (1998), Babcock and Pikitch (2001), McAllister et al. (2001) and Apostolaki et al. (2001, 2002). Berger (1985), Gelman et al. (1995) and Congdon (2001) provide a comprehensive treatment of Bayesian analysis. Hilborn and Walters (1992), Quinn and Deriso (1999) and Haddon (2001) provide more detailed treatment of parameter estimation issues.
10.6.5 Data quality
An extremely important principle of practical fisheries science identified by Hilborn and Walters (1992) and one often overlooked is that one cannot know exactly how a fish stock will respond to exploitation until the stock has been exploited. A good stock assessment depends as much on having an adequate model to describe the system dynamics as on the quality of the data that the model is fitted to. Data quality does not only refer to biases or errors, but also to the danger ofhow much information is embedded in the data. Historical variation in stock size and fishing pressure are needed if the data are to be used to estimate the parameters of the model with reliability. Otherwise the assessment may produce meaningless estimates that do not represent the stock dynamics well.
The most important quality of fisheries data is the degree of contrast imbedded in the data. To obtain good parameter estimates data must have high contrast. For example, in a SPM we should ideally have a data point at low stock sizes with low fishing effort (for information about r), data points at high stock sizes with low fishing effort (to estimate q and K) and data points at high fishing effort to estimate q. This is difficult to find in real fisheries because of the way most fisheries develop. Typically, low effort at large stock sizes is gradually increased to high fishing levels that usually lead to low stock sizes. Thus, one usually misses having a point of low fishing effort at low stock sizes. This common way in which fisheries develop leads to uninformative data and a typical case known as the “one way trip”in which the data show an increase in effort with time accompanied by a declining CPUE see (figure 10.5). This lack of contrast in the data makes for uncertain parameter estimates. In general, the standard deviation of such parameters is as large as, or larger than, the actual parameter values, and signales unreliable results. Under such circumstances management will be severely handicapped.
A hypothetical example of a ‘one-way’ trip type of data
(modified from Hilborn and Walters, 1992).
Data with better contrast can be obtained when a fishery shows a period of increased effort followed by a period when effort was reduced gradually such that the stock was allowed to rebuild after heavy exploitation. This case has been termed by Hilborn and Walters (1992) as ‘moving up and down the isocline’. Note how Figure 10.6 shows that there is a better scatter in the data instead of them all falling along one single line as before. These data have inherently more variation and contrast than the preceding example (the solid diamonds in the figure represent the start and finish points of the time series). Typically, in these cases the model parameters are more precisely estimated than for a ‘one-way trip’ case, but the slow pace of change in effort in these data still does not generally provide enough contrast for good precision. In cases like that in Figure 10.6, the standard deviation of the parameters is usually about half or less than the actual parameter estimates and although not good, it is better than in the previous example.
Hypothetical example of data with better contrast
(modified from Hilborn and Walters, 1992).
Data sets with high contrast have strong variations in data values, with relatively rapid changes back and forth between high and low effort. In these cases, parameters can be much more precisely estimated although other factors such as the total number of points in the time-series of data and the intrinsic variability of the data also influence the final precision of model parameter estimates.
In summary, when fitting models to fisheries data it is imperative to look at the uncertainty in the parameter estimates and not only at a single ‘goodness-of-fit’ measure such as the sum of squares. It is always advisable to apply different models to the same data set and compare the results between models, trying to validate results or to ask questions about why results might be different and what the implications of this are. In addition, it is important to learn how to use uncertain (‘bad’ ) results to improve the contrast in the data through carefully thought and well planned management regulations aimed at improving the quality of the data (such as large variations in effort over short periods of time).
10.6.6 The relationship between CPUE and abundance
There is an important assumption at the core of most fisheries models that use fisheries-dependent CPUE information (as most do) that the abundance of the fish stock (or other aquatic animal) has a direct relationship with CPUE, i.e. that CPUE is an index of abundance. This can be expressed mathematically for fisheries where the fishing season occurs as a single pulse or over a relatively short part of the year as:
Ct= qftBt ⇒ Ut = qBt
where Ut is CPUE in year t. According to this expression, CPUE is directly linked to biomass (abundance) by the coefficent q, the catchability coefficient. This model assumes that there is a linear relationship between CPUE and the abundance of the stock. This is a dangerous but necessary assumption of most fisheries models, but one that should be questioned and validated. Hilborn and Walters (1992) note that the relationship between CPUE and abundance can have at least two other forms apart from the linear form. Hyperdepletion occurs when the stock abundance decreases at a much slower rate than the CPUE. Thus, the CPUE signal tells us that the stock abundance is low when it is still high. If hyperdepletion is not detected, one would conclude that there was overexploition of the stock when in fact the stock might be in a good state. Hyperstability happens when the stock abundance falls more rapidly than the CPUE index, thus giving the opposite impression, that the stock abundance is still high when there may be dangerous overexploition of the resource.
Hyperdepletion can occur when the species is being exploited only over a relatively small part of its range, as when there are natural refuge areas (such as deeper waters or rougher grounds where the gear cannot fish). In such cases the exploited part of the stock will decrease rapidly but the overall abundance of the entire stock does not. Given that the abundance index (CPUE) is based only on the fishing grounds, it will show a faster decrease than if it was based on fishing over the entire geographical range of the stock.
Hyperstability is a well known phenomenon in fisheries for highly gregarious or schooling species such as herrings, sardines, anchovies and tunas. In these fisheries, searching for fish schools is highly efficient and once located, fishing an entire school is relatively quick and efficient. The remaining schools remain concentrated as the overall abundance of fish goes down.
Possible ways to detect a lack of proportionality between CPUE and effort include mapping and stratification of CPUE and effort data to analyse spatial patterns and depletion experiments to gain additional information. Overall, hyperstability is more common and more dangerous, as it leads to stock collapses. A way to avoid this is to obtain fishery-independent indices of stock abundance (see Section 12), either through research cruises or by coordinating efforts with fishermen to perform controlled experiments to fish in other areas or other ways than they usually do, such as following a systematic sampling design. Quinn and Deriso (1999) summarize different ways to model non-linear relationships between CPUE and abundance.
Finally, it should be mentioned that generalized linear models (GLMs) are becoming common methods for standardizing fishery-dependent CPUE data. These methods take account of the effect of various factors (such as environmental variables or fishery operational variables) on catch rates.
10.7 CONCLUSIONS AND RECOMMENDATIONS
Fisheries stock assessment in not usually a problem of the species or group under analysis but rather a problem of the approach used for the analysis. There are several methods available to perform stock assessment and some of them have been presented here in detail. However, keep in mind that there are three main rules for good stock assessment:
Table 10.4 presents a few examples of real elasmobranch fisheries with a list of their characteristics, the methods used in each case for stock assessment, the status of the fishery and major references. These examples can be reviewed more closely by those interested in more detailed analyses of real elasmobranch fisheries and the practice of their stock assessment and management.
A referenced selection of real shark fisheries, summarizing their main characteristics, the assessment methods in use and the state of management and the resource.
|Fishery||Species||Catch level||Management system||Stock Assessment Methods||Status||Main references|
|Southern Australian shark fish||Galeorhinus galeus, Mustelus antarcticus and other spp||2,800 t/y||Controls on amount of gear (licenses)||Surplus Production, Delay-difference and Age-structured models||Overexploited, under recovering regulations||Walker 1999|
|Canadian Porbeagle shark fishery||Lamna nasus||850 t/y||TAC (250 t), Fishing licenses plus fishing restrictions||Catch curves, catch rate trends, agestructured mode||Overexploited, under severe recovering regulations||Campana et al.1999, 2001|
|New Zealand shark fisheries||Galeorhinus galeus,|
Raja spp. Hydrolagus spp. and other 15 spp
|17,000 t/y||ITQs and TACs||None, quotas established through ad hoc methods (proportion of past catches)||Recovered after overxploitation or unknown||Francis and Shallard 1999|
|East coast of US shark fishery s||39 species mostly Carcharhinus||3,500 t/y||TAC||Bayesian Surplus Production Models||Overexploited, under recovering regulations||MacAllister and Pikitch. 1998a, b; Branstetter 1999|
|Gulf of Mexico shark fisheries||35 species mostly Carcharhinus||12,000 t/y||5 prohibited species and other simple regulations||None||Unknown, likely heavily overexploited||Bonfil 1997, Castillo et al. 1998|
|Argentinean shark fisheries||Mustelus schmitii, Galeorhinus galeus, Carcharhinus brachyurus and other 10 spp||30,000 t/y||None||None||Unknown, likely heavily overexploited||Chiaramonte 1998|
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