ICES Journal of Marine Science: Journal du Conseil

ICES Journal of Marine Science: Journal du Conseil Advance Access originally published online on November 6, 2006
ICES Journal of Marine Science: Journal du Conseil 2007 64(1):178-191; doi:10.1093/icesjms/fsl015

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Effects of fish density distribution and effort distribution on catchability

Nick Ellis¹ and You-Gan Wang²

¹ CSIRO Division of Marine and Atomospheric Research, PO Box 120, Cleveland, Queensland 4163, Australia
² CSIRO Mathematical and Information Sciences, Long Pocket Laboratories, 120 Meiers Road, Indooroopilly, Queensland 4068, Australia

Correspondence to Y-G. Wang: tel: +61 7 3214 2700; fax: +61 7 3214 2855; e-mail: you-gan.wang{at}csiro.au

Ellis, N., and Wang, Y-G. 2007. Effects of fish density distribution and effort distribution on catchability – ICES Journal of Marine Science, 64, 178–191.

The effects of fish density distribution and effort distribution on the overall catchability coefficient are examined. Emphasis is also on how aggregation and effort distribution interact to affect overall catch rate [catch per unit effort (cpue)]. In particular, it is proposed to evaluate three indices, the catchability index, the knowledge parameter, and the aggregation index, to describe the effectiveness of targeting and the effects on overall catchability in the stock area. Analytical expressions are provided so that these indices can easily be calculated. The average of the cpue calculated from small units where fishing is random is a better index for measuring the stock abundance. The overall cpue, the ratio of lumped catch and effort, together with the average cpue, can be used to assess the effectiveness of targeting. The proposed methods are applied to the commercial catch and effort data from the Australian northern prawn fishery. The indices are obtained assuming a power law for the effort distribution as an approximation of targeting during the fishing operation. Targeting increased catchability in some areas by 10%, which may have important implications on management advice.

Keywords: aggregation, catchability, catch and effort data, density distribution, effort distribution, stock assessment

Received 31 October 2005; accepted 25 August 2006; advance access publication 6 November 2006.

	Introduction

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 
Catchability is defined as the proportion of available fish in the population that would be caught by a unit of effort. If the fishing effort is randomly distributed in the sense that it is not related to fish abundance, catchability is also the probability of an individual fish being caught by a unit of effort. However, when the distribution of fishing effort depends on fish abundance (attributable to fish aggregation and fisher targeting), the overall catchability for the stock will be affected, depending on the degree of aggregation and targeting (Francis et al., 2003). One way to explain this is to attribute it to changes in overall catchability for the whole area of the stock. For convenience, we refer to this phenomenon as density-dependent catchability.

The simplest approach to catch and effort data is based on the well-known catch equation of Baranov (1918), which is described by Gulland (1983, p. 105) and Xiao (2005, 2006). The catch equation presumes that fishing effort is randomly distributed over a fishing ground in the sense that it is unrelated to fish abundance. This is clearly not the case for most fisheries because of (i) the aggregation behaviour of fish, and (ii) the targeting behaviour of fishers. In such cases, the traditional catch equation is inappropriate for describing the relationship between catch and effort (Crecco and Overholtz, 1990; Richards and Schnute, 1992; Paloheimo and Chen, 1993), because the overall catchability is affected by the extent of correlation between effort and abundance, resulting in the so-called density-dependent catchability. Paloheimo and Dickie (1964) proposed density-dependent catchabilities for Georges Bank haddock, and Crecco and Overholtz (1990) tested and supported their theory. Shardlow (1993) also found that the catchability increases with abundance in a salmon fishery. Furthermore, Paloheimo and Dickie (1964) argued that density-dependent catchability exists for most demersal and pelagic fisheries. Indeed, Swain and Wade (2003) found that abundance was positively correlated with effort for the snow crab fishery, and Hilborn and Walters (1992, pp. 175–192) discussed the challenges of obtaining abundance indices from spatial catch and effort data in the presence of hyperstability and hyperdepletion. The results from those authors suggest that density-dependent catchability should be incorporated into population models and into the estimation methods, to obtain less biased estimates of effective effort, which are essential for stock assessment models.

Walters (2003) has warned that "simple, nonspatial ratio (cpue) estimates should not be used". An alternative approach is to apply generalized linear models (GLMs) to take account of the many factors that affect catchability (Bishop et al., 2000; Campbell, 2004). The objective is to obtain a closer relationship between average catch per unit effort (cpue) that reflects the stock status for stock management. It is therefore essential to define an average cpue which, by taking account of how the allocation of effort responds to aggregation, is proportional to stock abundance. The constant of proportionality should then be independent of time and effort.

In Australian prawn fisheries, fishers use their knowledge of aggregation behaviour, together with their experience and modern searching technology, to enhance catch rates (). The extent of fisher targeting in general depends on the aggregation level and the cost of finding high densities of fish. Therefore, the overall catchability will depend on the distribution of fishing effort in relation to the distribution of the target species. In fact, Salthaug and Aanes (2003) showed that catchability of two migratory stocks was strongly related to the concentration of the fleet. Here we examine the effects of both fish distribution (aggregation) and effort distribution on catchability in a way similar to that of Swain and Sinclair (1994) and Swain and Morin (1996) and describe effort distribution using a parameter that measures the knowledge of fish density distribution. A change in catchability is a joint effect of the aggregation of fish and the knowledge of fishers. We also demonstrate our concepts by analysing the commercial catch and effort data for Australia's northern prawn fishery.

	Relative fish density distribution

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 
Following Swain and Sinclair (1994), assume that fish density at geographic position x is given by d(x)=Nf(x), where N is the abundance and f(x)>0 is given by a probability density function. We call f(x) the relative fish density distribution. (For a table of all symbols used herein, the reader is referred to Table 1.)

View this table: [in this window] [in a new window]   Table 1. Symbols used throughout the paper.

 
Let e(x) be the density of effort at position x, such that the total effort over area A is E=_Ae(x)dx. Then the catch density c(x) is given by

(1)

Quantity p is a constant independent of x called the instantaneous catchability. It is the proportion of fish at x that would be caught if subjected to a unit of effort density. Its units depend on the units of effort; for trawl effort, where the units are swept area (and effort density is dimensionless), p is the probability that a fish subjected to a single tow will be caught by the trawl gear. In contrast, for a hook fishery, where the units of effort are, for example, hook-minutes, p is the proportion of fish over a local area that would be caught by one hook-minute per square metre.

In general, let a be a unit of effort. The (spatial) catchability q(x) at position x is defined as

(2)

It is the expected proportion of the population caught by a unit of effort at x. It depends on position through the density of the fish.

The average catchability is the average of q(x) weighted by effort:

(3)

It is the proportion of the population caught per unit effort. The average catchability combines the distributions of both fish and effort. If fishing is assumed to be uniform over the entire area of fish stock A, we find that =pa/A, which differs from the instantaneous catchability by a constant factor. Quantity is the analogue of p at the scale of the fishery. For trawl effort, if we define the unit of effort as the singly swept-out area of the entire fishery, so that a=A, then, numerically, =p. For targeted trawling, one would expect /p>1.

In general /p, which we will refer to as the catchability index , may be used to quantify the effect of fish aggregation and fisher knowledge (or targeting) on the stock. It is usually much easier to estimate than to estimate either or p separately from catch and effort data.

	Effect of aggregation

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 
In the appendix, following Swain and Sinclair (1994), we consider one-dimensional examples of the function f(x) that include a spread parameter to quantify the degree of fish aggregation. We quantify the degree of aggregation in a way that is not dependent on such a form, so define a function G(r), the proportion of fish distributed in areas with fish density <r. This is given by

(4)

We can interpret G(r) as the cumulative density function of a random variable R. The actual fish densities in the sea can be thought of as samples from the probability density of R. Equivalently, the relative fish densities, f(x), can be thought of as realizations of R/N. We show that the variance of R/N provides a measure of the degree of fish aggregation. After some algebra, one can show that the nth moment of R/N is given by

(5)

When f(x) is the standard normal density in one dimension, we find, using the identity Var[X]E[X²]–E[X]², that

(6)

As this is inversely proportional to ², and is therefore a measure of fish aggregation, we propose using the variance of R/N for such an aggregation measure in general. In fact, we define a dimensionless aggregation index, thus:

(7)

which is the variance of the fish density relative to the mean fish density N/A.

	Effect of effort distribution

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 
We now consider the effect on catchability when the effort distribution varies for a given fish distribution. We assume that the effort distribution is a power function of the fish density, i.e. e(x)(f(x)). The constant of proportionality is E/(f(x))dx, to obtain a total effort of E. Without loss of generality, we will simply assume that E=1. We will refer to as the knowledge parameter. Higher values of provide more effort in higher density areas and result in greater catches. The three models discussed by Swain and Sinclair (1994) (appendix) correspond to the cases when =0, 1, and , respectively. The general expression for the catchability index is, from Equations (2) and (3),

(8)

For random fishing, =0, and the catchability index becomes

(9)

where N_A is the population within A. If the fish density is uniform within A, then, irrespective of , =₀. This implies that knowledge is ineffective at increasing catchability if there is no aggregation.

The increase in catchability relative to random fishing is given by the ratio

(10)

An alternative expression for l() can be found in terms of the fish density distribution, using the moment Equation (5); thus,

(11)

This form will prove useful in estimating l() assuming known .

In the case when f(x) is the normal density function, and A={x:|x| z},

(12)

in which z is the 100(1–/2) percentile value for a standard normal distribution. For example, z_0.05=1.96, and A becomes the highest density region containing 95% of the population.

Figure 1 shows how the catchability index changes with the knowledge parameter . We plot the catchability index for two different aggregation models (spread =1 and 2) and four different effort models (X=1, 10, , and z_0.05). The last case means that fishing is restricted to the most abundant region containing 95% of the population. Catchability always increases with knowledge up to an asymptote that depends on spread, but not on fishing area. This asymptotic value corresponds to fishing at the point of maximum fish density. Catchability decreases as the spatial limits widen. The case =0 corresponds to uniform fishing within(–X, X). Note that for an infinite area ₀=0, one must have a finite area to obtain a non-zero catchability for uniform fishing.

View larger version (10K): [in this window] [in a new window] [Download PowerPoint slide]   Figure 1. Catchability index vs. for two different aggregation models (spread = 1 and 2) and four different effort models (X=1, 10, , and z0.05). The units of are such that a=1. The fish distribution f(x) follows a normal distribution with mean zero and standard deviation , and the effort distribution is proportional to [f(x)] but truncated to be within (–X, X).

 

	Analysis of catch and effort data

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 
In the analysis of catch and effort data, we restrict attention to the part of the stock lying within the fishery area A, and treat it as a single stock confined to that area, so precluding migration into or out of the area. This is a nominal fishing area, because parts of it may be unfished or only rarely fished for some times. Abundance N is the same as N_A, the abundance within A, and f(x) is the density restricted to A.

Let (c_i,t, e_i,t) be the catch and effort data collected at time t and in grid (subarea) i of area A. If the abundance in grid i is n_i,t, we have, from (3),

(13)

We can estimate this by plugging in estimates for C and N. We can estimate n_i,t in fished grids, because from (1), E[c_i,t]=pn_i,t e_i,t/A, so that _i,t=u_i,tA/p, where u_i,t is the cpue in grid i at time t. In unfished grids, we assume that n_i,t equals the average over fished cells. We then have =_tA/p, where _t is the mean of u_i,t. For C, the expected total catch, the estimator is simply the observed total catch. Therefore, a plug-in estimator for is

(14)

where û_t is the overall cpue. Note that û_t can be thought of as the estimated cpue under a sampling scheme weighted by effort (because û=ue/e), whereas _t is the estimated cpue under a uniform sampling scheme. The abundance in unfished grids is actually likely to be less than in fished grids, so is over-biased and underbiased. Later, in Discussion, we suggest how this limitation might be overcome. As, from the previous paragraph, the fish density satisfies n_i,t/A=E[u_i,t]/p, the empirical density distribution of R at time t takes the set of discrete values of u_i,t/_k u_k,t at u_i,t/p. Therefore, E[R] can be estimated by

(15)

which, using Equation (11), leads to an estimator of l() as

(16)

The instantaneous catchability p, which links cpue to abundance, does not depend on effort distribution. We can therefore estimate it under the assumption of uniform fishing using the equation (0)=A_t/Np=1; then, p is simply the ratio of mean cpue to mean fish density N/A. This agrees with the relationship we obtained previously from Equation (3) for uniform fishing: pa=A.

The quantity l() measures the effect of aggregation on catchability when the knowledge parameter about the stock is fixed as . Substituting for p in Equation (16) gives

(17)

We can use the first and second moment estimates of R from Equation (5) to estimate the aggregation index , thus,

(18)

because Var[R]=E[R²]–E[R]².

The knowledge parameter is estimated by regressing log effort against log cpue and taking the coefficient of log cpue. Actually we use a GLM in which effort e_ijk follows a Poisson distribution with extra variation, and cpue u_ijk is used as a covariate, so is assumed known. (It might seem more logical to treat effort as the covariate, and cpue as the response, but then the coefficient to be estimated would be 1/, which is unbounded around = 0, the expected range of .)

The details of the model are

(19)

where e_ijk is the effort in grid i for year j and month k, u_ijk the cpue, the dispersion parameter, and an exponent to be estimated by inspection of the residuals. The dispersion parameter is estimated from the Pearson chi-squared statistic.

Estimates of the knowledge parameter can be plugged into expression (17) to give a second estimate, (), for the catchability index .

	Analysis of commercial catch and effort data from Australia's northern prawn fishery

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 
We now consider the commercial catch and effort data from the tiger prawn fisheries in the Gulf of Carpentaria (1970–1996), and we apply the theory of fish aggregation and effort allocation to see how the average cpue can be influenced. The Gulf is divided into large-scale management areas, shown in Figure 2. We focus on the four most important tiger prawn fishery areas: Groote, Vanderlins, Mornington, and Weipa. The finest practical scale on which to estimate fish density is the grid level (6 nautical mile squares), which are the small squares on Figure 2. For a temporal scale, we use month (i.e. we sum catch and effort within each month). This is because prawn distribution (and thus effort) can change markedly from one month to the next.

View larger version (69K): [in this window] [in a new window] [Download PowerPoint slide]   Figure 2. The Gulf of Carpentaria, Australia, showing large-scale management areas and small-scale grids of 6 nautical miles.

 
We also need to decide on the scale of the stock. Although the large-scale areas are in fact subdivided into smaller management areas, such areas are probably too small to contain a stock, and they have too few grids for reliable estimation. We therefore work on the large-scale areas (or amalgamations thereof): we assume that these areas completely contain the prawn stock whose catchability we are estimating.

The pattern of effort in Mornington shows a marked decline in September and October, suggesting that the fleet moves to neighbouring Vanderlins. We therefore carry out catchability analysis on amalgamated regions as well as on the basic areas. Such regions are SW Gulf (Mornington and Vanderlins), W Gulf (Groote, Mornington, and Vanderlins), and all four areas combined.

In 1980/1981, there was a major expansion of the fishery, so many new 6 nautical mile grids were visited. We restrict attention to the most important months for the tiger prawn fishery, namely August, September, and October. Before the seasonal closures were introduced in 1988, there was year-round effort, but this effort was still concentrated in the months August–November when the prawns are at their largest.

Catchability index
Because the northern prawn fishery is a trawl fishery, with effort measured in swept area, we define for convenience our units of effort such that a/A=1. We have computed the catchability index, =û_t/_t, for all three months within each year for each region. Grids with only a single day's effort have been excluded. Table 2 shows how many grids are included for each region, month, and year. The results are shown in Figure 3. The dotted horizontal line at =1 indicates no density-dependent effect.

View larger version (38K): [in this window] [in a new window] [Download PowerPoint slide]   Figure 3. Catchability index estimates, , for each month, year, and region in units where a=A. Also shown (dotted) is for the whole tiger prawn season.

 

View this table: [in this window] [in a new window]   Table 2. Number of grids in a region for which total effort is at least 2 d.

 
Some years are missing because of the lack of grids (Table 2). The dotted line is the index for all months combined. Groote, Vanderlins, and Mornington appear to have >1 (i.e. a density-dependent effect) since the period 1980–1985. This is quite clear when the regions are combined (W Gulf). For Mornington, there is a reduction in effort in October, leading to a more variable across months. This becomes less variable across months and years when we combine Mornington with Vanderlins (SW Gulf); we consistently get 1.1 since 1987. Groote has roughly the same mean, but greater variation across years. The estimate from the three months combined tends to be more consistent. Weipa has no consistent density-dependent effect.

Aggregation index
Estimates of from Equation (18) are shown in Figure 4 for each region and year. The estimates are on a logarithmic scale. Three curves are given per panel, one for each month. With the exception of Weipa, all regions show a steady decline in aggregation since 1980. As for Mornington, the estimates for October are highly variable.

View larger version (32K): [in this window] [in a new window] [Download PowerPoint slide]   Figure 4. Aggregation index estimates, , for each month, year, and region. Also shown (dotted) is for the whole tiger prawn season.

 
To explain how these estimates arise, Figure 5 shows details of the abundance distribution in two contrasting years for the W Gulf region. In the left-hand plots we have the relative fish density d(x)/(N/A) [or Af(x)] plotted against relative area x. The range under each curve equals the total area fished. In 1995, this area was about half that in 1985. The departure of the fish density from the average value, as indicated by the dashed line, is generally larger in 1985, which suggests that that year had greater aggregation. This is confirmed in the right-hand plots, which show the empirical density of R, as introduced before Equation (15), after gaussian smoothing. The variance is higher for 1985, so the distribution is indeed more aggregated. There is also slightly greater aggregation in August of both years.

View larger version (16K): [in this window] [in a new window] [Download PowerPoint slide]   Figure 5. (Left panel) Relative fish density d(x)/(N/A) plotted against fished area x, sorted by decreasing density, for each month in 1985 and 1995. (Right panel) Probability density of relative fish density for the same years. The area under the curve within a fish-density interval denotes the proportion of fish in areas with densities inside that interval.

 
However, this apparent decrease in aggregation with year may be largely an artefact of the shrinkage of the fishery, rather than an actual change in fish distribution. This is because the aggregation parameter, which is derived from the empirical distribution of R, is obtained on fished grids only. The fish density in unfished grids is likely to be less than in fished grids, so the fish distribution is more aggregated than it appears from the fished grids. This downward bias will be greater for the later years with fewer fished grids (Table 2). We revisit the issue of bias later, in Discussion.

Knowledge parameter
Using a range of values for in the GLM (19), = 2 produces the flattest variance–mean relationship in the Pearson residuals. Grids with a single day's effort can have huge influence on the estimates (Pollock et al., 1997), so we restricted the analysis to grids with 2 d effort. The analysis has been carried out using the SAS procedure GENMOD.

Figure 6 shows point estimates of _jkj¹+ß_k¹+_jk¹. The dotted line at =0 corresponds to no targeting. There appears to have been definite targeting since the mid-1980s in Groote, Mornington, and Vanderlins, but not in Weipa. The pattern tends to be fairly consistent across months for Groote and Vanderlins, and the consistency is improved for Vanderlins and Mornington when they are combined (SW Gulf).

View larger version (28K): [in this window] [in a new window] [Download PowerPoint slide]   Figure 6. Knowledge parameter estimates, , for each month, year, and region.

 
To assess the accuracy of the estimates, we show standard errors for one of the months (October in Weipa, August elsewhere) in Figure 7. The choice of October for Weipa is based on there being more available grids. It is clear that, before the mid-1980s, there was little evidence of targeting anywhere. Although the statistical power is low, this may actually be a real effect owing to the absence of GPS technology. On the other hand, the positive for areas in the W Gulf appears a significant effect. The dispersion parameter estimates range from 0.7 to 1.4, the larger values corresponding to "All areas" and Weipa. This further confirms that the relationship between cpue and effort differs between Weipa and the W Gulf.

View larger version (23K): [in this window] [in a new window] [Download PowerPoint slide]   Figure 7. Estimates of in August for each region and each year, with standard errors shown by bars. For Weipa, the estimate is for October.

 
The standard errors are governed by the number of grids fished in the region (Table 2). There was a lull in effort in Mornington and Vanderlins around 1975, so fewer grids were visited. Also the level of effort in Weipa was lower than in the other regions for most years. The fewer grids in earlier years may also be due to lack of reporting, because it was not compulsory to record grid-level position in the logbooks.

Using the estimated knowledge parameters for each month, year, and region, we evaluated the alternative estimates of the catchability indices, (), using Equation (17). Figure 8 shows the scatterplots of () against for each month, year, and region. They appear to agree, indicating that the power law for the effort distribution is a reasonable approximation of targeting in fishing.

View larger version (15K): [in this window] [in a new window] [Download PowerPoint slide]   Figure 8. Scatterplots of () against for each month, year, and region; the units of are such that a=A. Different months are denoted by a different symbol.

 

	Discussion

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 
We have demonstrated how a combination of stock clumping and fisher knowledge influences the relationship between catchability and stock size. The overall cpue based on lumped catch and effort over a stock's area is a good index of overall catchability, but the average of the cpues calculated from small units where the fishing is random is a better index for measuring stock abundance. In highly complex fisheries where there are numerous innovations in technology, it can be quite easy (it is almost inevitable) to overlook confounding effects even with a GLM approach with many parameters. Significant coefficients can still be misleading because of confounding (Bishop, 2006).

We further investigated the effect of aggregation and targeting on catchability, following Swain and Sinclair (1994), and suggested three indices to measure the extent of aggregation, targeting, and changes in catchability. Simple estimation methods are also proposed. As catch and effort data are commonly collected in fisheries, the proposed methods can be applied easily to those data to quantify possible effects on catchability.

The various indices presented here are based only on grids fished, because cpue information is not available for unfished grids. However, the population in reality extends over both fished and unfished grids, so the indices may be biased. To base indices on all grids requires assumptions about the cpue in unfished grids. The estimate of is unbiased if mean cpue for unfished grids equals _t, and the estimate of is unbiased if the fish density distribution is the same in both fished and unfished areas. Because fish density is likely to be less in unfished areas, both and are probably underestimated. The same estimates of would be obtained if the unfished grids were included with cpue set to zero. As fish density is likely to be somewhat greater than zero in those grids, is also probably underestimated; the same applies to l(). The effect of ignoring the unfished grids is consequently to underestimate all four indices. It is therefore of interest to further investigate the effect of these "missing" grids. Other factors, such as the interference competition effect reported by Swain and Wade (2003), may distort the validity of cpue as an abundance index. However, for the northern prawn fishery there is no evidence to suggest that interference competition is important.

This work has relevance to the estimation of fishing power. In studies by Robins et al. (1998), using regression models, and Bishop et al. (2000), using generalized estimating equations, the effect on fishing power of vessel characteristics such as gear size and the presence of GPS plotters during the transition period around 1990 was estimated. Those authors took account of density-dependent catchability by allowing catch to be proportional to a constant power of effort; both works reported the estimate as 1.07, implying that cpue is higher in regions of greater effort. We have confirmed this overall result, but we have also shown that the density-dependence varies both temporally and spatially. Our method filters out the effects attributable to intrinsic vessel characteristics, manifest through the instantaneous catchability p, and captures the effects attributable to improved search capability, manifest through . This allows us to build more informed models of fishing power, producing more reliable estimates and sounder advice to fisheries management ().

	Appendix

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 
Comparison with Swain and Sinclair's (1994) findings
Swain and Sinclair (1994) considered three models for fishing effort.

1. Uniform fishing, or equivalently fishing is random over the area (–x_T, x_T) where the fish density exceeds some threshold.
   
2. Fishing effort at position x is proportional to the density at x for x(–,).
   
3. Fishing is only at the location where the fish density is maximum, i.e. at x=0.
   

In model (i), it is assumed that fishers have no knowledge about the distribution of the fish stock (except that it exceeds some threshold fish density within a finite area). The corresponding catchability can therefore be regarded as a benchmark. In model (ii), the fishers have some knowledge about the distribution of the stock. Model (iii) is the extreme case in which the fishers know exactly where the hotspot is. We would expect to be smallest for model (i) and largest for model (iii). Model (ii) is probably the most realistic case.

Assuming f(x)=^–1(x/), the normal density function with a variance ², Swain and Sinclair (1994) obtained the corresponding catchability indices for the three effort models as

(A1)

Note that in (i) the fishing area (–x_T, x_T) depends on both relative fish density distribution and total abundance N. This is why Swain and Sinclair (1994) concluded that spatially uniform changes in abundance also influence catchability to some extent. In fact, such an effect is due to changes in distribution of fishing effort.

We now consider how the catchability changes when the fish density distribution changes for a given fishing pattern e(x). The e(x) is assumed to be

1. a constant over a fixed area (–X, X), where X> 0,
   
2. e(x)=(x),
   
3. e(x)=(x)/(2(x)–1) for x(–X, X).
   

Unlike case (i) considered by Swain and Sinclair (1994), the fishing area here is fixed in (a). Case (b) is slightly different from case (ii) of Swain and Sinclair (1994), in which the effort pattern changes as fish density changes. In the cases here, fixing the fishing effort function and allowing the fish density distribution to change, shows the effect of aggregation on catchability.

If we assume that f(x)=^–1(x/), we have the corresponding catchability indices for the above three fishing effort functions:

(A2)

Here, (·) is the standard normal cumulative density function. Similar algebra leads to

(A3)

We demonstrate how catchability varies with aggregation in Figure A1. Figure A1(a) shows the fish density distribution for a range of values of spread . More highly aggregated distributions have smaller values of . The effort pattern in (b) is identical to the distribution labelled 1.0. In (a) and (c), we let X=1, as shown by the dashed line. Figure A1(b) shows the catchability index plotted against spread for the three effort patterns. The catchability always decreases with spread. Case (a) is not very sensitive to for small spreads until X, because almost the entire population lies inside X. On the other hand, targeted patterns are more sensitive to . The catchability is higher for (c) than for (a) because, although both patterns cover the same area, the effort pattern (c) exploits the aggregation more. Similarly, _c is higher than _b because the effort used in (b) outside X is used more efficiently in (c) inside X. The catchability of (a) relative to (b) depends on X; for sufficiently large X, _a is less than _b, and for sufficiently small X, _a is greater than _b.

View larger version (7K): [in this window] [in a new window] [Download PowerPoint slide]   Figure A1. (a) Fish density distribution for a range of values of spread over the half-range x>0. Overlaid (dashed line) are the effort patterns (a)–(c). (b) catchability index vs. spread for each effort pattern. The units of are such that a=1.

 
The use of a one-dimensional function f(x) may appear overly restrictive. However, note that the results for f(x) would hold for any related distribution in which the x-axis was segmented and arbitrarily rearranged. One such distribution is that where the x values are sorted on density. The function f(x) is therefore a prototype for many actual spatial arrangements, so the particular form is not as restrictive and artificial as it may at first appear. Moreover, the x variable of the sorted form of f(x) can also be interpreted as cumulative area, so f(x) can represent two-dimensional distributions too.

	Acknowledgements

 
This research project was partly supported by the Fisheries Research and Development Corporation of Australia. We thank Janet Bishop and two referees for their helpful suggestions and comments on an earlier draft.

	References

Top Introduction Relative fish density... Effect of aggregation Effect of effort distribution Analysis of catch and... Analysis of commercial catch... Discussion Appendix References

 

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