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Effect of fish shape on gillnet selectivity: a study with Fulton’s condition factor
Fisheries Research 54 (2002) 153±170
Effect of ®sh shape on gillnet selectivity: a study with Fulton's condition factor M. Kurkilahtia,*, M. Appelbergb, T. Hesthagenc, M. Raskd a
Finnish Game and Fisheries Research Institute, RymaÈttylaÈ Fisheries Research Station, FIN-21150 RoÈoÈlaÈ, Finland b Institute of Freshwater Research, Fish Monitoring Group, S-178 93 Drottningholm, Sweden c Norwegian Institute for Nature Research, Tungasletta 2, N-7485 Trondheim, Norway d Finnish Game and Fisheries Research Institute, Evo Fisheries Research Station, FIN-16970 Evo, Finland Received 29 February 2000; received in revised form 29 September 2000; accepted 21 November 2000
Abstract The Eurasian perch, Perca ¯uviatilis (L.), was ®shed with Nordic multimesh gillnets, the mesh size combination of which was based on geometric series. From the test ®shing data the ®sh girth was found to be linearly related to the third root of ®sh weight and therefore the ®sh girth could be estimated indirectly with Fulton's condition factor K. Two different response surface models based on B-type gillnet selectivity model were ®tted to the empirical data: one model with Fulton's K and one without (simple model). The Fulton's K-model had narrower Monte Carlo simulated 95% con®dence interval than the simple model. Simulation studies showed that the traditional simple model without Fulton's K does not have a linear relationship between mesh size and ®sh length, which violates the classical principle of Baranov's theorem on geometric similarity of ®sh of different sizes. Therefore the simple model derived from one population is not applicable to another population if there are differences in ®sh condition. However, if as is the case for Nordic multimesh gillnets with a mesh size combination based on geometric series like Nordic multimesh gillnets, this error is minimal because adjacent mesh sizes cover each other and correct this error. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Condition factor; Multimesh gillnet; Geometric series; Net selectivity curves; Mathematical model
1. Introduction The girth is the main size factor determining the size of ®sh caught by different mesh sizes. Nevertheless, most gillnet selectivity models have been derived for ®sh length, not for the ®sh girth as they strictly should be. The main reason for this is that measuring girth is dif®cult, time-consuming, and therefore expensive
*
Corresponding author. Tel.: 358-20-57-51-731; fax: 358-20-57-51-731. E-mail address: [email protected] (M. Kurkilahti).
if compared to length and weight measurements of ®sh. Another reason is that we are usually interested in ®sh size, which is routinely measured as length, and therefore the ®sh girth based model would be adjusted to ®sh length anyway. Nevertheless, the length based gillnet selectivity models are justi®ed if the ®sh girth and ®sh length are linearly related to each other. This relationship is also important because the ®sh girth is the main factor in determining the optimum mesh size that will catch a ®sh of certain size and the mesh size/®sh length relationship is usually assumed to be linear (see Hamley, 1975, for example).
0165-7836/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 7 8 3 6 ( 0 0 ) 0 0 3 0 1 - 5
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Nomenclature g K Lj mi mjk
Maxjk Pijk W
factor to determine the peakedness of the net selectivity curve Fulton's condition factor, K 100W=L3 fish length in centimetres size of mesh i from knot to knot in millimetres optimum mesh size for fish length class j of condition factor class (Kk), which is expressed as a regression equation between the mesh size i of the observed maximum catch (Maxjk) of each fish size class j and the corresponding fish size class j and condition factor class k observed maximum catch of fish length class j from mesh size mi and condition factor class Kk relative efficiency of mi for fish of size class j of condition factor class Kk with constraint that max Pijk 1 fish weight in grams
Greek symbol p s constant s 1= 2p, from which follows that max Pijk 1 Another problem arises when different individuals of a ®sh species have equal length but different girth. This is a normal situation in nature; individuals are slim or fat and they differ within and between populations. Such ®sh have different probabilities of capture with a particular mesh size even if they have the same length, which will make the gillnet selectivity modelling more complex. When gillnet data are collected from the same population, this within-population variation causes inaccuracy in selectivity models: larger error terms in the least-squares models and smaller coef®cients of determination (r2). The between-population variation has, at least, two additional consequences. First, a gillnet selectivity model derived from one population might not be directly applicable to another population without taking the different shape into account in oneway or another. Second, combining data on ®shes of different shape causes dif®culties in modelling gillnet selectivity models can be more or less biased if the different shape is ignored.
If the girth (G) is linearly related to the third root of the ®sh weight (W), it can be used for indirect estimation of girth. However, the third root of the ®sh weight is not useful itself; but it must be adjusted ®sh length (L) to make ®shes of the samepshape but different length comparable. Therefore, if 3 W =G is linear then W=G3 is linear and supposing that G aL b, where a and b are estimates, this adjustment can be done, for example, by Fulton's condition factor (referred to as K here after), which has been shown to affect the modal length of ®sh in gillnet catches (Jensen, 1986, 1995). In this paper we study the linear relationships between ®sh length, girth and weight and the effect of Fulton's condition factor K on the mesh size selectivity of gillnets. We ®tted two different mathematical gillnet selectivity models, one with a condition factor and another without, for the same experimental catch data of perch (Perca ¯uviatilis (L.)) caught by Nordic multimesh gillnets and compare the effect of these two types of models. 2. Material and methods 2.1. Gillnet data Test ®shing with multimesh gillnets was carried out in Finland during the open water season (May±September) from 1994 to 1996 in a total of 19 lakes. Strati®ed random sampling with respect to depth zones was used. The ®shing was performed using the Nordic multimesh gillnets, which consist of 12 different mesh sizes varying between 5 and 55 mm from knot to knot. The size of the Nordic nets was 1:5 30 m2 , the size of a single mesh panel was 1:5 2:5 m2 and the order of mesh panels of different sizes was randomized originally and maintained permanently. A total of 2288 perch was caught in ®shing trials. The total length was measured to the nearest millimetre and weight to the nearest gram, and the mesh size in which each ®sh was caught was identi®ed. No distinction was made between gilled, tangled of wedged ®sh because in standard monitoring ®shing no difference are made between tangled, gilled and wedged ®sh. Therefore the same approach was used here. The length of perch varied from 5.0 to 40.4 cm and weights varied from 1 to 904 g.
M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
2.2. Girth data Data for girth/length/weight studies were collected in different ®shing trials with Nordic multimesh gillnets from 10 lakes in Finland in 1994 and 1996. Altogether 647 perch were caught and their total length and girth were measured to the nearest millimetre and weight to the gram. The girth was measured with a fabric gauge just in front of the dorsal ®n. The length range of ®shes was from 5.6 to 34.9 cm, the weight range from 2 to 595 g and the girth range from 27 to 215 mm. For example, for ®sh of 10 cm's length (from 9.6 to 10.5 cm) the mean girth was 53.9 mm (S.D. 6.6, range 47±69 mm). Corresponding values for ®sh of 20 and 30 cm's length were 110.2 mm (S.D. 8.1, range 96± 126 mm) and 173.6 mm (S.D. 6.4, range 166±188 mm), respectively. Direct girth measures would correspond to 13.5 mm (from knot to knot measure of mesh size corresponds to 14 of mesh perimeter, i.e. 53:9 4 13:5) (range 12±17 mm) mesh size for 10 cm's ®sh length and 27.5 mm's (range 24±31.5 mm) mesh size for 20 cm's ®sh length and 43.5 mm's mesh size (range 41±47 mm) for 30 cm's ®sh length. 2.3. The relationship between girth, length and weight The girth/length relationship was studied by ®tting a linear regression equation to observed data. For testing of linearity the data were divided into 2 cm length classes, i.e. groups, and for these grouped data a linear regression equation was ®tted. The deviations-fromlinearity SS was calculated as among-groups SS Ð regression SS (see Zar, 1996) and the corresponding degree of freedom (d.f.) was calculated as amonggroups d.f. Ð regression d.f. The deviations-fromlinearity MS was tested against within groups MS (error MS). Technically, this F-test-for-linearity is a one-way ANOVA for group means, the distances of which are tested against the ®tted regression line. The length/weight and the girth/weight relationships were studied by ®tting the logarithm-transformed form of the allometric equation, W aJ b
(1)
from which we get, ln W ln a b ln J
(2)
155
where W is the weight, J the girth or length and a and b are parameter estimates (Hayes et al., 1995). 2.4. Modelling gillnet selectivity The gillnet selectivity were modelled indirectly with a B-type gillnet selectivity curve, which means that the selectivity curve was based on comparing catches of one size class of ®sh by nets of several mesh sizes (see Hamley, 1975 for de®nition). We used the indirect method of Gulland and Harding (1961) combined with a mathematical model in which the relationships between mesh size, ®sh length, condition factor and the shape of the net selectivity curve were simultaneously solved for all mesh sizes, ®sh length classes and condition factor classes by using a nonlinear iterative method. With this model it is possible to seek the least-squares solution between the expected relative ef®ciency based on the model and the observed relative ef®ciency based on the gillnet catches (see Kurkilahti and Rask, 1996; Kurkilahti et al., 1998). The model is a response surface model within each condition factor class and the general approach is basically the same as described in Helser et al., 1998. The catch data were divided into 2 cm length classes and only distributions consisting of more than 30 ®sh were used for ®tting the net selectivity curves. In the model phase, 1294 ®sh were used in the analysis. 2.5. Assumptions of modelling The data used for model ®tting were pooled from a largenumberoflakesandusedasifthey were®shedfrom a single lake, even though soak time varied. Nevertheless, this does not affect the shape of type B curves, because the effect is the same for all mesh sizes in one lake and can therefore be ignored. This is valid as long as no mesh size is saturated during the time interval, which is a basic assumption of all studies of this kind. The second effect of pooling the data is that populations with different size structures were combined. The ®sh sample from one lake may consist of only small ®sh while that of another only of large ®sh. However, this has no effect. The only important assumption for pooling data from different lakes, and therefore from different samples, is that in each subsample each ®sh caught has the same probability of
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coming in contact with all the different mesh sizes. This assumption will be ful®lled when the sample size (number of ®sh caught) increases. The mathematical model is based on the assumption that the relative ef®ciency of mesh size i for ®sh length j is equal for those combinations of mi and Lj for which the ratio mi = a bLj is equal, assuming a and b are constant over all mesh sizes and ®sh lengths (®rst postulated by Baranov (1914) as cited by Hamley (1975)). We accept the foregoing with the extension that this is true when ®sh have similar Fulton's condition factor. 2.6. Model with Fulton's K The relationships between mesh size, ®sh length and condition factor were estimated from the nonlinear equation of the mesh size i of the observed peak catch (Maxjk) of each ®sh length class j on the corresponding ®sh length class j and condition factor class k. We used an equation, mik max a1 Kk a2 Ljk
a3 a4
(3)
where a1, a2, a3 and a4 are parameter estimates. Parameters a3 and a4 describe the X and Y co-ordinates, respectively, of the crossing point through which all the linear regression lines of different condition factor classes pass. If a3 and a4 are both 0, then all lines go through the origin (0, 0). From Eq. (3), the optimum mesh size mjk for each ®sh length class and condition factor class is calculated. The maximum possible catch of each length class of condition factor class (Maxjk from the hypothetical optimum mesh size mjk ) was estimated by using a normal probability density function, M^axjk
1=s
Maxjk p 2p exp 1=2 mjk =mi max 1=s2 (4)
The catch of each mesh size i by ®sh length class j and condition factor class k, expressed as the proportion of the estimated maximum catch M^axjk of that ®sh length class and condition factor class, is then plotted against the ratio of the best mesh size and the mesh used mjk =mi . For these data we ®tted a function originally based on the normal probability density function with an
additional parameter g " mjk Pijk 1 exp p mi
2 # 1 g21 ;
mjk 1 mi (5a)
or otherwise " Pijk 2 exp
mjk p mi
2 # 1 g22 ;
mjk >1 mi (5b)
where the parameters g1 and g2 affect the peakedness of the left and the right sides of the curve, respectively. For the extreme right limb of the net selectivity curve we de®ned a linear model Pijk 3 b1
mjk b2 mi
(5c)
which was used when Pijk 3 > Pijk 2 . This guarantees a better ®t of the model than the plain model based on the normal probability density function. 2.7. Initial parameters Initial parameters for mjk (a1 and a2) were obtained from the linear regression equation between the mesh size i of the maximum observed catch of ®sh size class j, the average length within the ®sh size class j and the average K within condition factor class k over the whole ®sh size class and condition factor class range (see Appendix A). The observed frequencies in different mesh sizes were used as weighting parameters for ®sh length and condition factor. The initial values of the parameters g1 and g2 were obtained by rearranging Eqs. (5a)±(5c): p ln Pijk =p mjk g1 ; 1 mi mjk =mi 1 or otherwise p ln Pijk =p g2 ; mjk =mi 1
1<
mjk
(6)
where h is 1.5. The initial values of parameters of Pijk(3) (b1 and b2) were obtained from linear regression equations ®tted for data while mjk =mi > 1:5. Parameters b1 and b2 were constrained in order to cause the
M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
regression slope of Pijk(3) to decline. So, the upper limit of b1 and the lower limit of b2 were set to 0 in the iteration phase. The nonlinear models were solved iteratively (with Newton method) by PROC NLIN in SAS (1990, 1997). The SAS version 6.12 estimates partial derivatives numerically and therefore no calculus for derivatives was needed. 2.8. Model without Fulton's K, i.e. simple model The relationship between mesh size and ®sh length was estimated from the linear regression equation of the mesh size i of the observed peak catch (Maxij) of each ®sh length class j and the corresponding ®sh length class j, mi max a1 Lj a2
(7)
where a1 and a2 are unknown. From this, the optimum mesh size mj for each ®sh length class was calculated. Eqs. (2)±(6) are identical for the simple model with the exception that all subscripts jk are changed to j, which denotes that all operations are done over the length class j only and not over both length class j and condition factor class k. This model is basically the same as the one used in Kurkilahti and Rask (1996). Initial parameters of mj (a1 and a2) were obtained from the linear regression equation between the mesh size i of the maximum observed catch of ®sh size class j and the ®sh size class j over the whole ®sh size class range. 2.9. Pooled relative ef®ciency For the simple model, the pooled relative ef®ciency (PRE) for a series of mesh sizes was calculated as the sum of Pij for all mesh sizes i with respect to length (L). For the K-model the PRE was calculated as the sum of Pijk for all mesh sizes i with respect to length (L) and condition factor (K). The PRE can be used, for example, for correcting the observed relative length frequency distribution due to the gillnet selectivity. The relative frequency (Rj) for each length class j is calculated for the simple model as Oj =PREj Rj P j Oj =PREj
(8)
157
and for the K-model as Oj =PRE Rj P P j k Ojk =PREjk
(9)
where Oj is observed frequency of length class j. 2.10. Resolution The basic problem with frequency data is that the resolution which can be used for model study depends directly on the smallest frequency observed. Here, when modelling the gillnet selectivity over several mesh sizes, ®sh length classes and condition factor class at the same time, the resolution is determined by the smallest peak frequency of the mesh size of the different length class-condition factor class combinations. Here the smallest peak frequency was 13 individuals and therefore the maximum possible resolution to be used in modelling was 0.076 1 (13 ). This means that all relative ef®ciencies were rounded to the nearest unit of 0.076's interval from 0 to 1, despite the observed peak frequency of any combination. 2.11. Con®dence limits and simulations The 95% con®dence interval for the PRE curve was calculated by Monte Carlo simulation (Johnson, 1987): for each species 5000 combinations of parameters were generated randomly from the multivariate normal distribution with the same estimates, standard errors and correlation structure between parameters as obtained from the model. This set of parameters was used to calculate the PRE curve over the same length range that the model was ®tted. Random vectors distributed as Np(m,s2,S) were generated from matrix equation, X AY m
(10)
where X is p dimensional multivariate normal distribution, A is provided by the Choleski factorization, which is the lower triangular matrix L for which LL0 S and S is p p dimensional correlation matrix of p parameters, Y is generated by p successive calls to a univariate normal generator with mean value of 0i and variance s2i (i 1 p), and m is a p 1 vector for parameter estimates (Johnson, 1987).
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M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
The 2.5% and 97.5 percentiles of simulated distribution of PRE values for each length class were chosen for lower and upper limits for the 95% con®dence interval. In a corresponding way the 95% con®dence interval was calculated for the corrected relative length frequency distribution (relative length frequency distribution is referred to as RLFD later on). The effect of pooling the data over condition factor classes was studied by simulation. We generated a data set with the parameters of the K-model over the length range from 11 to 31 cm (2 cm interval), condition factor range 0.8±1.3 and mesh sizes from 5 to 55 mm. This simulated data set was pooled over the condition factor and analysed with the simple model. The same number of observations was placed in all length classes observed in the original data set (see Appendix A) to be sure that each length class and condition factor combination would be equally weighted compared to the original data set. Because the length range of observed gillnet data was quite narrow, another simulation study for the effect of pooling was made by using the K-model. We generated a data set with the parameters of the Kmodel over the length range from 6 to 64 cm (2 cm interval). A randomly distributed vector for condition factor K was generated from equation, K N m; s2
(11)
where N is a normally distributed variate with mean value of m and variance of s2, and m a length b
(12)
This was derived from the relationship between Fulton's K and ®sh length (girth data set). A theoretical mesh size combination based on geometric series with factor 1.25 was used in this simulation. The theoretical
mesh size combination contained 23 mesh sizes from 1.049 to 142.108 mm including the exact 5 mm mesh size. The simulated data set was pooled over the condition factor and it was analysed with the simple model. With this data set the distance between the ends of the length range was large enough to guarantee that the distributions of the Fulton's K did not overlap. 2.12. Comparison of different models Differences between the simple model parameters of observed and simulated data sets were analysed by covariance analysis (by using dummy/indicator variables, Bates and Watts, 1988; Zar, 1996). The data behind the models of interest were lumped together (total data) and the reparametrized dummy variable model was ®tted for that data. In the reparametrized model the difference between the parameters of the observed data and the parameters of the simulated and pooled data was estimated at the same time. 3. Results 3.1. Length, girth, weight and Fulton's K The linear regression equation gave a good ®t for the girth/length data and the coef®cient of determination was high (r 2 0:972, Fig. 1A). Nevertheless, the test for linearity (the distance between the group means and the regression line) showed a statistically signi®cant deviation from this assumption (Table 1). Therefore we ®tted polynomial regression equations (2nd, 3rd, . . ., nth order) to the same data up to the order where
Table 1 Deviations from linearity in girth/length relationship Source of variation
d.f.
SS
Total
646
1273495.8
14 1 13 632
Among groups Linear regression Deviations from linearity (length groups) Within groups (error)
MS
F value
Pr > F
1234889.6 1230541.9 4347.6
334.4
5.47
0.0001
38606.3
61.1
M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
no further statistically signi®cant improvement was achieved. The second-order polynomial regression equation (Fig. 1A) resulted in a statistically signi®cant improvement (F1;644 18:16; p 2:34E 05)
159
but the third-order equation did not (F1;643 1:67; p 0:196 (Zar, 1996). Nevertheless, the linear regression equation (®rst-order polynomial regression equation) and the second-order equation were practi-
Fig. 1. Relationships between ®sh length, girth, weight and Fulton's K. The left side sub®gures represent empirical data and the right side sub®gures theoretical aspects. See text for detailed explanation.
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cally identical (Fig. 1A) and therefore we accepted the assumption of linearity between the girth and length. This result is also in accordance with classical geometry: the perimeter of the ball (2pr, here the ®sh girth) has the same dimension as the radius (r, here the ®sh length). The length/weight relationship was nonlinear and the exponent b of the allometric equation (Eq. (1)) differed from 3 (Fig. 1B, 95% con®dence limits: 3.1665±3.2086). As expected, given that the length/ girth relationship was linear, the girth/weight relationship was also nonlinear, and the exponent b in the allometric equation differed from 3 (Fig. 1C, 95% con®dence limits: 2.8268±2.8933). The length/Fulton's K relationship was linear although the coef®cient of determination was low r 2 0:354 indicating high variation within length class (Fig. 1D). If girth is linearly related to length, and weight is related to length according to an allometric function with parameters a and b (Eq. (1)), we can make following conclusions. Irrespective of the regression coef®cient between length and girth (Fig. 1E), the exponent b of Eq. (1) for length/weight (Fig. 1F) and for girth/weight (Fig. 1G) are the same as long as the intercept for the regression equation of the length/girth is 0 (Fig. 1E). The regression coef®cient between length and girth only affects the parameter a in Eq. (1) for girth/weight (Fig. 1G). If the intercept of length/girth is not 0, it does not affect the length/ weight relationship (Fig. 1E) but the girth/weight relationship. For example, an exponent b 3:1875 for the allometric length/weight relationship (Fig. 1B) and the intercept 8:7 for the linear length/girth relationship (Fig. 1A) result in an exponent b 2:86 and an intercept a 1 10 4 for the allometric girth/ weight equation, which are very close to the observed values 2.8601 and 1:1901 10 4 , respectively (see Fig. 1C). Fulton's condition factor K does not measure ®sh condition as it should according to its name, but rather the shape of ®sh (Cone, 1989, 1990). Strictly, it is a ratio between the observed ®sh weight and a hypothetical ®sh, the K of which is exactly 105 weight=length3 , where the weight is expressed in gram and the length in millimetre. If we keep the ®sh length constant (300 mm, for example, Fig. 1H) and let the Fulton's K vary from 0.7 to 1.8 with 0.1 unit interval, we will get ®sh weights from 189 to 486 g
with 27 g intervals. From the function of girth/weight W 4:6299 10 5 G3 we will get the inverse function for weight/girth G 27:8491 W 3 and values for the girth from 159.8 to 218.9. Numerical analysis between Fulton's K and girth shows an almost perfectly linear relationship (R2 0:9926, Fig. 1H). A second-order polynomial equation (K 0:0009 length2 0:0497 length 0:5011, R2 0:30, MSE 0:0139) described best the relationship between Fulton's K and ®sh length in the data set used for ®tting the gillnet selectivity models. Residuals were normally distributed. This equation was used in Monte Carlo simulations later on. It should be noted that the relationship between Fulton's K and ®sh length derived from the girth data set was linear with slightly smaller mean square error; 0.0136 (Fig. 1D). The equation from Fig. 1D was used in the theoretical mesh size combination simulation studies. 3.2. Gillnet selectivity models Four K-models were ®tted for the condition factor data set. All these models included b1, b2, g1 and g2 parameters but the part that described the relationship between mesh size and ®sh length differed. The model with four parameters (a1, a2, a3 and a4) gave the best ®t r 2 0:864 but it did not differ signi®cantly from the more simple model (a1, a2 and a4, r 2 0:862) (nested models, F1;119 1:9033, p 0:1703). The difference between the three parameter model (a1, a2 and a4) and the two parameter model (a1 and a2, r 2 0:855) was signi®cant (F1;118 5:5404, p 0:0202) and therefore the model with parameters a1, a2 and a4 (K-model later on) were chosen for further analysis. For comparison, another two parameter model (a2 and a4, r 2 0:831) was also ®tted; it differed clearly from the three parameter model (a1, a2 and a4) (F1;118 34:9183, p < 0:0001). A simple model (parameters a2 and a4) was ®tted for the data set that was pooled over the Fulton's condition factor class. Both chosen models ®tted well: the coef®cient of determination was 0.862 for K-model and 0.895 for the simpler model (Table 2). The residuals were symmetrically distributed along the ®sh length, the mesh size and condition factor (K) axes and no visual differences between any models ®tted were observed. Nevertheless, a slight positive correlation
M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
161
Table 2 Final parameters for the K and the simple modelsa Parameter
Estimate
S.E.
95% CI Lower
2
(1) Fulton's K-model: r 0:862, n 126, MSres a1 0.333 0.067 0.200 a2 1.1028 0.087 0.855 a4 1.153 0.521 2.184 0.066 0.038 0.142 b1 b2 0.257 0.079 0.101 g1 6.066 1.174 8.391 1.930 0.194 1.546 g2 (2) Simple model: r2 0:895, n 66, a2 1.435 0.034 a4 1.857 0.635 0.067 0.074 b1 b2 0.194 0.149 g1 6.706 1.322 1.693 0.149 g2
Asymptotic correlation matrix Upper 0:0231 0.466 1.201 0.121 0.009b 0.413 3.740 2.314
a1 1.000 1 0.475 0.020 0.008 0.362 0.270
MSres 0:0240 1.367 1.503 3.127 0.587 0.214 0.081b 0.104c 0.491 9.352 4.060 1.394 1.991
Simulation data based on the parameters of the K model (3) Simple model: r2 0:986, n 68, MSres 0:0016 1.486 0.013 1.460 1.513 a2 a4 2.743 0.214 3.170 2.315 b1 0.067 0.010 0.087 0.047 b2 0.267 0.024 0.219 0.315 5.823 0.368 6.558 5.089 g1 g2 1.802 0.069 1.665 1.939 (4) Difference model: r 2 0:937, n 134, MSres 1.435 0.025 1.386 a2 a4 1.857 0.466 2.779 b1 0.067 0.054 0.174 b2 0.194 0.109 0.023c 6.706 0.970 8.627 g1 g2 1.693 0.109 1.476 da2 da4 db1 db2 dg1 dg2
0.051 0.886 0.001 0.073 0.883 0.109
0.039 0.670 0.059 0.122 1.275 0.189
0.026 2.211 0.117 0.168 1.641 0.265
a2
a4
b1
b2
K1
K2
0.920 1 0.704 0.001 0.004 0.154 0.058
0.475 0.704 1 0.032 0.032 0.267 0.193
0.020 0.001 0.032 1 0.957 0.022 0.022
0.008 0.004 0.032 0.957 1 0.005 0.005
0.362 0.154 0.267 0.022 0.005 1 0.839
0.270 0.058 0.193 0.022 0.005 0.839 1
1 0.853 0.014 0.009 0.123 0.133
0.853 1 0.008 0.010 0.335 0.280
0.014 0.008 1 0.983 0.007 0.007
0.009 0.010 0.983 1 0.004 0.003
0.123 0.335 0.007 0.004 1 0.650
0.133 0.280 0.007 0.003 0.650 1
1 0.760 0.021 0.003 0.485 0.417
0.760 1 0.011 0.004 0.125 0.177
0.021 0.011 1 0.964 0.041 0.039
0.003 0.004 0.964 1 0.001 0.001
0.485 0.125 0.041 0.001 1 0.780
0.417 0.177 0.039 0.001 0.780 1
0:0082 1.485 0.935 0.041b 0.410 4.785 1.909 0.129 0.440 0.115 0.314 3.406 0.484
a Explanations of some symbols in the table: S.E.: asymptotic standard error of estimate, r2: coef®cient of determination, n: number of observations. The parameters (a1±a4) of the models describe the relationship between the mesh size m and the ®sh length L mj a1 Kk a2 Ljk a3 a4 . The other parameters describe the shape of net selectivity curves. In the difference model (4) parameters da2 dg2 measure the difference between the parameters of the models 2 and 3. Note that in the models 2±4 data were pooled over condition factor class. b The upper limit of b1 is de®ned to be 0. c The lower limit of b2 is de®ned to be 0.
was observed in all cases. Some outliers were detected: 11 out of 125 residuals of the K-model were larger than the studentized 95% con®dence range (1.980 with 119 d.f.) and correspondingly 4
out of 65 residuals of the simple model were larger than the studentized 95% con®dence range (2.000 with 59 d.f.). Both ratios were above the 5% level. All outliers were near the peak of the selectivity
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Fig. 2. Observed relative ef®ciencies and ®tted model with Monte Carlo simulated 95% con®dence bands for K-model (A) and simple model (B). Sub®gure C describes the simple model (black line) ®tted for the simulated data set based on the K-model (dashed line). Note that for the K-model the x-axis is the ratio of mjk =mi , for the simple level x-axis it is the ratio of mjk =mi .
curve indicating some inaccuracy in the shape of the model (Fig. 2). The shapes of the estimated B-type selectivity curves were very similar to each other and all the ``common'' parameters (a4, b1, b2, g1 and g2) that
estimate the same functional part of the model were within each others' 95% con®dence limits, respectively (Fig. 2, Table 2). Therefore, these parameters cannot be regarded as different. One reason for these similarities is that both curves were based on nearly
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163
Table 3 Parameters for the simple models of theoretical mesh size combination and simulation dataa Parameter
Estimate
S.E.
95% CI Lower
Upper
2
First-order model: r 0:942, n1 230, n2 54059, MSres 0:005936 a2 1.508 0.014 a4 2.768 0.120 b1 0.074 0.009 0.289 0.024 b2 g1 5.336 0.429 g2 1.767 0.079
1.481 3.005 0.093 0.242 6.182 1.611
1.535 2.532 0.056 0.335 4.491 1.922
Second-order model: r2 0:982, a0 a2 a4 b1 b2 g1 g2
0.004 1.300 1.359 0.085 0.267 6.318 1.571
0.004 1.343 1.018 0.064 0.320 5.268 1.733
n1 230, n2 54059, MSres 0:001921 0.004 0.000 1.322 0.011 1.188 0.087 0.075 0.005 0.293 0.014 5.793 0.266 1.652 0.041
a
The parameters a0 a4 of the models describe the polynomial relationship between the mesh size m and the ®sh length L mj a0 Lj2 a2 Lj a4 . The other parameters describe the shape of net selectivity curves.
the same model (parameters b1, b2, g1 and g2) and therefore the shape of the curve is already ®xed to some extent. Within both models the parameters g1 and g2 were not different for the same reason stated earlier (Table 2). When studying the pooling effect with the Nordic net mesh size combination, the model ®tted to the simulated data set was very close to the simple model ®tted to the original data. None of the parameters differed from each other; the 95% con®dence interval of parameters da2 dg2 that measure the difference between the parameters of the simple model and the simulated model overlapped zero (Fig. 2C, Table 2). Although the data set was generated without any random component, the pooling over the condition factor generated variation in the data, resulting in an r2 1 of 0.986. The MSE was less than 10 compared to the simple model ®tted to the original data. The biggest difference between the model curves and simulated ``observations'' was seen near the peak of the selectivity curve; the width of the peak was much wider according to the ``observations'' compared to the Kmodel from which the data set was generated or to the simple model that was ®tted. (Fig. 2C). This means that the shape of the peak is at least partly due to pooling. It should also be noted that the range of
observations within each length class was larger in the simulated data set than in the original one (see Appendix A). In the second simulation study for the pooling effect, we used the theoretical mesh size combination and a much wider length range for the data. The simple model gave a good ®t to the pooled simulated data and the residuals were symmetrically distributed over the length and mesh size axis (Table 3). The same kind of variation was observed as earlier in the ®rst simulation study of pooling (Fig. 2C). However, an additional effect was found when estimating the initial parameters for the iterative model ®tting; the relationship between length and mesh size was not linear but clearly curvilinear upwards. Therefore, we ®tted a secondorder polynomial function to describe this relationship and it was signi®cantly better than the simple model (F1;223 469:106, p < 0:0001) (Table 3). In practice, the ®rst- and second-order polynomial equations were identical over a length range from 6 to 44 cm and a clear difference was seen when ®sh length exceeded 50 cm (Fig. 3). We also tried to ®t the second-order polynomial model for the observed data set (length range 11±31 cm) but the parameter a0 (see Table 2) did not differ from 0 and therefore no additional improvement was achieved. This was expected because ®rst-
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Fig. 3. Relationship of ®sh length and mesh size in the simulated data set with a theoretical mesh size combination. First line describes the relationship with the simple model (®rst order) and second line describes the relationship with the second-order polynomial model.
and second-order functions were practically identical within that length range (Fig. 3). Because the selectivity curves of the K-model and the simple model were similar, the PRE curves of both
models were also similar and, correspondingly, the RLFD of both models were practically identical. For both models the 95% con®dence intervals of the PRE curves had the same pattern as the estimated curves (Fig. 4). Only the levels of the PRE curves were different; the average level of the PRE curve of the K-model was 1.08 times higher than the level of the simple model. Although the levels of the PRE curves were different, the RLFDs of both models were practically identical. Both the 95% con®dence range of the PRE curve and the 95% con®dence range of the RLFD for the K-model were 1.11 times wider than the ranges for the simple model (Fig. 4). However, the RLFD of different models are similar only when using the very same structure and distribution of Fulton's K of the population from which the both models were derived. In this case we used the second-order polynomial equation derived from the
Fig. 4. The PRE (with 95% con®dence interval) of Nordic multimesh gillnets for the K-model (upper) and the simple model (lower). The corrected RLFD was calculated for a hypothetical ®sh population with equal catch in each length class. The thick curves represent the K-model and thin curves represent the simple model. The upper and the lower limits of 95% con®dence interval are represented. The solid horizontal line represents the observed relative length distribution. The Fulton's K for each length class was calculated from the second-order polynomial equation derived from the original data set.
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165
Fig. 5. Effect of Fulton's K on PRE curve of Nordic net (upper) and relative ef®ciency of 14 and 40 mm individual mesh sizes (lower).
original data set. If two populations have different distributions of Fulton's K, the PRE curves and therefore the corrected RLFD are also different (Fig. 5), because the Fulton's K (a1) affects the regression coef®cient between ®sh length and mesh size (Eq. (3)). However, the difference increases with increasing ®sh length and the effect of Fulton's K depends on how the length interval under study and the used mesh size combination ®t each other. When using an individual mesh size of 14 mm and 5 cm length interval from 10 to 15 cm the difference between the areas that selectivity curves cover was only 4%. The corresponding difference with a similar 5 cm length interval from 25 to 30 cm but using a 40 mm mesh size was 46%. With Nordic mesh size combination the difference was 2% for length interval 10±15 cm and 1% for length interval from 25 to 30 cm (Fig. 5). 3.3. Effect of Fulton's K on 95% con®dence intervals Monte Carlo simulated 95% con®dence intervals of the simple model and the K-model were practically identical when using a ®xed K (Figs. 2, 4 and 6).
However, when taking into account the fact that the K is normally distributed within each length class and expanding the simulations (K having a standard deviation of 0.1181 and zero correlation with other parameters) we found that the 95% con®dence interval is on the average 1.8 times larger compared to con®dence interval with ®xed K (Fig. 6). Most of this increase of the 95% con®dence interval is due to deviation of K, because the simulated con®dence intervals with ®xed (predicted) model parameters and random K were 1.55 times larger than the simulated con®dence intervals with random parameters and ®xed K. Monte Carlo simulations with different standard deviations of K and 24 mm mesh size showed that the width of the 95% con®dence interval is directly dependent on the magnitude of the standard deviation of K in the population (Fig. 7A). The lower 95% con®dence interval increased about 5 mm and the upper 95% con®dence interval about 6 mm per 0.1 unit increase in the standard deviation of K. Due to parametrization of the K-model the con®dence interval of the optimum mesh size increases
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Fig. 6. Model predicted relative ef®ciency curves for 24 mm mesh size and model predicted optimum mesh sizes for both the simple model and the K-model. Monte Carlo simulated 95% con®dence intervals are shown with thin line, model predicted curves are shown with thick line. Linear regression equations estimating the upper and lower 95% con®dence bands and the predicted optimum mesh size of the K-model are shown in the lowest ®gure. Cut points of 24 and 35 mm mesh sizes are presented.
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167
when ®sh length increases (Fig. 6). Both the lower and the upper 95% con®dence intervals increase about 0.9 mm per 10 cm increase in ®sh length: at 22 cm both intervals are about 2 mm and at 45 cm both intervals are about 4 mm (Fig. 7B). Correspondingly, when studying con®dence intervals along the ®sh length axis, both the lower and the upper 95% con®dence intervals increase about 0.6 cm per 10 cm increase in ®sh length: at 15 cm both intervals are about 1 cm and at 46 cm both intervals are already about 3 cm (Fig. 7C). 4. Discussion
Fig. 7. The change of 95% con®dence intervals against the standard deviation of Fulton's K and ®sh length. In sub®gures A and C the con®dence interval is measured in ®sh length (cm) and in sub®gure B the con®dence interval is measured in mesh size (mm). Ninety-®ve per cent con®dence intervals for different standard deviations of K are calculated for 24 mm mesh size.
The B-type indirect gillnet selectivity curve is based on an assumption of the geometrical similarity of selectivity curves over ®sh length range. Mathematically this relationship can be described as the relative ef®ciency of mesh size (mi) for ®sh length (L) is equal for those combinations of mi and L for which the ratio mi = a1 L a2 is equal, assuming a1 and a2 are constant over all mesh sizes and ®sh lengths (®rst postulated by Baranov (1914) as cited by Hamley, 1975). This basic assumption is, in fact, based on the assumption of isometric growth of ®sh over ®sh length range, which is not strictly true for most ®sh species. In an allometric equation (W aLb , where W is weight and L length, a and b parameters) (Hayes et al., 1995) describing the length±weight relationship of ®sh the parameter b is usually >3, which means that longer ®shes are also relatively larger in other dimensions including girth. In practice this means that large ®sh have, on the average, bigger girth than expected according to the geometrical similarity assumption and small ®sh have smaller girth than expected. Parameters (b1, b2, g1 and g2) that estimate the same functional part of the B-type gillnet selectivity model were within each other's 95% con®dence limits, respectively (Fig. 2, Table 2) indicating that due to pooling the shape of the B-type curve did not change in practice. The relationship between ®sh length and mesh size was de®ned as linear within condition factor class from which follows that the overall relationship is nonlinear (Eq. (3)). This nonlinearity was also observed in the simulation study with wide length range (6±64 cm) and theoretical mesh size combination (Table 3, Fig. 3). Nevertheless, with the observed
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data set and in the simulation study with narrow length range this nonlinearity was not observed. It seems reasonable that within a short length range the linear simpli®cation of Eq. (3) is close enough to describe the true nonlinear phenomenon. However, the consequence from this is that no kind of extrapolation of the ``linear model'', i.e. simple model in this paper, can be done. A gillnet selectivity model derived from a certain length range cannot be shifted to another totally different length range. In ®sh stock assessment the effect of ®sh shape has some consequence. Usually a certain mesh size is chosen to be the smallest allowed in the ®sheries. This mesh size must be chosen so that it takes care of the seasonal and year-to-year variation in the ®sh shape, i.e. the condition factor. If regulation is based on a traditional approach, measuring the gillnet selectivity without taking care of the condition factor, the selected smallest mesh size might be several millimetre (from knot to knot) too small depending on the species and ®sh length and correspondingly a large number of under sized ®sh might be caught. Although the model derived from the simulated data set was very close to that of the original data, the simulated data had a much larger range of observations over the mesh size than the original data set (see Appendix A). Only in the condition factor class 1.2 were the original and observed data sets close. This indicates that some factor affecting the shape of the Btype gillnet selectivity curve is still missing. In a recent paper of Millar (2000) the modelling approach used in this paper is criticized to be biased in relation to parameter estimates which might be the reason for the difference between the observed data and the simulated data. However, we feel that the modelling approach we used is still valid to describe the relationship between mesh size, ®sh length and Fulton's condition factor on a general level. If we suppose that the B-type indirect gillnet selectivity curve is the same for both slim and fat ®sh type, which need not necessarily be true, the PRE of the slim population will have the same amplitude and also the same wave length on a log scale as the fat population. The overall levels of PRE curves of both populations are the same. The phase of the oscillation of the PRE is
different, because they have different coef®cients of regression between mesh size and ®sh length (Fig. 4). While regression coef®cient increases the optimum size of ®sh for certain mesh size shifts towards the left. A fat ®sh needs to be much shorter to achieve the same girth as a slim ®sh with the same girth (Jensen, 1995). When using a mesh size combination based on a geometric series (as the Nordic net here) this has no practical meaning because adjacent mesh sizes cover each other and correct this error. With the Nordic nets the error in the PRE curve is small; usually a few per cent, and it stays within the 95% con®dence interval due to parameter uncertainty of the model. Therefore, the observed catch per unit effort (CPUE) and the shape of length or age distributions are reliable from population to population with respect to condition factor. But when using individual mesh sizes or irregular or nongeometric series mesh size combinations the error in the PRE curve can be tens of per cent within a certain ®sh length range and therefore conclusions drawn from these kinds of studies can be seriously erroneous. Acknowledgements The study was ®nancially supported by the Foundation for the Study of Natural Resources in Finland, the University of Turku, Finland, the Finnish Game and Fisheries Research Institute, the Swedish Environmental Protection Agency, the Swedish National Board of Fisheries and the Norwegian Institute for Nature Research. We thank Harri Helminen, Jouko Sarvala, Toomas Tammaru and one anonymous referee for their helpful comments when preparing this paper. The paper is a contribution from the Nordic Freshwater Fish Sampling Group (NOFF).
Appendix A Data and starting values for iteration phase used for modelling the gillnet selectivity of perch. The bordered area shows the range of the simulated data set.
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Parameter
Start value K-model
a1 a2 b1 b2 g1 g2
169
1.9562 1.0145 0.0210 0.1337 2.6598 2.1822
Simple model 1.3432 1.9958 0.0265 0.1059 2.7541 2.0385
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References Baranov, F.I., 1914. The capture of ®sh by gillnets. Mater. Poznaniyu Russ. Rybolov. 3 (6) 56-99 (W.E. Ricker, partially Trans. from Russian). Bates, D.M., Watts, D.G., 1988. Nonlinear Regression Analysis and Its Applications. Wiley, New York, p. 365. Cone, R.S., 1989. The need to reconsider the use of condition indices in ®shery science. Trans. Am. Fish. Soc. 118, 510±514. Cone, R.S., 1990. Comments for the paper: the need to reconsider the use of condition indices in ®shery science (Cone, 1989). Trans. Am. Fish. Soc. 119, 1052±1058. Gulland, J.A., Harding, D., 1961. The selection of Clarias mossambicus (Peters) by nylon gillnets. J. Cons. Int. Explor. Mer. 26, 215±222. Hamley, J.M., 1975. Review of gillnet selectivity. J. Fish. Res. Bd. Can. 32, 1943±1969. Hayes, D.B., Brodziak, J.K.T., O'Gorman, J.B., 1995. Ef®ciency and bias of estimators and sampling designs for determining length± weight relationships of ®sh. Can. J. Fish. Aquat. Sci. 52, 84±92. Helser, T.E., Condrey, R.E., Geaghan, J.P., 1998. Estimating gillnet selectivity using nonlinear response surface regression. Can. J. Fish. Aquat. Sci. 55, 1328±1337.
Jensen, J.W., 1986. Gillnet selectivity and the ef®ciency of alternative combinations of mesh sizes for some freshwater ®sh. J. Fish. Biol. 28, 637±646. Jensen, J.W., 1995. Evaluating catches of salmonids taken by gillnets. J. Fish. Biol. 46, 862±871. Johnson, M.E., 1987. Multivariate Statistical Simulation. Wiley, New York. Kurkilahti, M., Rask, M., 1996. A comparative study of the usefulness and catchability of multimesh gillnets and gillnet series in sampling of perch (Perca ¯uviatilis L.) and roach (Rutilus rutilus L.). Fish. Res. 27, 243±260. Kurkilahti, M., Appelberg, M., Bergstrand, E., Enderlein, O., 1998. An indirect estimate of bimodal gillnet selectivity of smelt. J. Fish. Biol. 52, 243±254. Millar, R.B., 2000. Untangling the confusion surrounding the estimation of gillnet selectivity. Can. J. Fish. Aquat. Sci. 57, 507±511. SAS, 1990. SAS/STAT1 User's Guide, Version 6, Vol. 2, 4th Edition. SAS Institute Inc., Cary, NC. SAS, 1997. SAS/STAT1 Software: Changes and Exchangements Through Release 6.12. SAS Institute Inc., Cary, NC. Zar, J.H., 1996. Biostatistical Analysis, 3rd Edition. Prentice-Hall, Upper Saddle River, NJ, pp. 245±662.
Effect of ®sh shape on gillnet selectivity: a study with Fulton's condition factor M. Kurkilahtia,*, M. Appelbergb, T. Hesthagenc, M. Raskd a
Finnish Game and Fisheries Research Institute, RymaÈttylaÈ Fisheries Research Station, FIN-21150 RoÈoÈlaÈ, Finland b Institute of Freshwater Research, Fish Monitoring Group, S-178 93 Drottningholm, Sweden c Norwegian Institute for Nature Research, Tungasletta 2, N-7485 Trondheim, Norway d Finnish Game and Fisheries Research Institute, Evo Fisheries Research Station, FIN-16970 Evo, Finland Received 29 February 2000; received in revised form 29 September 2000; accepted 21 November 2000
Abstract The Eurasian perch, Perca ¯uviatilis (L.), was ®shed with Nordic multimesh gillnets, the mesh size combination of which was based on geometric series. From the test ®shing data the ®sh girth was found to be linearly related to the third root of ®sh weight and therefore the ®sh girth could be estimated indirectly with Fulton's condition factor K. Two different response surface models based on B-type gillnet selectivity model were ®tted to the empirical data: one model with Fulton's K and one without (simple model). The Fulton's K-model had narrower Monte Carlo simulated 95% con®dence interval than the simple model. Simulation studies showed that the traditional simple model without Fulton's K does not have a linear relationship between mesh size and ®sh length, which violates the classical principle of Baranov's theorem on geometric similarity of ®sh of different sizes. Therefore the simple model derived from one population is not applicable to another population if there are differences in ®sh condition. However, if as is the case for Nordic multimesh gillnets with a mesh size combination based on geometric series like Nordic multimesh gillnets, this error is minimal because adjacent mesh sizes cover each other and correct this error. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Condition factor; Multimesh gillnet; Geometric series; Net selectivity curves; Mathematical model
1. Introduction The girth is the main size factor determining the size of ®sh caught by different mesh sizes. Nevertheless, most gillnet selectivity models have been derived for ®sh length, not for the ®sh girth as they strictly should be. The main reason for this is that measuring girth is dif®cult, time-consuming, and therefore expensive
*
Corresponding author. Tel.: 358-20-57-51-731; fax: 358-20-57-51-731. E-mail address: [email protected] (M. Kurkilahti).
if compared to length and weight measurements of ®sh. Another reason is that we are usually interested in ®sh size, which is routinely measured as length, and therefore the ®sh girth based model would be adjusted to ®sh length anyway. Nevertheless, the length based gillnet selectivity models are justi®ed if the ®sh girth and ®sh length are linearly related to each other. This relationship is also important because the ®sh girth is the main factor in determining the optimum mesh size that will catch a ®sh of certain size and the mesh size/®sh length relationship is usually assumed to be linear (see Hamley, 1975, for example).
0165-7836/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 7 8 3 6 ( 0 0 ) 0 0 3 0 1 - 5
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Nomenclature g K Lj mi mjk
Maxjk Pijk W
factor to determine the peakedness of the net selectivity curve Fulton's condition factor, K 100W=L3 fish length in centimetres size of mesh i from knot to knot in millimetres optimum mesh size for fish length class j of condition factor class (Kk), which is expressed as a regression equation between the mesh size i of the observed maximum catch (Maxjk) of each fish size class j and the corresponding fish size class j and condition factor class k observed maximum catch of fish length class j from mesh size mi and condition factor class Kk relative efficiency of mi for fish of size class j of condition factor class Kk with constraint that max Pijk 1 fish weight in grams
Greek symbol p s constant s 1= 2p, from which follows that max Pijk 1 Another problem arises when different individuals of a ®sh species have equal length but different girth. This is a normal situation in nature; individuals are slim or fat and they differ within and between populations. Such ®sh have different probabilities of capture with a particular mesh size even if they have the same length, which will make the gillnet selectivity modelling more complex. When gillnet data are collected from the same population, this within-population variation causes inaccuracy in selectivity models: larger error terms in the least-squares models and smaller coef®cients of determination (r2). The between-population variation has, at least, two additional consequences. First, a gillnet selectivity model derived from one population might not be directly applicable to another population without taking the different shape into account in oneway or another. Second, combining data on ®shes of different shape causes dif®culties in modelling gillnet selectivity models can be more or less biased if the different shape is ignored.
If the girth (G) is linearly related to the third root of the ®sh weight (W), it can be used for indirect estimation of girth. However, the third root of the ®sh weight is not useful itself; but it must be adjusted ®sh length (L) to make ®shes of the samepshape but different length comparable. Therefore, if 3 W =G is linear then W=G3 is linear and supposing that G aL b, where a and b are estimates, this adjustment can be done, for example, by Fulton's condition factor (referred to as K here after), which has been shown to affect the modal length of ®sh in gillnet catches (Jensen, 1986, 1995). In this paper we study the linear relationships between ®sh length, girth and weight and the effect of Fulton's condition factor K on the mesh size selectivity of gillnets. We ®tted two different mathematical gillnet selectivity models, one with a condition factor and another without, for the same experimental catch data of perch (Perca ¯uviatilis (L.)) caught by Nordic multimesh gillnets and compare the effect of these two types of models. 2. Material and methods 2.1. Gillnet data Test ®shing with multimesh gillnets was carried out in Finland during the open water season (May±September) from 1994 to 1996 in a total of 19 lakes. Strati®ed random sampling with respect to depth zones was used. The ®shing was performed using the Nordic multimesh gillnets, which consist of 12 different mesh sizes varying between 5 and 55 mm from knot to knot. The size of the Nordic nets was 1:5 30 m2 , the size of a single mesh panel was 1:5 2:5 m2 and the order of mesh panels of different sizes was randomized originally and maintained permanently. A total of 2288 perch was caught in ®shing trials. The total length was measured to the nearest millimetre and weight to the nearest gram, and the mesh size in which each ®sh was caught was identi®ed. No distinction was made between gilled, tangled of wedged ®sh because in standard monitoring ®shing no difference are made between tangled, gilled and wedged ®sh. Therefore the same approach was used here. The length of perch varied from 5.0 to 40.4 cm and weights varied from 1 to 904 g.
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2.2. Girth data Data for girth/length/weight studies were collected in different ®shing trials with Nordic multimesh gillnets from 10 lakes in Finland in 1994 and 1996. Altogether 647 perch were caught and their total length and girth were measured to the nearest millimetre and weight to the gram. The girth was measured with a fabric gauge just in front of the dorsal ®n. The length range of ®shes was from 5.6 to 34.9 cm, the weight range from 2 to 595 g and the girth range from 27 to 215 mm. For example, for ®sh of 10 cm's length (from 9.6 to 10.5 cm) the mean girth was 53.9 mm (S.D. 6.6, range 47±69 mm). Corresponding values for ®sh of 20 and 30 cm's length were 110.2 mm (S.D. 8.1, range 96± 126 mm) and 173.6 mm (S.D. 6.4, range 166±188 mm), respectively. Direct girth measures would correspond to 13.5 mm (from knot to knot measure of mesh size corresponds to 14 of mesh perimeter, i.e. 53:9 4 13:5) (range 12±17 mm) mesh size for 10 cm's ®sh length and 27.5 mm's (range 24±31.5 mm) mesh size for 20 cm's ®sh length and 43.5 mm's mesh size (range 41±47 mm) for 30 cm's ®sh length. 2.3. The relationship between girth, length and weight The girth/length relationship was studied by ®tting a linear regression equation to observed data. For testing of linearity the data were divided into 2 cm length classes, i.e. groups, and for these grouped data a linear regression equation was ®tted. The deviations-fromlinearity SS was calculated as among-groups SS Ð regression SS (see Zar, 1996) and the corresponding degree of freedom (d.f.) was calculated as amonggroups d.f. Ð regression d.f. The deviations-fromlinearity MS was tested against within groups MS (error MS). Technically, this F-test-for-linearity is a one-way ANOVA for group means, the distances of which are tested against the ®tted regression line. The length/weight and the girth/weight relationships were studied by ®tting the logarithm-transformed form of the allometric equation, W aJ b
(1)
from which we get, ln W ln a b ln J
(2)
155
where W is the weight, J the girth or length and a and b are parameter estimates (Hayes et al., 1995). 2.4. Modelling gillnet selectivity The gillnet selectivity were modelled indirectly with a B-type gillnet selectivity curve, which means that the selectivity curve was based on comparing catches of one size class of ®sh by nets of several mesh sizes (see Hamley, 1975 for de®nition). We used the indirect method of Gulland and Harding (1961) combined with a mathematical model in which the relationships between mesh size, ®sh length, condition factor and the shape of the net selectivity curve were simultaneously solved for all mesh sizes, ®sh length classes and condition factor classes by using a nonlinear iterative method. With this model it is possible to seek the least-squares solution between the expected relative ef®ciency based on the model and the observed relative ef®ciency based on the gillnet catches (see Kurkilahti and Rask, 1996; Kurkilahti et al., 1998). The model is a response surface model within each condition factor class and the general approach is basically the same as described in Helser et al., 1998. The catch data were divided into 2 cm length classes and only distributions consisting of more than 30 ®sh were used for ®tting the net selectivity curves. In the model phase, 1294 ®sh were used in the analysis. 2.5. Assumptions of modelling The data used for model ®tting were pooled from a largenumberoflakesandusedasifthey were®shedfrom a single lake, even though soak time varied. Nevertheless, this does not affect the shape of type B curves, because the effect is the same for all mesh sizes in one lake and can therefore be ignored. This is valid as long as no mesh size is saturated during the time interval, which is a basic assumption of all studies of this kind. The second effect of pooling the data is that populations with different size structures were combined. The ®sh sample from one lake may consist of only small ®sh while that of another only of large ®sh. However, this has no effect. The only important assumption for pooling data from different lakes, and therefore from different samples, is that in each subsample each ®sh caught has the same probability of
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coming in contact with all the different mesh sizes. This assumption will be ful®lled when the sample size (number of ®sh caught) increases. The mathematical model is based on the assumption that the relative ef®ciency of mesh size i for ®sh length j is equal for those combinations of mi and Lj for which the ratio mi = a bLj is equal, assuming a and b are constant over all mesh sizes and ®sh lengths (®rst postulated by Baranov (1914) as cited by Hamley (1975)). We accept the foregoing with the extension that this is true when ®sh have similar Fulton's condition factor. 2.6. Model with Fulton's K The relationships between mesh size, ®sh length and condition factor were estimated from the nonlinear equation of the mesh size i of the observed peak catch (Maxjk) of each ®sh length class j on the corresponding ®sh length class j and condition factor class k. We used an equation, mik max a1 Kk a2 Ljk
a3 a4
(3)
where a1, a2, a3 and a4 are parameter estimates. Parameters a3 and a4 describe the X and Y co-ordinates, respectively, of the crossing point through which all the linear regression lines of different condition factor classes pass. If a3 and a4 are both 0, then all lines go through the origin (0, 0). From Eq. (3), the optimum mesh size mjk for each ®sh length class and condition factor class is calculated. The maximum possible catch of each length class of condition factor class (Maxjk from the hypothetical optimum mesh size mjk ) was estimated by using a normal probability density function, M^axjk
1=s
Maxjk p 2p exp 1=2 mjk =mi max 1=s2 (4)
The catch of each mesh size i by ®sh length class j and condition factor class k, expressed as the proportion of the estimated maximum catch M^axjk of that ®sh length class and condition factor class, is then plotted against the ratio of the best mesh size and the mesh used mjk =mi . For these data we ®tted a function originally based on the normal probability density function with an
additional parameter g " mjk Pijk 1 exp p mi
2 # 1 g21 ;
mjk 1 mi (5a)
or otherwise " Pijk 2 exp
mjk p mi
2 # 1 g22 ;
mjk >1 mi (5b)
where the parameters g1 and g2 affect the peakedness of the left and the right sides of the curve, respectively. For the extreme right limb of the net selectivity curve we de®ned a linear model Pijk 3 b1
mjk b2 mi
(5c)
which was used when Pijk 3 > Pijk 2 . This guarantees a better ®t of the model than the plain model based on the normal probability density function. 2.7. Initial parameters Initial parameters for mjk (a1 and a2) were obtained from the linear regression equation between the mesh size i of the maximum observed catch of ®sh size class j, the average length within the ®sh size class j and the average K within condition factor class k over the whole ®sh size class and condition factor class range (see Appendix A). The observed frequencies in different mesh sizes were used as weighting parameters for ®sh length and condition factor. The initial values of the parameters g1 and g2 were obtained by rearranging Eqs. (5a)±(5c): p ln Pijk =p mjk g1 ; 1 mi mjk =mi 1 or otherwise p ln Pijk =p g2 ; mjk =mi 1
1<
mjk
(6)
where h is 1.5. The initial values of parameters of Pijk(3) (b1 and b2) were obtained from linear regression equations ®tted for data while mjk =mi > 1:5. Parameters b1 and b2 were constrained in order to cause the
M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
regression slope of Pijk(3) to decline. So, the upper limit of b1 and the lower limit of b2 were set to 0 in the iteration phase. The nonlinear models were solved iteratively (with Newton method) by PROC NLIN in SAS (1990, 1997). The SAS version 6.12 estimates partial derivatives numerically and therefore no calculus for derivatives was needed. 2.8. Model without Fulton's K, i.e. simple model The relationship between mesh size and ®sh length was estimated from the linear regression equation of the mesh size i of the observed peak catch (Maxij) of each ®sh length class j and the corresponding ®sh length class j, mi max a1 Lj a2
(7)
where a1 and a2 are unknown. From this, the optimum mesh size mj for each ®sh length class was calculated. Eqs. (2)±(6) are identical for the simple model with the exception that all subscripts jk are changed to j, which denotes that all operations are done over the length class j only and not over both length class j and condition factor class k. This model is basically the same as the one used in Kurkilahti and Rask (1996). Initial parameters of mj (a1 and a2) were obtained from the linear regression equation between the mesh size i of the maximum observed catch of ®sh size class j and the ®sh size class j over the whole ®sh size class range. 2.9. Pooled relative ef®ciency For the simple model, the pooled relative ef®ciency (PRE) for a series of mesh sizes was calculated as the sum of Pij for all mesh sizes i with respect to length (L). For the K-model the PRE was calculated as the sum of Pijk for all mesh sizes i with respect to length (L) and condition factor (K). The PRE can be used, for example, for correcting the observed relative length frequency distribution due to the gillnet selectivity. The relative frequency (Rj) for each length class j is calculated for the simple model as Oj =PREj Rj P j Oj =PREj
(8)
157
and for the K-model as Oj =PRE Rj P P j k Ojk =PREjk
(9)
where Oj is observed frequency of length class j. 2.10. Resolution The basic problem with frequency data is that the resolution which can be used for model study depends directly on the smallest frequency observed. Here, when modelling the gillnet selectivity over several mesh sizes, ®sh length classes and condition factor class at the same time, the resolution is determined by the smallest peak frequency of the mesh size of the different length class-condition factor class combinations. Here the smallest peak frequency was 13 individuals and therefore the maximum possible resolution to be used in modelling was 0.076 1 (13 ). This means that all relative ef®ciencies were rounded to the nearest unit of 0.076's interval from 0 to 1, despite the observed peak frequency of any combination. 2.11. Con®dence limits and simulations The 95% con®dence interval for the PRE curve was calculated by Monte Carlo simulation (Johnson, 1987): for each species 5000 combinations of parameters were generated randomly from the multivariate normal distribution with the same estimates, standard errors and correlation structure between parameters as obtained from the model. This set of parameters was used to calculate the PRE curve over the same length range that the model was ®tted. Random vectors distributed as Np(m,s2,S) were generated from matrix equation, X AY m
(10)
where X is p dimensional multivariate normal distribution, A is provided by the Choleski factorization, which is the lower triangular matrix L for which LL0 S and S is p p dimensional correlation matrix of p parameters, Y is generated by p successive calls to a univariate normal generator with mean value of 0i and variance s2i (i 1 p), and m is a p 1 vector for parameter estimates (Johnson, 1987).
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The 2.5% and 97.5 percentiles of simulated distribution of PRE values for each length class were chosen for lower and upper limits for the 95% con®dence interval. In a corresponding way the 95% con®dence interval was calculated for the corrected relative length frequency distribution (relative length frequency distribution is referred to as RLFD later on). The effect of pooling the data over condition factor classes was studied by simulation. We generated a data set with the parameters of the K-model over the length range from 11 to 31 cm (2 cm interval), condition factor range 0.8±1.3 and mesh sizes from 5 to 55 mm. This simulated data set was pooled over the condition factor and analysed with the simple model. The same number of observations was placed in all length classes observed in the original data set (see Appendix A) to be sure that each length class and condition factor combination would be equally weighted compared to the original data set. Because the length range of observed gillnet data was quite narrow, another simulation study for the effect of pooling was made by using the K-model. We generated a data set with the parameters of the Kmodel over the length range from 6 to 64 cm (2 cm interval). A randomly distributed vector for condition factor K was generated from equation, K N m; s2
(11)
where N is a normally distributed variate with mean value of m and variance of s2, and m a length b
(12)
This was derived from the relationship between Fulton's K and ®sh length (girth data set). A theoretical mesh size combination based on geometric series with factor 1.25 was used in this simulation. The theoretical
mesh size combination contained 23 mesh sizes from 1.049 to 142.108 mm including the exact 5 mm mesh size. The simulated data set was pooled over the condition factor and it was analysed with the simple model. With this data set the distance between the ends of the length range was large enough to guarantee that the distributions of the Fulton's K did not overlap. 2.12. Comparison of different models Differences between the simple model parameters of observed and simulated data sets were analysed by covariance analysis (by using dummy/indicator variables, Bates and Watts, 1988; Zar, 1996). The data behind the models of interest were lumped together (total data) and the reparametrized dummy variable model was ®tted for that data. In the reparametrized model the difference between the parameters of the observed data and the parameters of the simulated and pooled data was estimated at the same time. 3. Results 3.1. Length, girth, weight and Fulton's K The linear regression equation gave a good ®t for the girth/length data and the coef®cient of determination was high (r 2 0:972, Fig. 1A). Nevertheless, the test for linearity (the distance between the group means and the regression line) showed a statistically signi®cant deviation from this assumption (Table 1). Therefore we ®tted polynomial regression equations (2nd, 3rd, . . ., nth order) to the same data up to the order where
Table 1 Deviations from linearity in girth/length relationship Source of variation
d.f.
SS
Total
646
1273495.8
14 1 13 632
Among groups Linear regression Deviations from linearity (length groups) Within groups (error)
MS
F value
Pr > F
1234889.6 1230541.9 4347.6
334.4
5.47
0.0001
38606.3
61.1
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no further statistically signi®cant improvement was achieved. The second-order polynomial regression equation (Fig. 1A) resulted in a statistically signi®cant improvement (F1;644 18:16; p 2:34E 05)
159
but the third-order equation did not (F1;643 1:67; p 0:196 (Zar, 1996). Nevertheless, the linear regression equation (®rst-order polynomial regression equation) and the second-order equation were practi-
Fig. 1. Relationships between ®sh length, girth, weight and Fulton's K. The left side sub®gures represent empirical data and the right side sub®gures theoretical aspects. See text for detailed explanation.
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cally identical (Fig. 1A) and therefore we accepted the assumption of linearity between the girth and length. This result is also in accordance with classical geometry: the perimeter of the ball (2pr, here the ®sh girth) has the same dimension as the radius (r, here the ®sh length). The length/weight relationship was nonlinear and the exponent b of the allometric equation (Eq. (1)) differed from 3 (Fig. 1B, 95% con®dence limits: 3.1665±3.2086). As expected, given that the length/ girth relationship was linear, the girth/weight relationship was also nonlinear, and the exponent b in the allometric equation differed from 3 (Fig. 1C, 95% con®dence limits: 2.8268±2.8933). The length/Fulton's K relationship was linear although the coef®cient of determination was low r 2 0:354 indicating high variation within length class (Fig. 1D). If girth is linearly related to length, and weight is related to length according to an allometric function with parameters a and b (Eq. (1)), we can make following conclusions. Irrespective of the regression coef®cient between length and girth (Fig. 1E), the exponent b of Eq. (1) for length/weight (Fig. 1F) and for girth/weight (Fig. 1G) are the same as long as the intercept for the regression equation of the length/girth is 0 (Fig. 1E). The regression coef®cient between length and girth only affects the parameter a in Eq. (1) for girth/weight (Fig. 1G). If the intercept of length/girth is not 0, it does not affect the length/ weight relationship (Fig. 1E) but the girth/weight relationship. For example, an exponent b 3:1875 for the allometric length/weight relationship (Fig. 1B) and the intercept 8:7 for the linear length/girth relationship (Fig. 1A) result in an exponent b 2:86 and an intercept a 1 10 4 for the allometric girth/ weight equation, which are very close to the observed values 2.8601 and 1:1901 10 4 , respectively (see Fig. 1C). Fulton's condition factor K does not measure ®sh condition as it should according to its name, but rather the shape of ®sh (Cone, 1989, 1990). Strictly, it is a ratio between the observed ®sh weight and a hypothetical ®sh, the K of which is exactly 105 weight=length3 , where the weight is expressed in gram and the length in millimetre. If we keep the ®sh length constant (300 mm, for example, Fig. 1H) and let the Fulton's K vary from 0.7 to 1.8 with 0.1 unit interval, we will get ®sh weights from 189 to 486 g
with 27 g intervals. From the function of girth/weight W 4:6299 10 5 G3 we will get the inverse function for weight/girth G 27:8491 W 3 and values for the girth from 159.8 to 218.9. Numerical analysis between Fulton's K and girth shows an almost perfectly linear relationship (R2 0:9926, Fig. 1H). A second-order polynomial equation (K 0:0009 length2 0:0497 length 0:5011, R2 0:30, MSE 0:0139) described best the relationship between Fulton's K and ®sh length in the data set used for ®tting the gillnet selectivity models. Residuals were normally distributed. This equation was used in Monte Carlo simulations later on. It should be noted that the relationship between Fulton's K and ®sh length derived from the girth data set was linear with slightly smaller mean square error; 0.0136 (Fig. 1D). The equation from Fig. 1D was used in the theoretical mesh size combination simulation studies. 3.2. Gillnet selectivity models Four K-models were ®tted for the condition factor data set. All these models included b1, b2, g1 and g2 parameters but the part that described the relationship between mesh size and ®sh length differed. The model with four parameters (a1, a2, a3 and a4) gave the best ®t r 2 0:864 but it did not differ signi®cantly from the more simple model (a1, a2 and a4, r 2 0:862) (nested models, F1;119 1:9033, p 0:1703). The difference between the three parameter model (a1, a2 and a4) and the two parameter model (a1 and a2, r 2 0:855) was signi®cant (F1;118 5:5404, p 0:0202) and therefore the model with parameters a1, a2 and a4 (K-model later on) were chosen for further analysis. For comparison, another two parameter model (a2 and a4, r 2 0:831) was also ®tted; it differed clearly from the three parameter model (a1, a2 and a4) (F1;118 34:9183, p < 0:0001). A simple model (parameters a2 and a4) was ®tted for the data set that was pooled over the Fulton's condition factor class. Both chosen models ®tted well: the coef®cient of determination was 0.862 for K-model and 0.895 for the simpler model (Table 2). The residuals were symmetrically distributed along the ®sh length, the mesh size and condition factor (K) axes and no visual differences between any models ®tted were observed. Nevertheless, a slight positive correlation
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161
Table 2 Final parameters for the K and the simple modelsa Parameter
Estimate
S.E.
95% CI Lower
2
(1) Fulton's K-model: r 0:862, n 126, MSres a1 0.333 0.067 0.200 a2 1.1028 0.087 0.855 a4 1.153 0.521 2.184 0.066 0.038 0.142 b1 b2 0.257 0.079 0.101 g1 6.066 1.174 8.391 1.930 0.194 1.546 g2 (2) Simple model: r2 0:895, n 66, a2 1.435 0.034 a4 1.857 0.635 0.067 0.074 b1 b2 0.194 0.149 g1 6.706 1.322 1.693 0.149 g2
Asymptotic correlation matrix Upper 0:0231 0.466 1.201 0.121 0.009b 0.413 3.740 2.314
a1 1.000 1 0.475 0.020 0.008 0.362 0.270
MSres 0:0240 1.367 1.503 3.127 0.587 0.214 0.081b 0.104c 0.491 9.352 4.060 1.394 1.991
Simulation data based on the parameters of the K model (3) Simple model: r2 0:986, n 68, MSres 0:0016 1.486 0.013 1.460 1.513 a2 a4 2.743 0.214 3.170 2.315 b1 0.067 0.010 0.087 0.047 b2 0.267 0.024 0.219 0.315 5.823 0.368 6.558 5.089 g1 g2 1.802 0.069 1.665 1.939 (4) Difference model: r 2 0:937, n 134, MSres 1.435 0.025 1.386 a2 a4 1.857 0.466 2.779 b1 0.067 0.054 0.174 b2 0.194 0.109 0.023c 6.706 0.970 8.627 g1 g2 1.693 0.109 1.476 da2 da4 db1 db2 dg1 dg2
0.051 0.886 0.001 0.073 0.883 0.109
0.039 0.670 0.059 0.122 1.275 0.189
0.026 2.211 0.117 0.168 1.641 0.265
a2
a4
b1
b2
K1
K2
0.920 1 0.704 0.001 0.004 0.154 0.058
0.475 0.704 1 0.032 0.032 0.267 0.193
0.020 0.001 0.032 1 0.957 0.022 0.022
0.008 0.004 0.032 0.957 1 0.005 0.005
0.362 0.154 0.267 0.022 0.005 1 0.839
0.270 0.058 0.193 0.022 0.005 0.839 1
1 0.853 0.014 0.009 0.123 0.133
0.853 1 0.008 0.010 0.335 0.280
0.014 0.008 1 0.983 0.007 0.007
0.009 0.010 0.983 1 0.004 0.003
0.123 0.335 0.007 0.004 1 0.650
0.133 0.280 0.007 0.003 0.650 1
1 0.760 0.021 0.003 0.485 0.417
0.760 1 0.011 0.004 0.125 0.177
0.021 0.011 1 0.964 0.041 0.039
0.003 0.004 0.964 1 0.001 0.001
0.485 0.125 0.041 0.001 1 0.780
0.417 0.177 0.039 0.001 0.780 1
0:0082 1.485 0.935 0.041b 0.410 4.785 1.909 0.129 0.440 0.115 0.314 3.406 0.484
a Explanations of some symbols in the table: S.E.: asymptotic standard error of estimate, r2: coef®cient of determination, n: number of observations. The parameters (a1±a4) of the models describe the relationship between the mesh size m and the ®sh length L mj a1 Kk a2 Ljk a3 a4 . The other parameters describe the shape of net selectivity curves. In the difference model (4) parameters da2 dg2 measure the difference between the parameters of the models 2 and 3. Note that in the models 2±4 data were pooled over condition factor class. b The upper limit of b1 is de®ned to be 0. c The lower limit of b2 is de®ned to be 0.
was observed in all cases. Some outliers were detected: 11 out of 125 residuals of the K-model were larger than the studentized 95% con®dence range (1.980 with 119 d.f.) and correspondingly 4
out of 65 residuals of the simple model were larger than the studentized 95% con®dence range (2.000 with 59 d.f.). Both ratios were above the 5% level. All outliers were near the peak of the selectivity
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Fig. 2. Observed relative ef®ciencies and ®tted model with Monte Carlo simulated 95% con®dence bands for K-model (A) and simple model (B). Sub®gure C describes the simple model (black line) ®tted for the simulated data set based on the K-model (dashed line). Note that for the K-model the x-axis is the ratio of mjk =mi , for the simple level x-axis it is the ratio of mjk =mi .
curve indicating some inaccuracy in the shape of the model (Fig. 2). The shapes of the estimated B-type selectivity curves were very similar to each other and all the ``common'' parameters (a4, b1, b2, g1 and g2) that
estimate the same functional part of the model were within each others' 95% con®dence limits, respectively (Fig. 2, Table 2). Therefore, these parameters cannot be regarded as different. One reason for these similarities is that both curves were based on nearly
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163
Table 3 Parameters for the simple models of theoretical mesh size combination and simulation dataa Parameter
Estimate
S.E.
95% CI Lower
Upper
2
First-order model: r 0:942, n1 230, n2 54059, MSres 0:005936 a2 1.508 0.014 a4 2.768 0.120 b1 0.074 0.009 0.289 0.024 b2 g1 5.336 0.429 g2 1.767 0.079
1.481 3.005 0.093 0.242 6.182 1.611
1.535 2.532 0.056 0.335 4.491 1.922
Second-order model: r2 0:982, a0 a2 a4 b1 b2 g1 g2
0.004 1.300 1.359 0.085 0.267 6.318 1.571
0.004 1.343 1.018 0.064 0.320 5.268 1.733
n1 230, n2 54059, MSres 0:001921 0.004 0.000 1.322 0.011 1.188 0.087 0.075 0.005 0.293 0.014 5.793 0.266 1.652 0.041
a
The parameters a0 a4 of the models describe the polynomial relationship between the mesh size m and the ®sh length L mj a0 Lj2 a2 Lj a4 . The other parameters describe the shape of net selectivity curves.
the same model (parameters b1, b2, g1 and g2) and therefore the shape of the curve is already ®xed to some extent. Within both models the parameters g1 and g2 were not different for the same reason stated earlier (Table 2). When studying the pooling effect with the Nordic net mesh size combination, the model ®tted to the simulated data set was very close to the simple model ®tted to the original data. None of the parameters differed from each other; the 95% con®dence interval of parameters da2 dg2 that measure the difference between the parameters of the simple model and the simulated model overlapped zero (Fig. 2C, Table 2). Although the data set was generated without any random component, the pooling over the condition factor generated variation in the data, resulting in an r2 1 of 0.986. The MSE was less than 10 compared to the simple model ®tted to the original data. The biggest difference between the model curves and simulated ``observations'' was seen near the peak of the selectivity curve; the width of the peak was much wider according to the ``observations'' compared to the Kmodel from which the data set was generated or to the simple model that was ®tted. (Fig. 2C). This means that the shape of the peak is at least partly due to pooling. It should also be noted that the range of
observations within each length class was larger in the simulated data set than in the original one (see Appendix A). In the second simulation study for the pooling effect, we used the theoretical mesh size combination and a much wider length range for the data. The simple model gave a good ®t to the pooled simulated data and the residuals were symmetrically distributed over the length and mesh size axis (Table 3). The same kind of variation was observed as earlier in the ®rst simulation study of pooling (Fig. 2C). However, an additional effect was found when estimating the initial parameters for the iterative model ®tting; the relationship between length and mesh size was not linear but clearly curvilinear upwards. Therefore, we ®tted a secondorder polynomial function to describe this relationship and it was signi®cantly better than the simple model (F1;223 469:106, p < 0:0001) (Table 3). In practice, the ®rst- and second-order polynomial equations were identical over a length range from 6 to 44 cm and a clear difference was seen when ®sh length exceeded 50 cm (Fig. 3). We also tried to ®t the second-order polynomial model for the observed data set (length range 11±31 cm) but the parameter a0 (see Table 2) did not differ from 0 and therefore no additional improvement was achieved. This was expected because ®rst-
164
M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
Fig. 3. Relationship of ®sh length and mesh size in the simulated data set with a theoretical mesh size combination. First line describes the relationship with the simple model (®rst order) and second line describes the relationship with the second-order polynomial model.
and second-order functions were practically identical within that length range (Fig. 3). Because the selectivity curves of the K-model and the simple model were similar, the PRE curves of both
models were also similar and, correspondingly, the RLFD of both models were practically identical. For both models the 95% con®dence intervals of the PRE curves had the same pattern as the estimated curves (Fig. 4). Only the levels of the PRE curves were different; the average level of the PRE curve of the K-model was 1.08 times higher than the level of the simple model. Although the levels of the PRE curves were different, the RLFDs of both models were practically identical. Both the 95% con®dence range of the PRE curve and the 95% con®dence range of the RLFD for the K-model were 1.11 times wider than the ranges for the simple model (Fig. 4). However, the RLFD of different models are similar only when using the very same structure and distribution of Fulton's K of the population from which the both models were derived. In this case we used the second-order polynomial equation derived from the
Fig. 4. The PRE (with 95% con®dence interval) of Nordic multimesh gillnets for the K-model (upper) and the simple model (lower). The corrected RLFD was calculated for a hypothetical ®sh population with equal catch in each length class. The thick curves represent the K-model and thin curves represent the simple model. The upper and the lower limits of 95% con®dence interval are represented. The solid horizontal line represents the observed relative length distribution. The Fulton's K for each length class was calculated from the second-order polynomial equation derived from the original data set.
M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
165
Fig. 5. Effect of Fulton's K on PRE curve of Nordic net (upper) and relative ef®ciency of 14 and 40 mm individual mesh sizes (lower).
original data set. If two populations have different distributions of Fulton's K, the PRE curves and therefore the corrected RLFD are also different (Fig. 5), because the Fulton's K (a1) affects the regression coef®cient between ®sh length and mesh size (Eq. (3)). However, the difference increases with increasing ®sh length and the effect of Fulton's K depends on how the length interval under study and the used mesh size combination ®t each other. When using an individual mesh size of 14 mm and 5 cm length interval from 10 to 15 cm the difference between the areas that selectivity curves cover was only 4%. The corresponding difference with a similar 5 cm length interval from 25 to 30 cm but using a 40 mm mesh size was 46%. With Nordic mesh size combination the difference was 2% for length interval 10±15 cm and 1% for length interval from 25 to 30 cm (Fig. 5). 3.3. Effect of Fulton's K on 95% con®dence intervals Monte Carlo simulated 95% con®dence intervals of the simple model and the K-model were practically identical when using a ®xed K (Figs. 2, 4 and 6).
However, when taking into account the fact that the K is normally distributed within each length class and expanding the simulations (K having a standard deviation of 0.1181 and zero correlation with other parameters) we found that the 95% con®dence interval is on the average 1.8 times larger compared to con®dence interval with ®xed K (Fig. 6). Most of this increase of the 95% con®dence interval is due to deviation of K, because the simulated con®dence intervals with ®xed (predicted) model parameters and random K were 1.55 times larger than the simulated con®dence intervals with random parameters and ®xed K. Monte Carlo simulations with different standard deviations of K and 24 mm mesh size showed that the width of the 95% con®dence interval is directly dependent on the magnitude of the standard deviation of K in the population (Fig. 7A). The lower 95% con®dence interval increased about 5 mm and the upper 95% con®dence interval about 6 mm per 0.1 unit increase in the standard deviation of K. Due to parametrization of the K-model the con®dence interval of the optimum mesh size increases
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Fig. 6. Model predicted relative ef®ciency curves for 24 mm mesh size and model predicted optimum mesh sizes for both the simple model and the K-model. Monte Carlo simulated 95% con®dence intervals are shown with thin line, model predicted curves are shown with thick line. Linear regression equations estimating the upper and lower 95% con®dence bands and the predicted optimum mesh size of the K-model are shown in the lowest ®gure. Cut points of 24 and 35 mm mesh sizes are presented.
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167
when ®sh length increases (Fig. 6). Both the lower and the upper 95% con®dence intervals increase about 0.9 mm per 10 cm increase in ®sh length: at 22 cm both intervals are about 2 mm and at 45 cm both intervals are about 4 mm (Fig. 7B). Correspondingly, when studying con®dence intervals along the ®sh length axis, both the lower and the upper 95% con®dence intervals increase about 0.6 cm per 10 cm increase in ®sh length: at 15 cm both intervals are about 1 cm and at 46 cm both intervals are already about 3 cm (Fig. 7C). 4. Discussion
Fig. 7. The change of 95% con®dence intervals against the standard deviation of Fulton's K and ®sh length. In sub®gures A and C the con®dence interval is measured in ®sh length (cm) and in sub®gure B the con®dence interval is measured in mesh size (mm). Ninety-®ve per cent con®dence intervals for different standard deviations of K are calculated for 24 mm mesh size.
The B-type indirect gillnet selectivity curve is based on an assumption of the geometrical similarity of selectivity curves over ®sh length range. Mathematically this relationship can be described as the relative ef®ciency of mesh size (mi) for ®sh length (L) is equal for those combinations of mi and L for which the ratio mi = a1 L a2 is equal, assuming a1 and a2 are constant over all mesh sizes and ®sh lengths (®rst postulated by Baranov (1914) as cited by Hamley, 1975). This basic assumption is, in fact, based on the assumption of isometric growth of ®sh over ®sh length range, which is not strictly true for most ®sh species. In an allometric equation (W aLb , where W is weight and L length, a and b parameters) (Hayes et al., 1995) describing the length±weight relationship of ®sh the parameter b is usually >3, which means that longer ®shes are also relatively larger in other dimensions including girth. In practice this means that large ®sh have, on the average, bigger girth than expected according to the geometrical similarity assumption and small ®sh have smaller girth than expected. Parameters (b1, b2, g1 and g2) that estimate the same functional part of the B-type gillnet selectivity model were within each other's 95% con®dence limits, respectively (Fig. 2, Table 2) indicating that due to pooling the shape of the B-type curve did not change in practice. The relationship between ®sh length and mesh size was de®ned as linear within condition factor class from which follows that the overall relationship is nonlinear (Eq. (3)). This nonlinearity was also observed in the simulation study with wide length range (6±64 cm) and theoretical mesh size combination (Table 3, Fig. 3). Nevertheless, with the observed
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data set and in the simulation study with narrow length range this nonlinearity was not observed. It seems reasonable that within a short length range the linear simpli®cation of Eq. (3) is close enough to describe the true nonlinear phenomenon. However, the consequence from this is that no kind of extrapolation of the ``linear model'', i.e. simple model in this paper, can be done. A gillnet selectivity model derived from a certain length range cannot be shifted to another totally different length range. In ®sh stock assessment the effect of ®sh shape has some consequence. Usually a certain mesh size is chosen to be the smallest allowed in the ®sheries. This mesh size must be chosen so that it takes care of the seasonal and year-to-year variation in the ®sh shape, i.e. the condition factor. If regulation is based on a traditional approach, measuring the gillnet selectivity without taking care of the condition factor, the selected smallest mesh size might be several millimetre (from knot to knot) too small depending on the species and ®sh length and correspondingly a large number of under sized ®sh might be caught. Although the model derived from the simulated data set was very close to that of the original data, the simulated data had a much larger range of observations over the mesh size than the original data set (see Appendix A). Only in the condition factor class 1.2 were the original and observed data sets close. This indicates that some factor affecting the shape of the Btype gillnet selectivity curve is still missing. In a recent paper of Millar (2000) the modelling approach used in this paper is criticized to be biased in relation to parameter estimates which might be the reason for the difference between the observed data and the simulated data. However, we feel that the modelling approach we used is still valid to describe the relationship between mesh size, ®sh length and Fulton's condition factor on a general level. If we suppose that the B-type indirect gillnet selectivity curve is the same for both slim and fat ®sh type, which need not necessarily be true, the PRE of the slim population will have the same amplitude and also the same wave length on a log scale as the fat population. The overall levels of PRE curves of both populations are the same. The phase of the oscillation of the PRE is
different, because they have different coef®cients of regression between mesh size and ®sh length (Fig. 4). While regression coef®cient increases the optimum size of ®sh for certain mesh size shifts towards the left. A fat ®sh needs to be much shorter to achieve the same girth as a slim ®sh with the same girth (Jensen, 1995). When using a mesh size combination based on a geometric series (as the Nordic net here) this has no practical meaning because adjacent mesh sizes cover each other and correct this error. With the Nordic nets the error in the PRE curve is small; usually a few per cent, and it stays within the 95% con®dence interval due to parameter uncertainty of the model. Therefore, the observed catch per unit effort (CPUE) and the shape of length or age distributions are reliable from population to population with respect to condition factor. But when using individual mesh sizes or irregular or nongeometric series mesh size combinations the error in the PRE curve can be tens of per cent within a certain ®sh length range and therefore conclusions drawn from these kinds of studies can be seriously erroneous. Acknowledgements The study was ®nancially supported by the Foundation for the Study of Natural Resources in Finland, the University of Turku, Finland, the Finnish Game and Fisheries Research Institute, the Swedish Environmental Protection Agency, the Swedish National Board of Fisheries and the Norwegian Institute for Nature Research. We thank Harri Helminen, Jouko Sarvala, Toomas Tammaru and one anonymous referee for their helpful comments when preparing this paper. The paper is a contribution from the Nordic Freshwater Fish Sampling Group (NOFF).
Appendix A Data and starting values for iteration phase used for modelling the gillnet selectivity of perch. The bordered area shows the range of the simulated data set.
M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
Parameter
Start value K-model
a1 a2 b1 b2 g1 g2
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1.9562 1.0145 0.0210 0.1337 2.6598 2.1822
Simple model 1.3432 1.9958 0.0265 0.1059 2.7541 2.0385
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M. Kurkilahti et al. / Fisheries Research 54 (2002) 153±170
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