A New, Improved Growth Model for Brown Trout, Salmo trutta

A New, Improved Growth Model for Brown Trout, Salmo truttaAuthor(s): J. M. Elliott, M. A. Hurley and R. J. FryerSource: Functional Ecology, Vol. 9, No. 2 (Apr., 1995), pp. 290-298Published by: British Ecological SocietyStable URL: http://www.jstor.org/stable/2390576 .

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Functional Ecology 1995 9, 290-298

290

A new, improved growth model for brown trout, Salmo trutta J. M. ELLIOTT, M. A. HURLEY and R. J. FRYER* NERC Institute of Freshwater Ecology, The Windermere Laboratory, Ambleside, Cumbria LA22 OLP, UK

Summary

1. A growth model for brown trout, developed almost 20 years ago, has been used to investigate growth potential in at least 40 populations over a wide geographical range. The chief disadvantages of the model are: it is based on growth data for only 55 hatchery trout kept in tanks without strict control of temperature and oxygen, it is not continuous and is restricted to the range 3-8-19-50C, it requires six parameters and only one of these can be interpreted biologically. 2. For the new model, growth data were obtained for an additional 130 trout bred from wild parents and kept in tanks at five constant temperatures (range ?0 1 or 0.2'C) and 100% oxygen saturation. The new model is continuous over the range 3.8-21.7'C and has five parameters, all of which can be interpreted in biological terms. It was fitted to growth data for individual fish and was an excellent fit (P < 0 001, R2 > 0.99) to the data for the 55 trout of the original experiment, the 130 trout of the new experiment and both experiments combined. The procedure for applying the model to field data is critically examined and a suitable test for maximum growth potential is described. The model ceases to be robust when mean temperatures are estimated over periods of more than 3 months. 3. Although parameter estimates for the new model are similar for the original and new experiments, they are significantly different. An iterative exercise, varying common and different parameters, showed this to be the result of slight differences between two parameters; the optimum temperature for growth and the growth rate of a 1-g fish at this temperature. Possible reasons for this are discussed and it is concluded that these differences have a negligible effect on values predicted from the model.

Key-words: Growth potential, optimum temperature, temperature limits, trout weight

Functional Ecology (1995) 9, 290-298

Introduction

It is almost 20 years since a predictive model was developed for the growth of brown trout, Salmo trutta L., fed on maximum rations of natural food (Elliott 1975a). This model has now been used to investigate growth potential in at least 40 populations of brown trout living in streams and rivers (Crisp 1977; Edwards, Densem & Russell 1979; Craig 1982; Crisp, Mann & Cubby 1983; Elliott 1984, 1985, 1988; Allen 1985; Mortensen, Geertz-Hansen & Marcus 1988; Preall & Ringler 1989; Jensen 1990). The model shows that water temperature and the initial size of the fish are often the chief factors affecting growth rate in populations of different densities, in

* Present address: SOAFD Marine Laboratory, PO Box 101, Victoria Road, Aberdeen AB9 8DB, UK.

streams with different environmental characteristics and over a wide geographical range. Use of the model has identified those populations, or periods in the life cycle, in which growth is restricted, presumably because of lack of food. There has also been a small number of cases in which the trout grew faster than the maximum rates predicted by the model, e.g. population in the Horokiwi stream, New Zealand (Allen 1985), and a few exceptions (11% of total data) in a comparative study of annual growth rates over several years for 12 Norwegian populations (Jensen 1990) (see also Elliott 1994 for a more detailed review).

Although this model has been widely used, it was developed from experimental data for a relatively small number of brown trout and requires a large number of parameters, only one of which has any biological significance. A synopsis of the weaknesses



291 in both the experimental methodology and the model Growth model is presented in the next section. This is followed by a for brown trout description of the new experimental methodology

used to obtain growth data. The new model is then introduced, is fitted to both the old and new data sets and a procedure developed for its application in the field. Finally, differences between the two models and the two data sets are discussed.

Critique of earlier experiment and model

A detailed description of the experimental tanks and feeding procedure is provided in Elliott (1975a). The fish were fed frequently to satiation, the chief food being freshwater shrimps, Gammarus pulex L. Although growth data were obtained at 15 mean temperatures (3-8, 4-2, 5-6, 6-8, 7-1, 95, 10-8, 12-8, 13-6, 15*0, 16-2, 17*8, 19-5, 20-4, 21.70C), there was no temperature control and therefore variation increased from ?0*30C at 3*80C to ?1-0C at 21*7'C. Some trials were abandoned because of marked changes in temperature. Major disadvantages of the experimental procedure were the inability to control water temperature and oxygen concentration. The latter was always greater than 85% saturation but often fluctuated between 90% and 100%, especially at higher temperatures. Forseth & Jonsson (1994) have recently suggested that such fluctuations could affect trout growth. Another disadvantage was the small number of trout in the experiment, individual growth rates being obtained for only 55 fish with 40 of these having a similar initial weight close to 50g. Finally, the fish were obtained from a commercial hatchery, not the wild, but this was not a major problem because the growth model provided an accurate description of the growth of wild trout fed to satiation within enclosures in a nearby stream (Elliott 1975a).

The growth model can be expressed in terms of the specific growth rate, Gw expressed as %, at a water temperature of TPC and at an instant in time when the live weight of the trout is Wt:

Gw = 100 (a + b2T) Wtb eqn 1

where the parameters a, b1 and b2 require separate estimates within the two temperature ranges 3 8- 12-8?C and 13-6-19-50C. In a second version, a growth equation was developed from equation 1 and integrated to provide fish weight at the end of a given period of time:

W. = [(a + b2T) b, t + Wb] l b, eqn 2

where W0 is the initial weight of the trout, W, is the final weight of the trout after t days at T0C and the parameters a, b1 and b2 require separate estimates within the same two temperature ranges as equa- tioni1.

In spite of its wide use, this 'prototype' growth model has some defects. For each of the two temperature ranges, only one parameter, the weight exponent (bl), has any biological significance. Six parameters are required for the complete temperature range for growth (3.8-19-5 'C), and it is assumed that growth is highest and constant within the range 12-8-13-6'C. The model is not continuous over the whole temperature range and cannot be applied to the experimental data obtained at temperatures above 19-5'C. All these problems are overcome in the new model.

New experimental methodology

Brown trout were reared in the hatchery of this laboratory, the eggs and sperm being obtained from sea-trout caught in a tributary of the River Leven in north-west England. The parr were kept in circular tanks (diameter 1 -83 m; depth range, 48cm at edge to 63cm at centre) made of glass-reinforced polyester resin. Water from Windermere was supplied at about 757litreh-1 from four jets so that there was a continuous circular flow. The trout were therefore maintained under exercise conditions and were in good physical condition. They were fed to satiation, usually twice a day, with a commercial pelleted food.

Trout growth was studied in constant-temperature tanks (width 31cm, length 48cm, depth 30 cm) arranged in two series, each with 10 tanks. Each tank contained 26 litre of filtered lake water which was recirculated continuously through a temperature control tank so that all 10 tanks were at the same temperature. Five constant temperatures were used: 5*0, 10-00C (range maintained at ?0.10C), 13-0, 15*0, 18-0C (?0 '2C). Oxygen concentration in the water always remained at 100% saturation. There was subdued, natural daylight with half of each tank covered by black polyethylene to provide some refuge for the trout. Only one fish was placed in each tank and all fish were fed to satiation on freshly killed, freshwater shrimps, the feeding procedure being the same as that used in the earlier experiment, except that all uneaten food was removed immedi- ately with a small air-lift extractor. When the trout were first placed in the tanks, they were kept at the same temperature as that in the hatchery. After 5 days, the temperature was increased or decreased at about 1Ch-l until the experimental temperature was attained. The fish were kept at this temperature for 3 days before the start of the period over which growth was measured. Fish were never used more than once.

Each trout was measured (from snout to tail fork to nearest mm) and weighed (wet weight to nearest 0-01g for fish < 80g and 0*lg for fish > 90g) at the start and finish of the growth period which always



292 lasted 42 days. For trout with initial weights close to J. M. Elliott et al. 1-3g, 5g, 12g and 50g, there were five fish of similar

size at each of five temperatures. The number of replicates was reduced from five to three at each temperature for fish with initial weight close to 100g and 300g. As there was only one fish per tank, the volume of the fish was always much smaller than that of the tank. It was therefore assumed that growth rate was unaffected by tank size. Estimates of growth were obtained for 130 trout. The size distribution of the fish was more even than that in the original experiment with its strong bias to 50g fish. Obvious advantages with the new tanks were the greater temperature control and the high oxygen concentra- tions. The trout in the new experiment were also obtained from wild parents.

New growth model

The new model was first developed as part of an unpublished thesis (Fryer 1989) but was tested on only the data obtained for the 55 fish in the original experiment. The basic logic behind the new model was similar to that used in the earlier model. Func- tional relationships were examined between specific growth rate and fish weight at each constant temperature, and between growth rate and temperature for fish of similar size. The first relationship was well described by a power function but the second could be described by different models. The new model was chosen largely on empirical grounds because it provided the best fit with the lowest number of parameters (see also discussion).

As in the earlier model, the first version of the new model is expressed in terms of the specific growth rate, Gw expressed as a %, at a water temperature of TPC and at an instant in time when the live weight of the trout is Wt [note that Gw is more correctly called the instantaneous relative growth rate but the shorter version is usually used, see Ricker (1979)]:

Gw = cWt (T- TLIM) I (TM- TLIM) eqn 3

where TLIM = TL if T ' TM or TLIM = TU if T > TM. Unlike the earlier model, the parameters can be

Table 1. Estimates of five parameters in equation 5 with SE in parentheses for experiment 1, experiment 2 and both experiments combined (n = number of fish, R2 = multiple coefficient of determination expressed as a percentage, P < 0-001 for goodness of fit for each data set)

Experiment 1 Experiment 2 Experiments 1 + 2

n 55 130 185 b 0-324 (0-010) 0-311 (0-003) 0-308 (0-002) c 3-077 (0-118) 2-802 (0-017) 2-803 (0-016) TM 13-37 (0-061) 13-07 (0-034) 13-11 (0-030) TL 3-57 (0-080) 3.57 (0.045) 3 56 (0-041)

To ~~19-41 (0-048) 19-53 (0-042) 19-48 (0.035) R2as % 99-981 99.994 99-993

defined in biological terms. The temperature for optimum growth is TM, and TL and Tu are the lower and upper temperatures at which growth rate is zero. The weight exponent b is the power transformation of weight that produces linear growth with time, and c is the growth rate of a 1 g trout at the optimum temperature. All five parameters remain constant for the temperature range 3-8-21V70C.

For the second version of the model, the procedure was similar to that used for the earlier model with integration providing an equation for fish weight at the end of a given period of time at constant temperature:

Wt = [Wo + bc (T- TLIM) tI{100 (TM - TLIM)}I 1/b

eqn 4

where WO is the initial weight of the trout, Wt is the final weight of the trout after t days at T0C and the five parameters are the same as those in equation 3 for the same temperature range of 3.8-21.7'C.

Equation 4 can be expressed in a statistical form with E representing the independent homogeneous random errors from the normal distribution:

ln(W,) = 1ln[Wb + bc (T- TLIM) ti

{100 (TM - TLIM)}I + E. eqn 5

Fryer (1989) showed that when equation 5 was fitted by non-linear least squares to the data for the 55 fish in the original experiment, it had small non-linearity according to the relative curvature measures of Bates & Watts (1980), the bias estimates of Box (1971), the skewness estimates of Hougaard (1985) and the asymmetry measures of Lowry & Morton (1983). Linear approximation standard errors obtained from the non-linear least squares fit can therefore be used to make inferences about the model parameters. Estimates of the parameters and their standard errors (SE) in equation 5 were obtained for the data from the original experiment, the new experiment and both experiments combined (experiments 1 and 2 in Table 1). The excellent fit of the model is illustrated by comparing the observed final weights of the trout in the experiments with expected values estimated from equation 5, using the actual initial weights of the fish (Fig. 1). A similar comparison of observed and expected growth rates illustrates well the effects of temperature and fish weight, as well as the excellent fit of the model (Fig. 2).

Although the parameter estimates for the two experiments were very similar, an analysis of residual variation, using an approximate F-test (Mead & Curnow 1983), revealed significant differences (F5,175 = 5029, P < 0.001). As the new experiment 2 included two weight classes (fish c.1*3 g, 5 g) of smaller trout that were absent from the original experiment 1, these weight classes were omitted in a new analysis. A smaller, but still significant,



293 Growth model (a) for brown trout 6

4

2

o XI I I I I I o 2 4 6

(b) 6

U))

-D~~~~

o 2 4 6

(c)/ 6_

4 ~ ~ ~

0 I I I o 2 4 6

Expected In weight

Fig. 1. Relationship between the observed final weights (g) of the trout and the expected weights estimated from equation 5 for: (a) experiment 1 (n = 55); (b) experiment 2 (n = 130); (c) experiments 1 and 2 (n = 185) (note that all weights are expressed as logs and that some values coincide on all figures).

difference was still apparent (F5,125 = 5 00, P < 0-001) and therefore the first result was not simply the result of differences in the fish sizes used in the two experiments.

Additional tests, treating the results for each experiment separately, showed that the parameter estimates for the combined experiments (Table 1) were significantly different from those for experiment 1 (F5,50 = 6-82, P < 0.001) but not those from experiment 2 (F5,125 = 0-88, P > 0.05). This similarity between the parameter estimates for the combined experiments and experiment 2 can be seen in Table 1. An iterative exercise investigated further the differences in the parameter estimates for experiments 1 and 2. By varying the number of common and different parameters, it was found that the best fitting model was obtained when values for the upper and lower temperatures (Tu, TL) and the weight exponent (b) were the same for the two data sets, but the optimum temperature (TM) and the growth rate of a 1-g fish at this temperature (c) were different. It is also clear from Table 1 that these two parameters exhibited the largest differences between the two experiments. It was therefore concluded, after con- sidering the results for all these analyses, that the parameter estimates for the combined experiments with their smaller standard errors should be used when the new growth model is used for predictive purposes. Possible reasons for the small, but significant, differences in the parameter estimates for experiments 1 and 2 will be discussed later.

Application of new model to the field

When the growth model is applied to wild brown trout in the field, the null hypothesis is that the fish are growing at their maximum potential. Use of the model assumes that: (1) fish weight is measured without error, (2) mortality is random, (3) the five parameters in the model are estimated exactly without error, (4) all fish grow according to the model with no variability in growth rate between fish and (5) water temperature is constant for the growth period. These assumptions are presented in an approximate order of probability from highest to lowest. The least probable assumption of a constant temperature will be examined later in more detail.

As stated above, the weight exponent b is the power transformation of weight that produces linear growth with time. This can be seen by rearranging equation 4 thus:

Wb = Wo + bc(T - TLIM) t/{100 (TM - TLIM)} eqn 6a

where the intercept is Wo and the five parameters define the gradient.

For a sample of n1 fish at the start of a growth season, the live weights are W01f, W02, * * * W0n1 and the mean weight on the power transformation scale is



294 (WO). If these fish grow for t days at T0C, then the J. M. Elliott et al. expected live weights at the end of the growth period

are Wt1, W,2,. . . Wt, and the expected mean weight on the power transformation scale is E (Wbt). The latter is obtained by substituting (W$) for Wo in equation 6a because all fish grow at the same rate on the power transformation scale.

Unless the growth period is short, the assumption of constant temperature will be fulfilled rarely. However, if the mean temperature is known for short periods of time, then the expected values from equation 6a can be calculated thus:

E (Wt) = (Wo) + 0C [(T1 - TLIM) t1/

(TM- TLIM) + (T2 - TLIM) t2/(TM - TLIM) + (Tk - TLIM) tk/(TM - TLIM)I

eqn 6b

Fish weight (a) ~~~~~~~~~~~~~C. 11 g

2 2

08 c. 50 g .. . ... . . . . C. 90 g

c. 250 g 0.4

0

i1 -04

o 0 5 10 15 20

(b) c. 1-3g

2

c. 5g

c. 12 g

c. 50 g C. 100 g

c. 300 g

0 0 5 10 15 20

Temperature (00)

Fig. 2. Relationship between the observed growth rates of the trout and the water temperature for trout of different live weights in: (a) experiment 1; (b) experiment 2 (solid lines are values estimated from the new model).

where T1, T2 . .. Tk are the mean temperatures over t1, t2 . . . tk days respectively.

If a sample of n2 fish at the end of the growth season, with live weights of W,1, W,2, . . W,,n2, is changed to the power transformation scale, then the observed mean weight is Obs(W,) and the sample variance is s2. A two-sample t-test can now be used to determine if the observed and expected values of (W') have the same population mean and therefore the fish are growing at their maximum potential as stated in the original null hypothesis. This test should, of course, always be preceded by a variance ratio F-test between s0 and s2.

If the latter indicates no significant difference between variances, then the usual t-test, using a pooled variance, is appropriate. However, the latter test may indicate a significant difference between variances. For example, in a long-term study of a brown trout population (see references in Elliott 1993), it was found that the coefficient of variation (CV = lOOs/W) for untransformed fish weight remained fairly constant (?5% of mean value) for most of the life cycle within each year-class and so s was proportional to W. Therefore, on the power transformation scale, S2 was always significantly greater than s2. Such a relationship would be expected if there is a multiplicative error on the growth rate and the new model for growth is applicable (see proof in Appendix I). Thus a significant difference between variances is indicative of a failure of assumption (4). The simplest solution is to do a t-test using two unequal variances and adjust the degrees of freedom accordingly. An example of this procedure, using data from the long-term study, is summarized in Table 2 and shows that the null hypothesis is not disproved (data are provided in Appendix II so that the calculations can be followed for self-instruction). These trout were therefore growing at their maximum potential over the period between 23 May and 23 August 1973.

In the above example, the expected final weight was calculated incrementally (equation 6b), using the mean water temperature for periods ranging from 8 to 16 days (see data in Appendix II). The effect of increasing the time period over which mean temperatures are estimated is illustrated by comparing estimated final weights for two year-classes from the long-term study (Table 3). The 1973 year-class was one of the better year-classes for growth whereas the 1983 year-class was one of the worst (Elliott 1993). However, for both year-classes the estimated weights increased with increasing time periods. This effect was small for an increase from 2 weeks to a month, especially for the 1973 year-class, but was marked for an increase from 3 to 6 months. The increase would, of course, be even greater if annual mean temperatures were used. Values for the 2-week periods were closest to the observed weights obtained in the field but it is perhaps surprising that use of longer periods



295 of 1 or 3 months would lead to only a slight increase in Growth model expected weights. If these examples are representa- for brown trout tive, they suggest that predictions from the model are

robust when mean temperatures are estimated over periods of less than 3 months, and especially less than 1 month, but become markedly inaccurate when the time period exceeds 3 months.

Discussion

The new model is an obvious improvement on the original. It covers a wider temperature range, is continuous over this range and has one less parameter. All the parameters in the new model can be interpreted biologically with three defining the optimum temperature and temperature limits for growth, the fourth being the growth rate of a 1-g trout at the optimum temperature and the fifth being the weight exponent that is also the power transformation of weight to produce linear growth with time. A comparison of values predicted by the two models shows them to be virtually identical for the original experiment, the new experiment and both experiments combined (Fig. 3).

Table 2. Example of the application of the new model to a wild population of brown trout growing in the first summer of their life cycle, i.e. from 23 May to 23 August 1973 (n = number of fish in the sample, other values are defined in the text, original data are given in Appendix 2)

23 May 23 August

n 38 n 30

W0 0-717 Obs(Wb) 1-377 E(W') 1399 95% CL ?0 031 95% CL ?0066 5 2 0000874 Obs S2 0-0314 so 0 0935 Obs s, 0-177 F(s2/s2) 3.59

(P<0.001) t -0-616

(P> 0.05) df 44.7

Degrees of freedom (df) = (s2 + 52)2 / {[(52)2/(n1 - 1)] + [(52)2/(n2 - F-test null hypothesis: all fish grow according to the model with no variability in growth rate between fish. t-test null hypothesis: mean weight of fish in population agrees with value predicted by the model.

Table 3. Estimated weights of trout for the 1973 and 1983 year-classes, using different time periods over which mean temperatures were estimated for use in equation 6b

Expected weight (g) on: 15 Mar 1974 15 Sep 1974 15 Mar 1975 30 Apr 1975

1973 year-class T(0C) every 2 weeks 4-58 29-22 50-09 57-15 T(0C) every month 4-60 29-24 50-27 57-30 T(0C) every 3 months 4-56 30-47 52-08 59-36 T(0C) every 6 months 6-76 38-93 64-15 72-53

1983 year-class T(0C) every 2weeks 2-97 20-21 32-68 37 90 T(0C) every month 3-33 22-12 35-25 40-71 T(0C) every 3 months 3-38 22-59 35 93 41-42 T(?C) every 6 months 6-81 44-08 64-49 72-65

(a)

6

4

2

o 2 4 6

(b) 6

C C

24

0)

o 2 4 6

(c)/ 6_

4_

0 / 1 I I o 2 4 6

Expected In weight (original model)

Fig. 3. Comparison of expected weights (g) of trout estimated from the new model (equation 5) and the original model (equation 2) for: (a) experiment 1 (n = 55); (b) experiment 2 (n = 130); (c) experiments 1 and 2 (n = 185) (note that all weights are expressed as logs and that some values coincide on all figures).



296 The new model includes a power function that J. M. Elliott et al. accounts for the relationship between specific growth

rate and fish weight. Such a functional relationship is now well established for fish species, especially salmonids, and the weight exponent usually lies in the range 03-0-4, as in the present study (see reviews by Brett 1979; Jobling 1983; Jobling et al. 1993; Elliott 1994). There is less information and agreement on the functional relationship between specific growth rate and temperature. A linear relationship was chosen for the new model and was presented so that the slope was positive and negative below and above the optimum temperature respectively (equation 3, Fig. 2).

Data on other fish species (see Ricker 1979 and reviews cited above) suggest that the relationship is sometimes dome shaped rather than triangular. It is likely, however, that the apparent degree of curvature of the growth response to temperature is exag- gerated in plots of empirical growth rate against temperature for a fixed weight class of fish. This is because, as fish become larger, their growth rate is usually reduced and the mean growth rate over an experimental period is underestimated for a nominal weight class. Underestimation will be greatest for maximum growth rate at the dome peak, thus increasing the apparent curvature of the dome.

In his thesis, Fryer (1989) developed a hyperbolic extension to the model for growth rate, incorporating an extra parameter controlling the degree of curvature. When this parameter was set at zero, the hyperbola model simplified to the triangular form. As the hyperbola model is smooth in the region of maximum growth, it would appear to be more realistic biologically than the triangular model. However, when the hyperbola model was fitted to the growth rates for the replicates in the original experiment using common parameters for the four weight classes, the extra parameter for curvature (estimated as 0.002) was not significantly different from zero (F1,24 = 1.82). Because the triangular model was perfectly adequate for the observed data, fitting a more complex model would lead to problems of co-linearity and hence unstable parameter estimates and predictions. Hence a triangular change-point model appears necessary if the experimental data are to be described in a parsimonious way.

As recorded in the introduction, the original model has been used to investigate growth potential in at least 40 populations of brown trout over a wide geographical range. The new model would be equally applicable to these populations. It has been shown (Table 3) that the new model is robust if water temperatures are estimated over periods of 1 month or less and is fairly robust for longer periods of up to 3 months. Serious overestimates can occur if longer time periods, exceeding 3 months, are used in the estimation of mean water temperatures. Such a source of error could explain some of the discrep-

ancies obtained in previous comparisons, using the original model (e.g. Craig 1982).

When the parameter estimates for the combined experiments (experiment 1 + 2 in Table 1) were used in the new model, there was excellent agreement between the observed final weights obtained in both experiments and the expected weights estimated from the model (Fig. 1c). When the data from experiments 1 and 2 were analysed separately, different parameter estimates were obtained (cf. values for separate experiments in Table 1), but the expected weights obtained in each case were virtually identical (Fig. 1). Therefore, for all practical purposes, the parameter estimates for the combined experiments are recommended when the new growth model is used for predictive purposes.

In spite of this conclusion, there remains the problem of explaining why small significant differences were obtained between the parameter estimates in the two experiments. The iterative exercise showed that two parameters were chiefly respon- sible; the optimum temperature for growth being slightly higher at 13 370C in experiment 1 than that of 13*07'C in experiment 2, and the closely related growth rate of a 1-g fish at this temperature also being slightly higher at 3*077 in experiment 1 than that of 2*802 in experiment 2 (Table 1). Differences between the two experiments could possibly explain these discrepancies. The most obvious is the poorer control of constant temperature and oxygen concentration in experiment 1 compared with experiment 2. Water temperatures varied from ? 0*3'C at 3*8'C to ? 1*00C at 21*70C and oxygen concentration from 85% to 100% saturation in the earlier experiment, whereas temperatures were always maintained at ? 020C or less and oxygen concentration at 100% in experiment 2. Although these differences could explain the discrepancies in the parameter estimates, they also show the negligible effects of such differences on the expected weights estimated from the model (Fig. 1). For example, it would appear that fluctuations in oxygen concentration between 85% and 100% saturation do not have a biologically significant effect on growth rates, contrary to the proposal of Forseth & Jonsson (1994).

Another obvious difference between the experiments is the source of the trout. Fish in experiment 1 were all from a commercial hatchery but the similarity between their growth rates and those of wild trout fed to satiation supported the conclusion that their growth was typical of wild brown trout (Elliott 1975b). Fish in experiment 2 were reared from wild migratory brown trout (sea-trout) and were heavier than fish of similar lengths from experiment 1 for lengths less than 30cm. For example, estimated live weights from log weight on log length relationships were 10*2g, 81*5g, 274 5g for trout from experiment 1 and 11-7g, 85 8g, 274*5g for trout from experiment 2 for lengths of 10 cm, 20 cm, 30cm respectively. Such



297 differences could be the result of the larger migratory Growth model trout females producing larger eggs, alevins and fry for brown trout than the hatchery females (see also Elliott 1985,

1988), and/or to genetic differences between the wild and hatchery trout. Once again, such differences appear to have a negligible effect on the expected weights estimated from the new, improved growth model.

The latter therefore provides a robust model that appears to be applicable to brown trout from widely different stocks. It also provides a testable model not only for investigating growth potential in different populations but also for exploring possible genetic differences in growth between populations of brown trout. As all the parameters in the model can be interpreted in biological terms, further work is now required to explore how they change in response to other variables such as energy intake. Similar models, obviously with different parameter values, may be applicable to other fish species, especially the salmonids, and this possibility also provides scope for further work.

Acknowledgements

We wish to thank Trevor Furnass for producing the figures. This work is financed jointly by the Natural Environment Research Council and the Atlantic Salmon Research Trust. We are especially grateful to the latter for their support. R. J. Fryer was also a recipient of a NERC Research Studentship.

References

Allen, K.R. (1985) Comparison of the growth rate of brown trout, Salmo trutta, in a New Zealand stream with experimental fish in Britain. Journal of Animal Ecology 54, 487-495.

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Box, M.J. (1971) Bias in nonlinear estimation (with discussion). Journal of the Royal Statistical Society, B 33, 171-201.

Brett, J.R. (1979) Environmental factors and growth. Fish Physiology, vol. 8 (eds W. S. Hoar, D. J. Randall & J. R. Brett), pp.599-675. Academic Press, New York.

Craig, J.F. (1982) A note on growth and mortality of trout, Salmo trutta L., in afferent streams of Windermere. Journal of Fish Biology 20, 423-429.

Crisp, D.T. (1977) Some physical and chemical effects of the Cow Green (Upper Teesdale) impoundment. Fresh- water Biology 7, 109-120.

Crisp, D.T., Mann, R.H.K. & Cubby, P.R. (1983) Effects of regulation of the River Tees upon fish populations below Cow Green Reservoir. Journal ofApplied Ecology 20, 371-386.

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Received 14 February 1994; revised 4 July 1994; accepted 29 July 1994



298 Appendix 1. Multiplicative error on growth J. M. Elliott et al. rate

If Gi is the growth rate of an individual fish and Gw is the 'population' growth rate for fish of weight W. g at time t and temperature T, then:

lnGi= lnGw+ ln5 eqnAl

where ln 8 - N (0, r2). For the individual fish:

dWt W. Gw 8 dt 100

Because Gw 100 = cWy" k (T), where k (T) = (T - TLIM) / 100 (TM - TLIM) as in equations 4 and 5 in the text, then:

dWt=W Wi-b dt -

ck (7)

and in integral form:

W. = [Woo + bc k(T) t8] 1/b. eqn A2

Therefore a multiplicative error on the growth rate leads to each fish growing linearly on the power b scale but with a slightly different gradient for each fish. If equation A2 is applicable for growth periods in which temperature can be assumed to be constant (as in equation 6b), and W. - O at t = 0, then equation A2 simplifies to:

W. = [bc k (T) t8] /b eqn A3

and hence:

lnWt=-ln[bck(T)] +-lnt+E b b eqn A4

whereE N (0, c 2/b2) represents the independent homogeneous random errors from the normal distribution, i.e. there is additive homogeneous variance on the ln weight scale.

For the population as a whole, a regression of In W. on lnt will have a gradient of 1/b and an intercept of (1/b) ln[bck (T) ], assuming constant temperature as before. So for a fish population that does not have its growth variance disrupted (e.g. by size-selective mortality), a constant CV over the life cycle would be expected. This conclusion remains true when changes in weight are calculated incrementally, using equation 6b.

Appendix 2. Original data for estimates in Table 2

Weights (g) Weights (g) Sampling 23 May 23 August dates t(days) T(0C)

0-119 0-386 1-016 3*859 23 May 0-120 0-391 1-020 4-014 8 12-8 0-144 0-393 1-322 4-095 31 May 0-169 0-399 1-773 4-434 15 12-5 0-209 0-439 1-858 4-523 15 June 0-218 0-453 1-860 4-726 15 14-7 0-219 0-471 1-929 4-863 30 June 0-227 0-481 1-949 5-075 15 14-7 0-229 0-520 2-068 15 July 0-239 0-527 2-400 16 14-1 0-240 0-530 2-434 31 July 0-282 0-554 2-605 15 13-6 0-291 0-570 2-841 15 August 0-306 0-595 2-886 8 13-6 0-317 0-605 3-029 23 August 0-319 0-612 3*034 0-334 3-069 0-339 3-090 0-355 3-292 0-356 3-348 0-361 3-492 0-363 3.739



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