a modelling framework for analyzing anthropogenic stresses on brook trout (salvelinus fontinalis)...

E L S E V I E R Ecological Modelling 80 (1995) 171-185

E(OLOGI(DL mODELLInG

A modelling framework for analyzing anthropogenic stresses on brook trout ( Salvelinus fontinalis) populations

M. P o w e r a, . , G. P o w e r b

a Dept. of Agricultural Economics, University of Manitoba, Winnipeg, Man. R3T 2N2, Canada b Dept. of Biology, University of Waterloo, Waterloo, Ont. N2L 3G1, Canada

Received 17 November 1993; accepted 23 March 1994

Abstract

Modelling provides a convenient and useful means of assessing the possible risks contaminant stresses pose for many populations. While abundance has traditionally been considered an appropriate modelling endpoint, and has been shown to work well at broad levels of aggregation, it is nonetheless deficient for analyzing the population-level risks posed by sublethal contaminant stresses. That is because abundance-based models fail to consider important changes in population size-structure brought about by many sublethal stressors. An individuals-based modelling framework incorporating the specifics of life-history information for brook trout (Salvelinus fontinalis) populations is discussed as an alternative to abundance models. The developed model is validated against field data and the projections produced by the Leslie-matrix-based approach, RAMAS. Comparisons suggest critical differences exist in the projections made by the two approaches because of the lack of size-structure data contained in the Leslie-matrix-based approach. As an example of how critical such differences can be, the potential consequences of sublethal toxaphene exposures on brook trout population abundance and size-structure are examined.

Keywords: Fish; Leslie models; Pollution, water

1. Introduct ion

The advent of environmental risk assessment and management has led environmental policy makers to shift emphasis from the measurement and discussion of the organism-level impacts of contaminant and environmental stresses to the measurement and discussion of population-level impacts of those same stresses (Emlen, 1989). The shift in emphasis has created a need for

* Corresponding author.

appropriately constructed and calibrated population-level response measurement tools. Key among these tools are population-level mathematical demography models that trace the changes in age-structured populations through time. The use of models for predicting the impacts of human activity on populations, however, does not guarantee either their accuracy or utility. To be truly useful models must be properly constructed and validated. Emlen (1989), Bartell et al. (1992) and Cairns (1994), among others, have stipulated a number of necessary characteristics that population-level models designed to

0304-3800/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-3800(94)00058-P

172 M. Power, G. Power/Ecological Modelling 80 (1995) 171-185

assess the effects of anthropogenic stresses should possess. These include: 1 the production of data outputs consistent with

the endpoints of regulatory interest; 2 the ability to use available toxicological infor-

mation; 3 the ability to use available, or obtainable, popu-

lation-level data; 4 recognition of the importance of variability

among individuals within a population. One approach with a long history of population- level modelling, the Leslie-matrix approach (Leslie, 1945), appears appropriate to the task of analyzing the population-level effects of anthropogenic stresses because of the simple, intuitive manner in which it represents the key population processes of birth, aging, reproduction and death. This apparent simplicity, however, belies a significant amount of mathematical complexity that can preclude the models from being easily adapted for use in environmental risk assessment. Emlen (1989), however, describes the Leslie-matrix approach as appropriate because it specifically con- siders population age-structure. At the same time, he has also criticized it for not having included density-dependence in the modelling mechanics used to make population-level predictions. Begon and Mortimer (1986) note that temporal changes in fecundity and survival can be incorporated into the matrix structure by varying the elements of the transition matrix. Furthermore, the notion of density-dependence can be built into the modelling framework by varying the fecundity and survival elements of the transition matrix in rela- tion to population size. Ferson et al. (1989) out- line a Leslie-matrix-based approach (RAMAS), for use in environmental risk assessment, aimed directly at measuring the consequences of human activities on stochastic age-structured populations exhibiting density-dependence. While the approach is capable of predicting abundance in the time domain, it was not designed to produce information on the population-level endpoints favoured by many field biologists and ecotoxicolo- gists. These include: rates of growth, length- frequency distributions and condition factors (Munkittrick and Dixon, 1989).

Despite the fact that the Leslie-matrix-based

approach fails to produce many of the required population-level endpoints, or to be readily able to incorporate available toxicological information in a meaningful way, Emlen (1989) concluded that the models "work well over broad circumstances". Are broad circumstances sufficient? The attention that has been paid to the identification and quantification of sublethal contaminant responses (NRCC, 1985) and the complexities associated with accounting for the interaction of environmental factors on whole-organism responses to contaminant stress would argue that broad circumstance approaches are deficient. Under field conditions stresses rarely act selectively on a single species (Wedemeyer et al., 1984). Lethal effects elicited in a single species are often paral- lelled in the lethal and sublethal effects elicited in other species inhabiting the same ecosystem. When coupled with the population-level endpoint environmental effects monitoring and risk-based assessment demands of field biologists and eco- toxicologists, this suggests a need for population models which produce more than the age-structured population abundance measures typical of Leslie-matrix-based approaches. As Cairns et al. (1984) have argued, environmental effects monitoring and risk assessment endeavours to connect the observed changes in populations to the sum- mation of stresses, lethal and sublethal, that im- pinge upon them. Only when consistent abundance and population-level size-structure descrip- tions of the effects of anthropogenic stresses can be produced will models fulfil the promise of improving the practice of environmental effects monitoring and risk assessment.

In an attempt to meet that need, this paper describes an individuals-based modelling approach designed specifically for the population- level analysis of brook trout (Salvelinus fontinalis) that: 1 incorporates available biological life-history

data; 2 can be adapted to include consideration of

anthropogenic stress, e.g. exploitation and contaminant loading;

3 compares favourably with the established Leslie-matrix-based modelling approach in predicting abundance;

M. Power, G. Power/Ecological Modelling 80 (1995) 171-185 173

[ Determine number of eggs produced

Determine egg survival us ng a R cker mode

Determine recruitment in all other age-groups Logic

sequence

Assign sex & length to age 0 recruits]

Increment individual lengths using mean annua growth va ues

Census population by age in current year (t)

[ Shift time to year (t+l) [ V / i

Fig. 1. A flow diagram of the individuals-based modelling framework for analyzing the fluctuations in brook trout (Salvelinus fontinalis) population abundance and size structure characteristics. The sequence of calculations depicted are for a single year and scaled for a population inhabiting a representative 0.25-ha habitat area. The sequence is repeated for the number of years (n) for which model output is required.

4 produces the required predictions of the effects of contaminant stresses on population size- structure.

2. Mode l structure

The individuals-based model described below was based on the data contained in the McFad- den et al. (1967) study of a brook trout population in a stable stream environment. The study was carried out over a 14-year period on a 1.59-ha surface area of Hunt Creek in south central Montmorency County, Michigan. Details of the sampling protocol, topography and stream mor- phology are given in McFadden et al. (1967). The size of the potential egg deposition (132000) and of the age 0 population ( > 4800) required scaling for reasons of computational efficiency. Accord- ingly, the model schematic given in Fig. 1 describes a model suited to the analysis of a population contained in a representative 0.25-ha area.

The model begins all calculations in year t = 0 with the deposition of eggs for emergence the

following spring. The number deposited in years t = 0 to t = 3 are determined by a random variable drawn from a normal distribution with mean and standard deviation set equal to the scale adjusted annual egg production data reported in McFadden. This method of determining egg production was used as a means of priming the population model by allowing all age-group vectors to fill before making egg production endoge- nous to the model. Furthermore, as the rate of convergence to the steady-state is a function of the initial conditions used to prime the simulation (Law and Kelton, 1991), the selection of conditions conducive to the quick convergence of the model can significantly reduce model running times. While not necessarily optimal, the use of the adjusted annual egg production data reported in McFadden significantly reduced model convergence times and improved computational efficiency.

Once endogenously determined, potential egg production is modelled as a function of individual female length, (cm), using the average female egg content data contained in McFadden, table IX, as follows:

Eggs / female = 65.43 e 0'095 length(cm) (1)

The results of individual egg production are summed to arrive at total egg production. Vari- ability in adult female abundance is thus directly reflected in variability in total egg production and the observed stochasticity in age 0 and age 1 abundance. The adequacy of the model was judged on the basis of its r 2 value (0.989), parameter p-values, all < 0.001, and a careful examination of the residuals. Tests of the residuals included plots of the residuals against both fitted and regressor variables, normal probability plot- ting and formalized normality testing (Bates and Watts, 1988). On the basis of the above, the model was judged as statistically adequate and incorporated into the individuals-based modelling framework.

Once egg production has been determined, the model calculates recruitment from the egg to age 0 and age 0 to age 1 life-stages using a Ricker (1954) stock-recruitment model. Consistent with the findings of Elliott (1985), the Ricker model

174 M. Power, G. Power~Ecological Modelling 80 (1995) 171-185

proved to be the best description of the recruitment phenomenon at all life-stages. The percentage of variance in recruitment (R) that could be explained by variations in the parent stock (S), however, declined as the age of the parent stock rose. Thus while the Ricker model could explain 81% of the variance in R for egg to age 0 recruitment, it could only explain 6% of the variance in R for age 2 to age 3 recruitment. Accord- ingly, the Ricker model was used to explain recruitment only in the early life stages, egg to age 0 and age 0 to age 1. Because the model was intended for the analysis of populations initially in an equilibrium state, the parameters for the Ricker model were estimated iteratively from the McFadden data such that the age-specific recruitment at age x equalled the mean age-specific recruitment implied by the scaled McFadden data given the appropriate McFadden mean parent stock data. This resulted in the following recruitment models:

Egg-age 0: R = 0.14585 S e -°'°°°°648 s

Age 0-age 1: R = 1 . 4 7 0 7 0 S e -°°°16715s (2)

The progressive degeneracy in the explanatory power of the Ricker stock-recruitment model argued against its use as an appropriate representa- tion of the recruitment phenomenon in later life- stages. Elliott (1985) found no support for the notion of constant proportional survival in an examination of the recruitment data available for a 1966-83 study of migratory brown trout, Salmo trutta, in the English Lake District. Odum et al. (1979), Simberloff (1982) and Ferson et al. (1989) note that variability is a normal and fundamental component of ecological processes. Empirical evidence (Allen, 1973; Hennemuth et al., 1980; Gar- rod, 1983) and theoretical arguments (Waiters and Hilborn, 1976; Peterman, 1978) exist for as- suming that the recruitment of fish is lognormally distributed and the product of independent, non density-dependent influences on survival. Taken together, the evidence argues for the inclusion of both stochasticity and lognormality in any repre- sentation of the later life-stage recruitment phenomenon. Accordingly, recruitment for the age 1 to 2, age 2 to 3 and age 3 to 4 life-stages was

modelled probabilistically using lognormal survival distributions. The means of the distributions were defined by the data reported in McFadden and the standard deviations of the distributions were selected to produce a distribution of survival values bounded by the high and low survival values reported in McFadden. Thus, for the later life-stages the proportion of individuals surviving from one life-stage to the next in any given year is generated as a random variable drawn from the appropriate life-stage survival distribution. The modelling framework is thus able to incorporate the notions of density-dependent mortality in the early life-stages for which there is empirical support both in the McFadden data and the literature (McFadden, 1961; Elliott, 1984a) and density-independent mortality in the later life-stages caused by environmental perturbations (e.g. tem- perature changes, predation, disease, etc.) for which there is both theoretical support in the literature (see above) and species-specific evidence (Power, 1980).

Within age-groups mortalities were assumed to be uniformly distributed by size and sex and between age-groups mortalities, as implied by the McFadden data, declined exponentially as a function of age for all age 1 and older individuals. That is:

Survival = 0.976 e-°845 age. (3)

All parameter p-values < 0.003 and r z = 0.998. Immigration and emigration effects are not considered by the model which implies that mortalities represent only natural losses to the population. The average probability of mortality for an individual, (Pi), before reaching the ith age-group can be expressed as:

R i P~ = 1.0 - - - (4)

S i - 1

where R i expresses the number of average recruits that will survive to reach the ith age-group given the candidate stock, S i_ 1, in the previous age-group. In any given year each individual in the population is assigned a uniformly distributed random number on the interval [0,1]. Those assigned values between 0 and P / a r e removed from the population, while those assigned values

M. Power, G. Power~Ecological Modelling 80 (1995) 171-185 175

greater than Pi survive to reach the next age- group.

Survivors to age 0 are assigned sex and length characteristics. Though variable, the ratio of males to females was set at h l and, consistent with McFadden, was assumed to remain constant for all age-groups. Males and females were treated as having equal probabilities of surviving from one life-stage to the next and any differences arising in the resulting sex ratio result from the randomness inherent in the modelling framework. The age 0 length characteristic for each individual is sex dependent . The mean length of males in Hunt Creek exceeded that of females at age 0 with the difference increasing as a function of age. The lengths of both male and female individuals were generated as normally distributed random variates with means set equal to the mean age 0 male and female fork-lengths reported in McFadden. Representat ive variation for each distribution was based on available data. The resulting mean and standard deviation values for the normal distributions of male and female age 0 fork-lengths (cm) were: 8.3 _+ 0.83 male and 8.0 _+ 0.80 female.

Growth in subsequent years is also sex dependent. Growth in each year for each sex was set equal to the mean annual increments in fork- length implied by the McFadden data (Table 1). This results in the individual differences in fork- length existing at age 0 being maintained throughout the life-cycle. The postulated significance of age 0 fork-length in determining subse-

Table 1 Annual increments in fork-length by age and sex

Growth Increment in Increment in period male size (cm) female size (cm)

Age 0 to 1 5.80 5.00 Age 1 to 2 4.40 3.80 Age 2 to 3 8.10 7.20 Age 3 to 4 6.00 5.20

Mean annual increments in fork-length by age and sex for brook trout in Hunt Creek over the 14 years of the McFadden et al. (1967) study on numerical changes and population regulation in brook trout. The values given above were incorporated directly into the M O D E L and used to model individual annual increments in fork-length by age and sex.

[ ii

(~ 8 10 12 14 16 18 20 22 24 Length (cm)

Fig. 2. A histogram of the simulated length-f requency distribution at the end of the growing season for the studied brook trout population. Consistent with field observation, as the population ages, the age-specific length-f requency distributions increasingly overlap.

quent size is consistent with Elliott (1984b) who found alevin size and water tempera ture to be the chief determinants of growth in migratory brown trout. Furthermore, use of the data contained in McFadden allows the model to accurately reflect the impact of sexual maturat ion on growth for both sexes. The marked decrease in growth from age 1 to age 2 is associated with sexual maturation. Power (1980) reports that al- most all egg production in southern streams, such as Hunt Creek, is provided by two-year-old females. The increased use of energy for sexual maturat ion limits growth potential and explains the associated decline in observed growth. An example of the resulting population-level length- frequency distribution is given in Fig. 2. Charac- teristic of many populations, the variation in length increases as individuals age, resulting in an increasing overlap between the portions of the distribution describing successive age-groups (E1- liott, 1984b). Finally, the model takes a census of the existing population in each age-group and calculates the required summary statistics includ- ing abundance, length-frequency and egg production measures. The information is stored for later use before the information contained in each of the age-group vectors is shifted up a year to represent an increment in time. The model then returns to the top of the logic sequence defined in Fig. 1 to begin calculations for the next year.


3. Validating the model

Critical to any modelling exercise is the validation process. If accurate, or meaningful, results are to be obtained, and used, it is necessary to know how much confidence can be placed in model results. Validation can be generically defined as the measurement of how well model-generated and real system data compare. Of particular interest for the purposes of the present modelling exercise is the notion of structural validity. Models are considered to be structurally valid if they replicate the behaviour of the studied system in a manner that can be regarded as being reflec- tive of the operating characteristics of the real system (Power, 1993). In completing the validation exercise for the modelling framework proposed above, a variety of techniques were used. Consistent with Van Horn (1971) the structural correspondence between the modelling framework and the trout population dynamics system being studied was assured through the use of input validation. Wherever possible the published data of McFadden et al. (1967) were used to model critical life-stages, e.g. the fecundity-size relationship and growth. Furthermore, the structural correspondence between the modelling approach and real world dynamics was assured by seeking verification of the selected modelling approach in the scientific literature, e.g. the preva- lence of density-dependent mortality in the early life-history stages, and the suitability of the lognormal recruitment model.

While the use of approaches supported in the literature does much to establish model credibil- ity, the best evidence of model validity comes from the use of predictive validation as this allows an all-ways comparison between model outputs, alternative model outputs and observations from the real system. To complete the predictive validation of the model, comparisons of model outputs to: (1) the equilibrium values implied by the McFadden field data, and (2) the predictions made by the commercially available Leslie-matrix approach (RAMAS) were carried out. The comparison of the individuals-based modelling approach (MODEL) to actual field data helps to establish that the long-term dynamics inherent in

Table 2 Confidence interval measures for the all-ways comparison of differences in MODEL, RAMAS and actual abundance measures by age-group

Age-group MODEL MODEL RAMAS vs actual vs RAMAS vs actual

Age 0 -75.58, 71.92 -10.98, -8 .93 -64.62, 81.87 Adults -33.33, 33.33 7.67, 9.65 -41.99, 24.67 All -97.48, 93.09 -3.77, -0 .62 -94.42, 96.15

The 95% confidence intervals for the differences in MODEL-actual, RAMAS-actual and MODEL-RAMAS mean abundance measures by age-group. The MODEL-RAMAS intervals were constructed using a paired-t interval approach due to the availability of an equal number of observations. MODEL-actual and RAMAS-actual intervals were constructed using the Welch (1938) solution for the Fisher-Be- hrens problem. Confidence intervals containing zero indicate no significant differences between considered alternatives at the a = 0.05 level of significance.

the model adequately replicate observed field data. While the correspondence of field and MODEL results cannot prove the validity of the model per se, it can nonetheless do much to establish confidence in model predictions. Fi- nally, the demonstration that the modelling framework predicts at least as well as an established alternative, here the Leslie-matrix approach, is both required by the objectives initially set down for the modelling exercise and by the requirements of good forecasting practice (Pindyck and Rubenfeld, 1981). For any proposed change in modelling approaches must be able to predict as least as well as the next best alternative.

Table 2 presents 95% confidence interval measures for comparisons of the MODEL, RAMAS and actual data points for the mean age 0, adult (age 1 and older) and total population abundance measures. The choice of age-groupings was im- posed by the lack of age-group abundance detail available for the RAMAS model outputs. The actual data represents the scaled 14-year averages for each of the reported age-groupings from the McFadden data. Data for the MODEL and RA- MAS results were taken from the averages of 50 replications of 50-year population trajectories for the hypothesized trout population. The required initialization survival and fecundity data for RA-

M. Power, G. Power~Ecological Modelling 80 (1995) 171-185 177

MAS were taken from McFadden table XIII. The fecundity data were entered directly. Survival data were transformed such that the probability at birth of survival to age x data given in McFadden were compatible with the probability of survival from time t to time t + 1 data required by RA- MAS. Abundance data included in the initialization routine for the RAMAS model were the scaled age-group abundance data given by Mc- Fadden. The parameters used for the Ricker stock-recruitment routine in RAMAS were those developed for use in the individuals-based modelling framework. Finally, the RAMAS model was run using the minimum stochasticity options as a means of ensuring that the resulting solu- tions were not unduly biased by the induction of excessive model variability. Data required for the i n d i v i d u a l s - b a s e d m o d e l l i n g f r a m e w o r k (MODEL) were obtained from McFadden as out- lined above.

The 95% confidence intervals for the mean abundance measures for the MODEL-actual and RAMAS-actual comparison pairs were calculated using the Welch (1938) solution for the Fisher- Behrens problem. In each case n I 4= n 2 and the variances of the M O D E L and RAMAS and actual abundance observations, were assumed to be unknown, though equal. The 95% confidence intervals for the observed differences were calculated as:

/ [ Sm - Sa ] __"~ tC: l _ct /2 V n---~ "~- Fl a

interval approach (Law and Kelton, 1991) The approach requires that n, = n 2 but does not re- quire that the MODEL, (XMj) and RAMAS (Xnj) observations used to form the differences be independent or that Var(XMj) = Var(XRj). The interval is constructed by forming the differences Z j = ( X M j - - X n ~ ) for j = l , 2 ..... n and applying the following equation:

+ t,,_ ,,, _,/2~/Var[ Z ] (6)

where Z defines the mean of the Z~ differences and Var[Z] defines the variance of Z. The resulting confidence intervals do not contain zero, indi- cating significant differences exist between the two modelling frameworks. While neither was significantly different from the actual data values, the two were statistically different from each other due to the tendency of the M O D E L to under- predict mean abundance and of RAMAS to over-predict mean abundance.

The indication of statistically significant differences between the modelling frameworks raises questions of statistical bias and accuracy. The existence of bias is evidence that the constructed models make systematic predictive errors. Models with good predictive properties will have bias measures close to zero (Power, 1993) where bias is measured as:

( rnii - a j) (5) Bias ~--- i = 1 ( 7 )

n

where "~m and -~a define the appropriate model and actual age-specific mean abundance measures, Sm z and S~ define the appropriate model and actual age-specific variances and t3 defines the degrees of freedom. Confidence intervals, given in Table 2, containing zero indicate no significant differences be tween e i ther the MODEL-actual or the RAMAS-actual mean abundances at the a = 0.05 level of significance.

The differences in mean abundance between the M O D E L and RAMAS frameworks were tested for significance using a paired-t confidence

and mii defines the model-specific ith abundance prediction for the j th age-specific abundance measure and a i defines the j th age-specific mean abundance measure implied by the McFadden data (see Table 4). Accuracy was measured using mean square error (MSE). As with bias, good predictive models should produce MSE values close to zero with better models having lower MSE values. A test for model accuracy may be obtained by noting that the predictive errors should be normally distributed with zero mean and positive variance (tr2). If the errors, ( m i i - a j), are so distributed then the variable (mji-


a j)2/0- 2 will follow a chi square distribution (Hogg and Craig, 1978) and predictive accuracy may be assessed by comparing the predictive sum of squared errors to the X 2 distribution with n degrees of freedom (Abraham and Ledolter, 1983) as follows:

( m j i - aj) 2 Q = i=1

0-2 (8)

Here 0 -2 was estimated with the use of the standard normal distribution as the value that would describe an error distribution in which 99% of all errors were within + 5% of the age-specific abundance measures, a j, defined in the McFadden data. While arbitrary, the method provides a convenient means of establishing whether model error measures possess the desired degree of predictive accuracy, as large values of Q would lead to the rejection of the null hypothesis concerning the sufficiency of model accuracy.

The results of the bias, accuracy and accuracy testing are given in Table 3. The table confirms the pattern of results seen in Table 2. The M O D E L displays negative bias in all age-groups. The -0 .00 adults M O D E L bias entry results from rounding the computed results and has been included in Table 3 with its sign to indicate the direction of the slight bias that does exist. RA- MAS, on the other hand, displays positive bias in the age 0 and total age-groups and negative bias in the adult age-group. The results indicate that, in comparison to the MODEL, RAMAS tends to over-predict age 0 and total abundance and to under-predict adult abundance. In the age 0 and adult age-groups the bias and MSE measures for the MODEL are superior to those for RAMAS. The combination of over- and under-prediction in the age 0 and adult age-groups for RAMAS, however, produces the anomalous result of superior RAMAS bias and 'MSE measures for the total abundance category. Statistical testing of the observed accuracy measures indicated that while M O D E L inaccuracies, as measured by the MSE, were statistically insignificant, RAMAS inaccuracies for the age 0 and adult age-groups were

Table 3 Predictive validation measures for the M O D E L and RAMAS modelling frameworks

Age 0 Adults Total

M O D E L Bias - 1.33 - 0.00 - 1.33 MSE 14.30 5.76 19.76 AD statistic 0.246 0.325 0.273 Q statistic 4.48 7.37 2.77 RAMAS Bias 8.62 - 8.66 0.86 MSE 82.52 80.20 12.60 AD statistic 0.577 0.522 0.191 Q statistic 25.86 102.69 1.77

Predictive validation measures for the mean abundance measures produced by the M O D E L and RAMAS frameworks. Lower bias and MSE measures are indicative of superior predictive capabilities. Q statistics > 67.505 indicate rejection of the null hypothesis, at the t~ = 0.05 level of significance, that model predictions are accurate to within +_5% of the age-specific abundance measures reported in McFadden. An- derson-Darling (AD) statistics < 0.752 indicate, at the a = 0.05 level of significance, normally distributed predictive errors and are indicative of models with appropriate predictive properties. The - 0.00 adults abundance M O D E L bias results from rounding and the sign has been included to indicate the directive tendency of the bias.

statistically significant at the a = 0.05 level of significance. The pattern of results provides an explanation for the observed array of results in Table 2. Differences in MO D EL and RAMAS predictions arise because of statistically significant predictive biases on the part of RAMAS, and not on the part of the MODEL. Finally, model predictive errors are required by good statistical practice to be normally distributed (Draper and Smith, 1981). The Anderson-Darling statistical test for normality was applied to the predictive errors of both models under the as- sumption that neither the true population mean or variance were known. Results indicated that the predictive errors of both were normally distributed as required.

Ecology typically does not attach any meaning to over- or under-prediction biases. Nonetheless there are relevant costs attached to predictive errors of any kind. Economic studies of over- and under-prediction of other natural resource bases (NEB, 1986) have argued that the costs of over-


prediction are potentially more severe, either because they retard the taking of appropriate reme- diative actions or because they result in the misal- location of the resource base. Accordingly, while ecological principles may fail to provide any rea- son for preferring under- to over-prediction, economic principles would argue that where re- sources are concerned under-prediction is prefer- able. Furthermore, as Cairns (1994) has argued, environmental risk assessment studies must bear in mind policy relevance. The possible costs associated with predictive error and their inevitable consideration in the calculus of social decision- making strengthen the need for selecting one form of predictive bias over another.

Table 4 presents a detailed percentage difference comparison of the MODEL-actual and RA- MAS-actual predictions by age-group. The age 0 and total figures were taken directly from the MODEL and RAMAS simulations. Age 1, 2, 3 and 4 data for the MODEL also come directly from model simulations. For RAMAS, however, these data were developed from the adult mean abundance data using the mean year over year survivorship values implied in McFadden and used as inputs for the completion of the RAMAS simulations. The percentage differences were calculated as actual-model data values expressed as a percentage of the actual. No particular pattern of predictive errors was evident for the age 1, 2, 3, and 4 mean abundance predictions made by

the MODEL. RAMAS, however, over-predicts age 0 abundance and then under-predicts abundance in all other age-groups. The tendency is undoubtedly attributable to the approach taken to modelling species life-history. Recruitment is modelled as density-dependent only from the egg to age 0 life-stage. Evidence from the literature, however, suggests that for brook trout density-dependent mortality continues into later life-stages (Power, 1980). The inclusion of a second density- dependent recruitment relationship in the individuals-based model regulates adult recruitment more effectively, allowing the MODEL to com- pensate for fluctuations in age 0 abundance in a manner consistent with species life-history pres- sures. Admittedly the inclusion of a second density-dependent recruitment function is a species- specific modelling adjustment. However, in its defence Bartell et al. (1992) have argued that risk analysis frameworks must incorporate what is known about the life history dynamics o f the species being studied. Furthermore, the pattern of differential effects of contaminants on species argues that species-specific adjustments will be crucial to the development of reliable and credible studies of the impacts of lethal and sublethal stresses on affected populations.

Finally, to validate MODEL length-frequency and annual egg production data, the computed means of MODEL generated lengths by age- group and annual egg production were compared

Table 4 Percentage difference comparisons for MODEL, RAMAS and actual data by age-group

Age-group Actual MODEL Percent RAMAS Percent difference difference

Age 0 757.86 756.53 - 0.18 766.48 1.14 Age 1 314.23 314.02 - 0.07 306.97 - 2.31 Age 2 56.01 56.23 0.38 54.72 - 2.30 Age 3 4.60 4.59 - 0.16 4.51 - 2.06 Age 4 0.09 0.10 6.67 0.07 - 18.61 Total 1132.79 1131.46 - 0.12 1133.65 0.08

Differences by age-group between the MODEL, RAMAS and actual data. Figures for age 0 and totals were taken directly from each model simulation. Age 1, 2, 3, and 4 data for the MODEL were also taken directly from simulations. Age 1, 2, 3, and 4 data for RAMAS were developed from adult mean abundance data using the mean year over year survivorship values used to complete the simulations.

180 M. Power, G. Power /Ecological Modelling 80 (1995) 171-185

to those obtained from the McFadden data using a two-sample-t 95% confidence interval approach. The approach constructs a confidence interval for the observed difference in means (Hogg and Craig, 1978) as follows:

~ Sm2i S m i - X a i = t n - l ' l - a / 2 rl (9)

tends the applicability of population-level modelling frameworks to the analysis of environmental hazards beyond the prediction of abundance, to address the issues associated with analyzing the subtle changes in population age and size- structure that are frequently the harbinger of the adverse effects of human activities on fish populations (Munkittrick and Dixon, 1989).

where X m i and X a i are the ith age-group, or egg production, means of the M O D E L and McFad- den data respectively and S m 2 is the variance of the M O D E L ith age-group length, or egg production, data. Table 5 reports the results. All the constructed 95% confidence intervals for the differences in M O D E L and actual data contain zero. Accordingly, there is no evidence for the rejection of the null hypothesis concerning the similarity of M O D E L and McFadden length and annual egg production data.

Taken together, the data contained in Tables 2, 3, 4 and 5 indicate that the individuals-based trout modelling framework provides credible and realistic population abundance estimates. The M O D E L compares favourably with both actual data and with predictions made by the Leslie-matrix-based approach. Where differences in the two approaches exist, the M O D E L produces data with superior statistical prediction properties. Furthermore, the M O D E L is capable of producing length-frequency information consistent with observed field data. As such, the M O D E L ex-

4. Model experimentation

Despite the acceptance of abundance as a regulatory endpoint for the purposes of environmental risk assessment, there are reasons to question both its utility and appropriateness. Endpoints for ecological risk assessment need not focus directly on changes in population abundance. This has been evidenced in the environmental risk assessment literature with discussions about the selection of early warning indicators of adverse environmental change (Suter, 1993). The selection of sentinel species, the development of advanced water quality assessment protocols and population monitoring programs all suggest there is much to be gained by developing our ability to respond to the impacts of anthropogenic stress before the effects of such stress become irreversible. Modelling stands to play a potentially important role in this process. To do so, however, models must be capable of incorporating, using and predicting the biological effects data touted

Table 5 Comparison of MODEL predicted and actual mean length and egg production data

Age-group MODEL McFadden Percent Confidence mean mean difference intervals

Age 0 8.15 8.13 0.23 -0.11, 0.15 Age 1 13.56 13.54 0.12 -0.17, 0.21 Age 2 17.67 17.60 0.40 - 0.25, 0.39 Age 3 25.43 25.30 0.51 - 0.47, 0.72 Eggs 22321 20773 7.45 - 1532, 4629

Comparisons of MODEL and actual age-specific mean fork-lengths (cm) and egg production. The confidence intervals are for the differences in MODEL and McFadden data and were constructed using a two-sample-t approach. They show no significant differences at the a = 0.05 level of significance. The data confirm the ability of the MODEL to realistically replicate critical population-level size structure information.


as having the potential of acting as early warning indicators. Models based solely on abundance fail to do this because they aim to predict the path of population trajectories through time and not the precursors to changes in those trajectories. As Macek (1968) has argued, to fully understand the impact on populations of chronic exposure to sublethal levels of a contaminant, it is necessary to evaluate both the ability of populations to survive and their ability to carry on other fundamental life processes such as growth and reproduction.

Changes in abundance can be induced by both acute and chronic stresses. Whereas acute stress will induce an immediate mortality response, many chronic (sublethal) stresses will not. The changes in abundance that are seen inevitably act through changes induced in the growth or reproductive capacity of the population. A fact which implies that abundance-based models will typically be incapable of appropriately analyzing the population-level implications of sublethal stresses. By failing to recognize the subtle physiological changes occurring in the ability of populations to grow and reproduce, abundance-based models will tend to under-estimate the effects of such stresses on the affected population. This can lead to unfortunate errors in both risk assessment and management resulting from the failure to reject the experimental no-impact null hypothesis when it should be rejected. And, if the results of com- mitting type II error in the decision-making process include irreversible damage to economically important species, the costs of such errors to society will be large.

As an example, consider the effects of toxaphene on brook trout growth. Although the use of many organochlorines has been reduced in recent years, significant quantities are still applied to agricultural crops. The acute and sublethal effects of toxaphene on aquatic organisms have been reported by a number of investigators. The potential effects on trout abundance and population size-structure in the long-term, however, have not been reported in the primary literature. Mehrle and Mayer (1975) report experimental results on the effects of toxaphene exposure on the growth and development of continu-

ously exposed brook trout. Data for the 139 ng/1 concentrations contained in Table 3 (Mehrle and Mayer, 1975) were used for the purposes of the M O D E L experimentation reported below. At this concentrat ion toxaphene showed significant, though non-lethal, effects on trout growth patterns. Differences were established using a multi- pie means comparison (least significant difference) test at the 0.05 level of significance. The data in Mehrle and Mayer were converted to differences and expressed as percentages of the observed control lengths before being modelled as a linear response function. The model express- ing the percentage difference between the control and the 139 ng/1 exposure growth patterns was estimated as follows:

Difference in growth = 0.21 time. (lO)

The model was initially estimated with an intercept term. Testing of the intercept indicated it was not significantly different from zero at the a = 0.05 level of significance (t = - 0 . 2 9 , p = 0.80). As there was no indication that the no intercept model was inappropriate and as theory would suggest regression through the origin (e.g. no-exposure, no effect on growth) the model in Eq. 10 above was estimated. The r 2 and p-values were respectively; 0.989 and < 0.001.

The model was then used to estimate the 90-day continuous exposure impact on trout abundance and population size-structure using the individuals-based modelling framework discussed above. A no-exposure and experimental cases were run using common random number streams. Each case was run for a total of 60 years, with the first 10 years being eliminated from the data collection routines using the method of Welch (1983) to avoid problems with transient phase observations biasing results. Furthermore, each case was replicated 50 times and the results averaged as a means of guaranteeing normality in the distributions describing each of the observations. The resulting plots of the age 0 and adult abundance for the no-exposure and experimental cases are given in Figs. 3 and 4. The no-exposure data is plotted as a solid line and the experimental data is plotted as squares in both figures.


780-

76O"

'0

740" <

720-

70(0

• • • • • • m m m • -,..',. - , - . - - . ' , . . . . . ' , .

m m mmm

mm • • •

10 20 30 40 50 Time (years)

Fig. 3. Plots of the age 0 mean abundance data for the no-exposure (solid line) toxaphene simulation and the 90-day continuous exposure (squares) experimental simulation. Expo- sure results in significant declines in average length at sexual maturity and a consequent decline in annual egg production and age 0 abundance.

3 8 5

~= 381 t c

~ 3n - i

. 3 7 3

3 6 9

3 6 5 10 20 30 40 50

Time (years)

Fig. 4. Plots of the adult mean abundance data for the no-exposure (solid line) toxaphene simulation and the 90-day continuous exposure (squares) experimental simulation. While exposure results in significant declines in age 0 abundance, compensatory mortality implied by the density-dependent mortalities of the recruitment curve result in lower mortalities and slight, though statistically significant, increases in adult abundance.

Ninety-five percent confidence intervals for the differences in the no-exposure and experimental cases were constructed using a paired-t confidence interval approach (Law and Kelton, 1991) as discussed above. Results of the computations are given in Table 6 and, with Figs. 3 and 4, indicate a significant decrease in the abundance of age 0 trout and a slight, though statistically significant, increase in the abundance of adult trout as a result of exposure to toxaphene. Toxaphene reductions in growth result in declines in abundance of age 0 trout as a result of the declines in annual egg production implied by the fecundity-size relationship. Compensatory

mortality mechanisms imbedded in the stock-recruitment relationships for the egg to age 0 and age 0 to age 1 life-stages, however, allow the recovery of the adult population to levels slightly in excess of the pre-exposure equilibrium. Re- duced competition and density-dependent mortality among young of the year increase recruitment and allow the increase in adult abundance to be achieved. While adult abundance figures show no worsening of the situation for the considered population as a result of exposure to toxaphene, changes in age 0 abundance and the

Table 6 Differences between the no-exposure and experimental data

Abundance Mean Confidence interval Confidence interval Percent measure difference lower bound upper bound difference

Age 0 27.37 * 25.15 29.59 3.62 Age 1 - 1.85 * - 2.45 - 1.24 - 0.59 Age 2 - 0.08 - 0.44 0.28 - 0.14 Age 3 0.02 - 0.09 0.13 0.37 Age 4 0.00 - 0.01 0.02 3.75 Adults - 1.91 * - 2.60 - 1.21 - 0.51

Differences in mean abundance by age-group for the no-exposure and experimental 90-day toxaphene exposure simulations of the MODEL. Confidence intervals for the observed differences were completed using a paired-t approach. Significant differences are marked with asterisks. Age 0, age 1 and adult numbers show significant differences. The Age 0 and age 1 differences are directly attributable to the effects of toxaphene on growth, hence reproductive capacity. Compensatory mortality induced by the density-dependence relationships allows adult abundance to rebound to levels slightly in excess of the no-exposure case.

M. Power, G. Power / Ecological Modelling 80 (1995) 171-185 183

°.7° T ] | ~ ~ No-exposure :a~ /

°"°1 II / • °"°1 | /

II 21 0.10

7.5 9.5 11.5 13.5 15.5 >17.5 Length (cm)

Fig. 5. Histograms of the relative frequencies by length intervals for all individuals over all years and replications of the no-exposure and toxaphene exposure cases. Shifts in the length-f requency distribution toward increases in the preva- lence of smaller fish can be seen in the rise in the relative frequency of 7.5-cm length interval individuals and the decline in the relative frequency of > 13.5-cm length interval individuals in the toxaphene exposure case. Rises in the relative frequency of 11.5-cm length interval individuals represents the reduced growth potential of individuals that would normally have been in the 13.5-cm length interval in the absence of exposure to toxaphene.

population length-frequency distribution both indicate the adverse effects of chronic stress. Leslie-matrix-based models, such as RAMAS, that tend to over-estimate age 0 abundance (+1 .14%) would tend to mask the declines in abundance in this critical age-group while show- ing little or no effect on adult numbers.

Fig. 5 presents the histograms for the length- frequency distributions constructed using the length information generated for all individuals over all years and replications of the no-exposure and toxaphene exposure cases. The no-exposure and exposure length interval frequencies are rep- resented respectively by the solid and open bars. Fig. 5 clearly depicts a pat tern of slower growth in the exposure case with a leftward shift in the observed length-frequency distribution. There is a significant increase in the frequency of individuals in the 7.5-cm length interval and a shift in the frequency of age 1 individuals from the 13.5-cm length interval in the no-exposure case to the l l .5-cm interval in the exposure case. The leftward shift in the length-frequency distribution is

further confirmed by declines in the relative frequencies of individuals in all length intervals > 13.5 cm in the exposure case. Toxaphene exposure acts to retard growth and shifts the mean of the observed length-frequency distribution from 10.15 cm in the no-exposure case to 8.89 cm in the experimental case, a decline of some 12.39%. This represents a significant difference in mean lengths between the two cases (t > 1000 for a = 0.05).

From the perspective of a field biologist, such changes would be indicative of a stressed population and a potential warning of long-term declines in population abundance. In the individuals-based modelling framework discussed above, the impact on growth is reflected immediately in egg production through the fecundity-size relationship. As average spawning size declines, so does egg production. At the levels of exposure used in the experimental case the decline in average spawning size was sufficient to have an impact on age 0 abundance. The compensatory population mortality mechanisms imbedded in the Ricker stock-recruitment model, however, were elastic enough to mitigate against the decreases in age 0 abundance through increases in survival over the egg to age 0 and age 0 to age 1 life-stages. Without such physiologically-based relationships, however, the Leslie-matrix modelling approach cannot incorporate the sublethal effects of contaminant stress on either population abundance or size-structure in a meaningful way. The approach, in presuming no impacts on reproduction, and without having specific knowledge of the fecundity-size response, would tend to over- predict age 0 abundance. Such predictions would create the impression of no observable effects and lead to the erroneous acceptance of any hypothesized no-effect postulate. That is not to suggest, however, that Leslie-matrix-based approaches, such as RAMAS, are wholly inade- quate for assessing environmental risks. Differ- ences in the matrix-based and individuals-based projects are, at worst, minor. The failing, if any, is one of not having recognized the need to develop tools capable of analyzing, in an integrated manner, a suite of representative population-level responses to chronic anthropogenic stresses.


5. Conclus ion

The individuals-based modelling approach out- lined above at tempted to meet several stringent criteria. The model successfully addresses endpoints of regulatory concern by producing both population abundance and size-structure information. It is capable of incorporating available toxicity data as witnessed by the toxaphene exper- iment. Furthermore, it is able, indeed relies upon, the available population information routinely produced as part of species life-history studies. Finally, because of its individuals-based nature it is able to accurately and appropriately reflect natural variations among individuals within the population. The strengths of the modelling approach lie in the simplicity with which it views life-history events. Individuals are born, age, grow, reproduce and die. The recognition of growth as a critical population-level process increases the flexibility of the modelling framework vis-h-vis matrix-based approaches, qualifying it for use in the analysis of sublethal contaminant stresses on populations.

The effects of stressors on fish populations can be direct, acting on growth, reproduction a n d / o r survival. As Emlen (1989) has pointed out, models specifically addressing contaminant stress have tended to assume damage will be reflected in survival but not reproduction. Unhealthy individuals, however, do not reproduce as efficiently as healthy individuals, a fact suggesting a more ex- tensive approach to modelling population-level impacts than focusing on abundance allows. Munkittrick and Dixon (1989) have argued that " there is a need to develop methodologies capable of early detection of environmental degrada- tion, methodologies which would be responsive to a wide range of environmental perturbations". It is our contention that precisely because modelling allows the examination of a wide range of stresses on populations that it is a potentially powerful methodological tool. However, to be fully useful, models must advance beyond focusing purely on abundance measures and begin to consider other fundamental changes in population-level characteristics indicative of contaminant stress. Only then can modelling-based ap-

proaches be viewed as appropriate and scientifi- cally credible means of completing environmental risk assessments.

Finally, the modelling approach discussed above can easily be extended to include consideration of a variety of other anthropogenic and environmental stresses. Exploitation, habitat al- teration and loss could also be addressed. Other lethal and sublethal stresses can easily be incorporated into the modelling framework providing that a suitable exposure-response curve can be generated. Such stresses include those that affect mortality, growth, fecundity and reproductive success.

Acknowledgements

We thank Drs. S. McKinley and J. Reist for helpful comments during the development of the model. The suggestions and comments of the anonymous reviewers were appreciated and helped us improve the manuscript. Natural Sci- ences and Engineering Research Council of Canada grants to the authors supported this work.

References

Abraham, B. and Ledolter, J., 1983. Statistical Methods for Forecasting. John Wiley and Sons, New York, NY, 405 pp.

Allen, K.R., 1973. The influence of random fluctuations in stock-recruitment relationships on the economic return from salmon fisheries. In: B.B. Parrish (Editor), Fish Stocks and Recruitment, Proceedings of a Symposium held in Aarhus, 7-10 July, 1970. Rapp. P.-V. Reun. Cons. Int. Explor. Mer, 164: 350-359.

Bartell, S.M., Gardner, R.H. and O'Neill, R.V., 1992. Ecolog- ical Risk Estimation. Lewis Publishers, Chelsea, MI, 252 PP.

Bates, D.M. and Watts, D.G., 1988. Nonlinear Regression Analysis and its Applications. John Wiley and Sons, New York, NY, 365 pp.

Begon, M. and Mortimer, M., 1986. Population Ecology: A Unified Study of Animals and Plants, 2nd edition. Black- well Scientific Publications, Oxford, 200 pp.

Cairns, J., 1994. Ecosystem health through ecological restora- tion: barriers and opportunities. J. Aquat. Ecosys., 3: 5-14.

Cairns, V.W., Hodson, P.V. and Nriagu, O.J., 1984. Contami- nant Effects on Fishes. Advances in Environmental Sci-

M Power, G. Power~Ecological Modelling 80 (1995) 171-185 185

ence and Technology, Vol. 16. John Wiley and Sons, Toronto, 333 pp.

Draper, N. and Smith, H., 1981. Applied Regression Analysis, 2nd edition. John Wiley and Sons, New York, NY, 709 pp.

Elliott, J.M., 1984a. Numerical changes and population regulation in young migratory trout, Salmo trutta, in a Lake District Stream, 1966-83. J. Anita. Ecol., 53: 327-350.

Elliott, J.M., 1984b. Growth, size, biomass and production of young migratory trout, Salmo trutta, in a Lake District Stream, 1966-83. J. Anim. Ecol., 53: 979-994.

Elliott, J.M., 1985. The choice of a stock-recruitment model for migratory trout, Salmo trutta, in an English Lake District Stream. Arch. Hydrobiol., 104: 145-168.

Emlen, J.M., 1989. Terrestrial population models for ecological risk assessment: a state-of-the-art review. Environ. Toxicol. Chem., 8: 831-842.

Ferson, S., Ginzburg, L. and Silvers, A., 1989. Extreme event risk analysis for age-structured populations. Ecol. Model., 47: 175-187.

Garrod, D.J., 1983. On the variability of year-class strength. J. Cons. Int. Explor. Mer, 41: 63-66.

Hennemuth, R.C., Palmer, J.E. and Brown, B.E., 1980. A statistical description of recruitment in eighteen selected fish stocks. J. Northwest. Atl. Fish. Sci., 1: 101-111.

Hogg, R.V. and Craig, A.T., 1978. Introduction to Mathemati- cal Statistics, 4th edition. Macmillan, New York, NY, 658 pp.

Law, A.M. and Kelton, W.D., 1991. Simulation Modeling and Analysis, 2nd edition. McGraw-Hill, New York, NY, 759

PP. Leslie, P.H., 1945. On the use of matrices in certain popula-

tion mathematics. Biometrika, 33: 183-212. Macek, K.J., 1968. Reproduction in brook trout (Salvelinus

fontinalis) fed sublethal concentrations of DDT. J. Fish. Res. Board Can., 25: 1787-1796.

McFadden, J.T., 1961. A population study of the brook trout, Salvelinus fontinalis. Wildlife Monographs 7, 73 pp.

McFadden, J.T., Alexander, G.R. and Shetter, D.S., 1967. Numerical changes and population regulation in brook trout, Salvelinus fontinalis. J. Fish. Res. Board Can., 24: 1425-1459.

Mehrle, P.M. and Mayer, F.L., 1975. Toxaphene effects on growth and development of brook trout (Salvelinus fontinalis). J, Fish. Res. Board Can., 32: 609-613.

Munkittrick, K.R. and Dixon, D.G., 1989. Use of white sucker (Catostomus commersoni) populations to assess the health of aquatic ecosystems exposed to low-level contaminant stress. Can. J. Fish. Aquat. Sci., 46: 1455-1462.

NEB (National Energy Board of Canada), 1986. Canadian Energy Supply and Demand 1985-2005, Ottawa, 352 pp.

NRCC (National Research Council of Canada), 1985. The Role of Biochemical Indicators in the Assessment of Ecosystem Health - Their Development and Validation. Publ. No. NRCC 24371, National Research Council of Canada, Ottawa, 119 pp.

Odum, E.P., Finn, J.T. and Franz, E.H., 1979. Perturbation theory and subsidy-stress gradient. BioScience, 29: 349- 352.

Peterman, R.M., 1978. Testing for density-dependent marine survival in pacific salmonids. J. Fish. Res. Board Can., 35: 1434-1450.

Pindyck, R.S. and Rubenfeld, D.L., 1981. Econometric Mod- els and Economic Forecasts, 2nd edition. McGraw-Hill, New York, NY, 630 pp.

Power, G., 1980. The brook charr, Salvelinus fontinalis. In: E.K. Balon (Editor), Charrs: Salmonid Fishes of the Genus Salvelinus. Junk Publishers, The Hague, pp. 141-203.

Power, M., 1993. The predictive validation of ecological and environmental models. Ecol. Model., 68: 33-50.

Ricker, W.E., 1954. Stock and recruitment. J. Fish. Res. Board Can., 11: 559-623.

Simberloff, D., 1982. A succession of paradigms in ecology: essentialism to materialism to probabilism. In: E. Saarinen (Editor), Conceptual Issues in Ecology. Reidel, Dordrecht, pp. 63-99.

Suter, G.W., 1993. Ecological Risk Assessment. Lewis Pub- lishers, Chelsea, MI, 538 pp.

Van Horn, R.L., 1971. Validation of simulation results. Man- age. Sci., 17: 247-258.

Waiters, C.J. and Hilborn, R., 1976. Adaptive control of fishing systems. J. Fish. Res. Board Can., 33: 145-159.

Wedemeyer, G.A., McLeay, D.J. and Goodyear, C.P., 1984. Assessing the tolerance of fish and fish populations to environmental stress: the problems and methods of monitoring. In: V.W. Cairns, P.V. Hodson and J.O. Nriagu (Editors), Contaminant Effects on Fishes. Advances in Environmental Science and Technology, Vol. 16. John Wiley and Sons, Toronto, pp. 163-195.

Welch, B.L., 1938. The significance of the difference between means when the population variances are unequal. Biometrika, 25: 350-362.

Welch, P.D., 1983. The statistical analysis of simulation re- suits. In: S.S. Lavenberg (Editor), Computer Performance Modeling Handbook. Academic Press, New York, NY, pp. 268-329.

a modelling framework for analyzing anthropogenic stresses on brook trout (salvelinus fontinalis)...

Documents