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OIKOS 92: 501–514. Copenhagen 2001

Application of a model of scale dependence to quantify scale domains in open predation experiments Go¨ran Englund, Scott D. Cooper and Orlando Sarnelle

Englund, G., Cooper, S. D. and Sarnelle, O. 2001. Application of a model of scale dependence to quantify scale domains in open predation experiments. – Oikos 92: 501–514. We use a model of open predation experiments to define scale domains that differ in terms of the controlling processes and scale dependence of predator impacts. For experimental arenas that are small compared to the movements of the prey (small scale domain) the model predicts that predator impacts are scale independent and controlled by prey movements. For arenas of intermediate scale we predict that predator impacts are scale dependent and controlled by both prey movements and direct predation, and for the largest scale domain we predict weak scale dependence and predation control. We propose that the scale-domain concept is useful when designing and interpreting field experiments. As an illustration we apply the concept to experiments examining predator effects on the stream benthos. First, we test two key assumptions of the underlying model: that area-specific prey migration rates decrease with increasing size of experimental arenas and that predation rates are independent of arena size. For this purpose we used published estimates of prey emigration and predator consumption rates for nine studies examining the effects of stream predators on benthic prey. We found that prey per capita emigration rates but not predation rates decreased with increasing arena length. Second, we demonstrate a method for identifying the scale domain of real experiments. The model of predation experiments was parameterized using experimental data and the expected spatial and temporal scale dependence of predator impacts on prey in these experiments was simulated. The simulations suggest that the studies conducted in the largest arenas (length 15 – 35 m) should be classified as large-scale, consumption-controlled experiments, whereas the experiments conducted in smaller arenas (length 1.5– 6 m) should be classified as small or intermediate-scale. We also attempted to determine the scale domain of the experiments in a large data set, including results from most published stream predation experiments. The majority of arenas used in these experiments (73%) were smaller than 1 m in length. Our data on the scale dependence of predation and prey migration rate suggest that experiments in this scale range ( B1 m) should be classified as small-scale, movement-controlled experiments for most prey taxa. G. Englund, Dept of Ecology and En6ironmental Science, Uni6. of Umea˚, SE-901 87 Umea˚, Sweden ([email protected]) (also at: Umea˚ Marine Sciences Centre, Norrbyn, SE-910 20 Ho¨rnefors, Sweden). – S. D. Cooper, Dept of Ecology, E6olution and Marine Biology and Marine Science Inst., Uni6. of California, Santa Barbara, CA 93106, USA. – O. Sarnelle, Dept of Fisheries and Wildlife, Michigan State Uni6., East Lansing, MI 48824, USA.

Field experiments have been an important tool in the study of population and community dynamics since the pioneering work of Connell (1961) and Paine (1966). To

date, a large number of field experiments, performed in a diverse array of systems, have been published (Hairston 1989, Wilson 1991, Gurevitch et al. 1992,

Accepted 27 October 2000 Copyright © OIKOS 2001 ISSN 0030-1299 Printed in Ireland – all rights reserved OIKOS 92:3 (2001)

501

Wooster 1994, Osenberg et al. 1999). Typically, the densities of one or several potentially important species are manipulated within a small portion of habitat and the density responses of species of interest are monitored. For logistical reasons, the spatial and temporal scales of experiments are often much smaller than those of the system of interest; however, meaningful interpretation requires that results can be extrapolated from small to large scales. It is thus critical that we understand how experimental results change with the spatial and temporal scales of manipulation. Experimental and theoretical studies have identified two broad categories of effects that may result from changes in the size of experimental arenas. First, arena size per se can influence properties such as species richness, food-chain length and complexity, response time to perturbations, stability, and the amount of biotic and abiotic heterogeneity, and such changes in system structure and function may affect experimental outcomes (Schoener 1989, de Roos et al. 1991, Englund and Olsson 1996, Spence and Warren 1996, Cooper et al. 1998). For example, manipulations of productivity may interact with food-chain length to affect the biomasses of different trophic levels (Oksanen et al. 1981, Wootton and Power 1993), and the increase in the amount of prey refuge that often occurs when arena size is increased can influence prey responses to predator manipulations (Englund and Olsson 1996, Sarnelle 1997). Second, an alteration of arena size typically alters the perimeter to area ratio. Thus, whenever perimeter length influences the processes of interest, we may expect the outcome of experiments to be scale dependent. A particularly important effect operates in open experiments, i.e., those that allow movements into and out of experimental units by the species of interest (Wiens et al. 1986, Hall et al. 1990, Lancaster et al. 1991). In such experiments, scale dependence arises because the influence of migration on densities of the organism of interest decreases with an increase in the size of experimental units (Stamps et al. 1987, Englund 1997, Wooster et al. 1997, Cooper et al. 1998). Englund (1997) used mathematical models to analyze this type of scale dependence in predation experiments where prey, but not predators, can move into and out of enclosures. The models predict that the scale dependence in predator impacts is strongly influenced by the movement responses of prey to predators (increasing or decreasing prey emigration rates, Sih and Wooster (1994)). Much of our current understanding of the scale dependence of experimental results stems from experiments where the same treatments have been applied to experimental units of different size or shape (Luckinbill 1974, Spence and Warren 1996, Petersen et al. 1997, Sarnelle 1997, Cooper et al. 1998, Schindler 1998). To further increase our understanding of the scale dependence of experimental outcomes it is important to develop theoretical tools that can be used to explain 502

why, and predict when, treatment effects are scale dependent. This information can then be used to determine the degree to which results can be extrapolated across scales. In this paper we examine if scale domains, a concept originally proposed by Wiens (1989), can function as such a tool. Wiens suggested that the scale dependence of processes and patterns are discontinuous; there are scale ranges, called scale domains, with weak or monotonic scale dependence within which the same processes control observed patterns. In between scale domains there are transition zones with strong or changing scale dependence and a change in the identity of the controlling processes. To examine the usefulness of this idea we first use predictions from a model derived in Englund (1997) to identify scale domains that differ in predicted spatial and temporal scale dependence and in the processes controlling prey densities (prey movements vs predator consumption). We then validate the underlying model with data from eleven experiments examining effects of predators on densities of stream benthos. These data are also used to illustrate methods for identifying the scale domain of experiments, including parameterizing the model and predicting the scale dependence of predator impact on different prey taxa in each of the experiments. Finally, we examine the implications of this analysis for the interpretation of the data in a much larger data set consisting of most published experiments addressing predator effects on stream benthos.

Models The models that we use to define scale domains are only given a brief presentation as a detailed derivation is given elsewhere (Englund 1997). Parameter definitions and model equations are summarized in Tables 1 Table 1. Parameters used in models of predation experiments. Model equations are given in Table 2. Description

Parameters

Unit

Prey density in control units, predator units and outside the units Area-specific prey migration rate for control units, predator units and outside the units Consumption rate Time No of individuals emigrating or immigrating Area of experimental unit Length of experimental unit Constant

nc, np, no

inds. area−1

mc, mp, mo

time−1

q t E, I

time−1 time inds. time−1

A

area

l

length

k

length time−1

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Table 2. Equations used to model the scale dependence of predator impacts in experiments allowing prey migration. Parameters are given in Table 1. Description

Equations

No.

Change in prey density in control and predator units when there are predators in the background habitat Change in prey density in control and predator units when the background habitat is predator-free habitat Equilibrium PI in habitats with and without predators in the background PI as a function of time in habitat with and without predators in the background

dnp /dt= −qnp dnc /dt= mono−mcnc

(1) (2)

dnp /dt= mono−(mp+q)np dnc /dt= mono−mcnc

(3) (4)

PI= ln{mp /(mc−q)} PI= ln{(mp+q)/mc }

(5) (6)

PI(t) =ln[{mp−(mp−mc+q) (7) −(mc+q)t ×e }/(mc−q)] PI(t) =ln[(mp+q)/ (8) {mc+(mp+q−mc ) ×e−(mp+q)t}]

Scale dependence of m = k/l area-specific migration rate

(9)

and 2. The models are meant to describe short-term experiments where the densities of predators are manipulated in enclosures and prey are allowed to move into and out of enclosures. Thus, prey densities are determined by prey migration in control cages and by both migration and predator consumption in predator cages (eqs 1–4 in Table 2). Thus, it is assumed that prey do not reproduce over the short time scales of experiments and that there are no sources of mortality other than predation. Moreover, it is assumed that per capita predator consumption and prey migration rates are independent of prey density. Separate models were derived for experiments in predator-free habitats and for those with predators in the background habitat. When the background has predators we assumed that prey were consumed at the same rate in both predator units and in the background habitat. As the response variable quantifying the predator impact on prey we used a predator impact index (PI) given by the natural logarithm of the ratio between the densities in control and predator units, i.e. PI =ln(nc /np ). Both models were solved for equilibrium PI (eqs 5 and 6). However, in most cases experiments are run for a fixed period rather than to equilibrium; consequently, we also expressed PI as a function of time (eqs 7 and 8). Scale dependence is introduced by assuming that migration rates, but not consumption rates, depend on arena size. The measure of migration rate that we use, area-specific migration rate (m), has proven to be OIKOS 92:3 (2001)

difficult to explain and we recommend Englund (1997: 2317– 18, 2321) for a detailed derivation. It is a constant that relates the numbers emigrating (E) or immigrating (I) per unit arena area to the density of organisms in the source habitat. For emigration out of an arena it collapses to per capita emigration rate, i.e., m=E/Ni, where Ni is the number of organisms in the patch. For immigration it has a less intuitive definition (m =I/Ano ). The area-specific migration rate is proportional to the ratio of perimeter to area (see Englund 1997). The experiments that we analyze here were performed in artificial channels, stream pools, or stream sections, where prey can only move into and out of arenas at the upstream and downstream ends. For such arenas the ratio of permeable perimeter and area is inversely proportional to the arena length (eq. 9, Table 2). To evaluate the effect of arena length on predator impact we substituted eq. 9 into eqs 5 –8 (Table 2).

Model predictions and scale domains Here we show predictions for the scale dependence of predator impacts (PI) at large and intermediate spatial scales that were derived in Englund (1997), as well as derive new predictions for the spatial scale dependence of PI at very small spatial scales and for the temporal scale dependence of PI across different spatial scales. Because the presence of predators in the background habitat has little influence on these relationships for the parameter values used, we only show predictions generated from eq. 7 for a habitat with predators in the background. The model (eq. 7) predicts a sigmoid relationship between spatial scale and PI (Fig. 1). This relationship shows two scale domains within which PI is scale independent and an intermediate scale range with strong scale dependence. Properties of the scale domains are summarized in Table 3. In the small scale domain, PI is controlled by prey movements. A positive PI, i.e., depressed prey densities in predator cages, is predicted if prey emigration rates are higher in predator than control cages (e.g., mp = 2mc in Figs 1 and 2a). No effect is predicted when migration rates are independent of the presence of predators (mp = mc ), and a negative PI is expected if prey freeze in the presence of predators (e.g. mp = 12mc in Figs 1 and 2a). At these small scales migration rates are very high and PI quickly reaches the equilibrium given by eqs 5 and 6 (Fig. 2a). The asymptotic value of PI is found by letting prey movement rates approach infinity in eqs 5 and 6; hence, PI approaches ln(mp /mc ). The direct extrapolation of results to larger scales (i.e., without transformations) is possible within this domain but not across domains (Fig. 1). 503

Fig. 1. Predicted relationships between predator impact and spatial scale for experiments performed in a habitat with predators in the background. On top we have indicated approximate boundaries between the scale domains described in the text and Table 3. The predator impact index is given by PI= ln(nc /np ), where nc and np are prey densities in control and predator treatments. mp = 2mc refers to a situation where area-specific migration rates are higher in predator than control cages (avoidance), mp = mc means that emigration rates are equal in the two types of cages, and mp = 12mc refers to a situation where prey decrease emigration rate in the presence of predators (freezing). Predictions were generated using eqs 7 and 9, a consumption rate (q) of 0.01 per unit time, an area-specific emigration rate in control cages (mc ) of 0.05 per unit time, and a time (t) of 20 units.

In the intermediate scale range PI is controlled both by predator consumption and prey movements and increasing the scale causes a change from movement control to consumption control. Particularly strong scale dependence in PI is expected when the PI generated by movements alone is very different from the PI generated by consumption alone. Under these conditions it is not possible to extrapolate results within or between domains (Fig. 1) and the time to equilibrium is intermediate (Fig. 2b). Note, however, that PI is scale independent for the special case when the PI caused by movements is equal to the PI caused by consumption. In the large scale domain the movement behavior of prey has negligible influence on prey density and PI is controlled by the consumption of prey by predators. Direct extrapolation to larger scales is possible because PI is predicted to be scale independent within this domain. The values of PI at very large scales are found by letting area-specific migration rates approach zero. In eqs 7 and 8, PI then approaches qt, i.e., the predator effect is mediated entirely by direct consumption and is

Fig. 2. Predicted changes in PI through time for experiments conducted in a habitat with predators in the background at three different spatial scales. a) Small scale, arena length = 0.03 units. b) Intermediate scale, length =1 unit. c) Large scale, length =30 units. Predictions were generated using eqs 7 and 9. Parameter values are given in Fig. 1.

predicted to increase linearly over time (Fig. 2c). Thus, under the simplifying assumptions of the model, it is predicted that PI will not equilibrate at large scales.

Table 3. Properties of predation experiments performed at different spatial scales as predicted by eqs 7 and 8 (see also Figs 1 and 2). Scale domain

Relative size of arena Controlling process Scale dependence of predator impact within the domain Direct extrapolation to larger scales possible Time to equilibrium

504

I

II

III

Small Movements No No Short

Intermediate Move.+Cons. Yes No Intermediate

Large Consumption No Yes Long

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Testing assumptions and validating the model Two fundamental assumptions that underlie the scale domains derived here are a) that area-specific prey migration rates decrease with increasing arena size and b) that predator consumption rates per prey are scale independent. The concept of different spatial domains dominated by different processes (prey migration vs predator consumption) would be invalid if prey migration rates were scale independent or if consumption and movement rates were equally scale dependent. To test these assumptions we examined the scale dependence of consumption and emigration rates for experiments estimating the effects of stream predators on benthic prey. We validate our modeling approach by comparing observed predator impacts for different experiments and prey taxa with the predator impact predicted by the models. Predictions were generated using consumption and migration rates reported in these experiments.

Identifying the scale domain of real experiments A straightforward method to identify the processes dominating at different scales in predation experiments is to compare the magnitudes of predator consumption and prey migration rates. For example, if area-specific migration rates are much larger than consumption rates (m q) we can conclude that the experiment is a smallscale experiment as defined by our model. We use this rule of thumb to determine the scale domain of experiments in a large data set of published studies that examined the effects of stream predators on benthic prey (see below Data compilation; Large data set). A more detailed picture of the position of an experiment in relation to the scale domains predicted by our models can be obtained by parameterizing the appropriate equation (eq. 7 or 8) using reported consumption and migration rates, then simulating spatial and temporal scale dependence for individual experiments and prey taxa. Such simulations also show the strength and the sign of any dependence of PI values on scale. This method is demonstrated using data from stream predation experiments that provided estimates of prey migration and predator consumption rates.

Data compilation We searched the literature for all studies published between 1980 and 1998 where open experiments were used to examine the influence of stream predators on benthic prey. Seven studies, including nine experiments and a total of 24 prey taxa, provided data that enabled us to calculate instantaneous rates of predator conOIKOS 92:3 (2001)

sumption per prey and area-specific prey emigration (Table 4). We included all prey taxa reported in these seven studies, except for Bechara et al. (1993) and Dahl (1998a, b). The latter studies reported data for a large number of taxa, many of which occurred in low numbers; consequently, we chose to analyze only the five most abundant taxa in each of these three studies. In Sih et al. (1992) and Englund (1999), the surrounding habitat lacked predators whereas the background habitat contained predators in the other five studies. Consumption rates were estimated with three different methods. In five studies, consumption rates were calculated from data on gut contents (Lancaster 1990, Bechara et al. 1993, Forrester 1994, Dahl 1998a, b). Converting gut content data to consumption rates requires a measure of gut passage time. For Forrester (1994) and Dahl (1998a, b) the calculations were based on experimentally estimated gut evacuation rates. Such data were not available for Lancaster (1990) and Bechara et al. (1993). In these two studies, we assumed that 50% of 24-h consumption of prey by predators was detected in predator stomach contents. To examine how sensitive our conclusions were to this assumption we also ran tests assuming that this proportion was 25% or 100%. For one study, predator consumption rates were estimated from the survival of marked prey in the presence and absence of predators (Sih et al. 1992). Finally, Englund (1999) used budget calculations, i.e., decreases in prey density that could not be attributed to net migration, or to background mortality estimated from control enclosures, were regarded as losses to predator consumption. Area-specific migration rates were estimated as per capita emigration rates, i.e., the number of prey emigrating per time unit divided by prey numbers in the experimental unit. Data on the numbers emigrating were obtained from drift samples taken at the end of the experiments when benthos also was sampled. Exceptions were Lancaster (1990) and Englund (1999) where daily drift samples and frequent benthic samples were taken. In these two studies, we used the arithmetic means of per capita emigration rates across dates in calculations. Bechara et al. (1993) only sampled drift during six hours around dusk. Drift of the two mayfly genera (Baetis and Ephemerella) common in their experiments often show a sharp peak at dusk, although the proportion of daily drift that occurs during the peak is highly variable between species and between seasons for the same species (Waters 1962, Bishop 1969, Elliott 1969). Chironomid drift may also peak at dusk (Ali and Mulla 1979, Stoneburner and Smock 1979), but sometimes shows no periodicity (Brusven 1970). Thus, lacking more precise information we assumed that 50% of 24-h drift occurred during the sampled period. To examine the robustness of patterns to this assumption we also ran statistical tests assuming that the proportion was 25% (no periodicity) or 100%. 505

506

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1.5

1.5

6

7 8 15 35

Dahl 1998a

Dahl 1998b

Bechara et al. 1993

Englund 1999

Sih et al. 1992

Lancaster 1990

Forrester 19944 50

253

322

6

281

32

31

Duration (d)

Green sunfish (Lepomis cyanellus) Green sunfish (Lepomis cyanellus) Stonefly larvae (Doroneuria baumanni ) Brook char (Sal6elinus fontinalis)

(Sal6elinus fontinalis)

Brook char

Leech (Erbopdella octoculata)

Trout (Salmo trutta)

Bullhead (Cottus gobio)

Trout (Salmo trutta)

Predator taxon

0.337 0.015

Tanypodiinae Baetis

0.008 0.002 0.071

Paraleptophlebia 0.001 Stenonema 0.003 Eurylophella 0.008

0.070 0.004

0.056

0.004 0.017

0.038 0.007

0.570

0.023 0.190 0.035 0.030

0.189

0.400 0.778

0.400 0.040 2.00

0.020 0.040

0.020 1.25 0.800

0.800

Area-specific emigration rate in control cages

0.002 0.001 Baetis Ephemerella

Baetis Chironomidae

Ambystoma

0.002 0.074 0.005 0.170

0.000 0.003 0.070

Tanypodiinae Gammarus Ephemerella

Chironominae Ortocladiinae Tanypodiinae Cambarus

0.0004 0.012

Orthocladiinae Gammarus

0.003

0.002 0.000 0.419

Orthocladiinae Simuliidae Baetis

Ephemerella

0.105

Consumption rate

Baetis

Prey taxon

0.008 0.002 0.004

0.011 0.009

0.053 −0.001

0.042

−0.006 −0.080 0.010 0.021

−0.053

1.40 0.102

−0.100 −0.005 6.00

−0.002 0.140

0.010 0.200 3.40

1.40

Avoidance rate

0.22 −0.09 0.00

0.88 0.18

0.53 0.41

2.08

0.18 −0.12 0.06 0.88

−0.26

0.78 0.37

0.61 0.13 1.50

−0.30 0.88

−0.08 1.25 2.17

1.07

Observed PI

2

0.29 0.22 0.16

0.45 0.32

0.71 0.09

2.40

−0.05 −0.05 0.16 1.00

−0.29

3.21 0.14

−0.29 −0.03 1.42

−0.04 1.42

0.26 0.14 2.40

1.15

Predicted PI

Stomach data were only available for day 16. Density and drift data are from day 14–15. Density and drift data are for day seven because the estimated consumption rate supposedly reflected conditions early in the experiment (A. Sih pers. comm.). 3 The experiment lasted 25 d but we ignored the first 10 d when a flood apparently obscured any predator effects. 4 Data are for the treatment with medium predator density.

1

Arena length (m)

Study

Table 4. Consumption rates per prey, area-specific migration rates and predicted predator impacts in experiments examining effects of predators on stream benthos. Consumption and migration rates are instantaneous rates per day (unit =d−1). Avoidance rate is the difference between the area-specific emigration rates in predator and control treatments. A negative avoidance rate indicates prey freezing behavior, whereas a positive value indicates increased prey emigration in response to predators. A positive PI value indicates that prey densities were higher in control than predator treatments. Predicted predator impacts for local manipulations were calculated using eq. 7, except for Sih et al. (1992) and Englund (1999) where the background did not have predators and eq. 8 was used.

Large data set The size distribution of arenas used in published stream predation experiments was obtained from the COMPLETE data set used in Englund et al. (1999a), which is available in electronic form at Ecological Archives (Englund et al. 1999b). This data set includes 41 studies published between 1980 and 1996 that examined the effects of lotic fish or invertebrate predators on benthic invertebrates where at least one treatment included predators and another lacked predators (control). A detailed description of the dataset and a discussion of the criteria used to select data can be found in Englund et al. (1999a). Eight studies, including nine experiments, provided PI values for multiple sampling dates. For each study we recorded density data for two categories of prey, i.e., chironomids (order Diptera, family Chironomidae) and epibenthic prey. The latter category primarily consisted of mayflies (order Ephemeroptera) and stoneflies (order Plecoptera), taxa that dominate the epibenthic fauna of most stony streams and that are very common components of the diets of both fish and invertebrate predators. Chironomids were examined separately because they are small and cryptic, and often numerically dominant.

Statistical analyses The estimates of emigration rates in Bechara et al. (1993) and the estimates of consumption rates for this study and Lancaster (1990) were particularly uncertain. To examine the robustness of our conclusions to variation in these estimates we performed tests when these rates were 50%, 100%, or 200% of the value we regarded as most plausible. This procedure was used for the following correlations: observed PI vs predicted PI, arena length vs emigration rate, and arena length vs consumption rates. For these tests we present the correlation coefficients as ranges. The non-independence of observations is a potential problem in our data set because the responses of several taxa were estimated from the same experiment. Aggregating data, e.g., using the average response for all taxa in an experiment, can decrease the inter-dependence of observations, but carries a cost as information about differences among taxa are lost. There are no formal criteria for judging whether or not the non-independence of data is a serious problem and for some tests we found it hard to judge the appropriate units for analyses. This was the case for correlations between arena length and migration and consumption rates, so we used both aggregated data (experiments as replicates) and non-aggregated data (taxa as replicates) to test for the effect of non-independence of observations on our results. OIKOS 92:3 (2001)

A different type of non-independence occurred when testing for relationships between observed and predicted predator impacts. The same abundance data were used to estimate both observed PIs and per capita consumption and movement rates used to calculate predicted PIs. For example, observed PI is given by PI= nc /np and per capita consumption rate by q = prey consumption by predators per unit area and time/np. This means that we should expect observed and predicted PIs to be positively correlated even if the predictions are based on random emigration and consumption data. Thus, the appropriate null hypothesis is not that the observed correlation is zero, but that it is equal to some value larger than zero. To find this null expectation and p-values for observed correlations we sorted the original data on numbers eaten and emigrating in control and predator treatments in random order, and calculated predicted PIs and Spearman’s rank correlations between predicted and observed PIs. This procedure was repeated 1000 times to generate a frequency distribution of correlation coefficients. The mean of this distribution, and thus the expected correlation under the null hypothesis, was 0.26– 0.28. The p-value for an observed correlation coefficient was calculated as the proportion of the randomly generated correlation coefficients that were larger than the observed correlation coefficient. Separate tests were run when examining the robustness of this correlation to variation in emigration and consumption rates. In this test we excluded data for Englund (1999) because the estimate of predator consumption rate was based on the difference between prey densities in control and predator treatments.

Results Test of assumptions and validation of the model To examine the validity of the models and the data, we predicted PI for each taxon in Table 4 using eq. 7 or 8. Observed and predicted PIs were significantly correlated (Fig. 3, rs = 0.61– 0.70, N =23, p= 0.003– 0.021, randomization test), and a linear regression yielded a slope indistinguishable from unity and an intercept close to zero (slope 9SE =1.009 0.19, intercept = 0.0799 0.17). This suggests that the measurements of predator consumption and prey movements were accurate and that the assumptions of the models were reasonable (Fig. 3). Area-specific emigration rates in control arenas (mc ) and the lengths of experimental arenas were negatively correlated (rs = − 0.88– − 0.89, N =7, pB0.01) (Fig. 4), and a linear regression of ln(mc ) on ln(arena length) produced a slope of − 1.23 (SE =0.26), comparable to the slope of − 1 predicted from eq. 9. Using individual taxa as replicates produced the same conclusion (rs = − 0.64– 0.67, N= 20, pB 507

Fig. 3. Observed versus predicted predator impacts for the taxa listed in Table 4. Predictions were calculated using eqs 7 and 8, the duration of each experiment and observed migration and consumption rates.

0.01, slope of regression = − 1.13, SE =0.28). Consumption rates were not significantly related to the length of experimental arenas (experiments as replicates: rs = −0.01–0.20, N= 9, NS; taxa as replicates: rs = −0.05–0.09, N=24, NS) (Fig. 4).

Simulation of spatial and temporal scale dependence Simulations of spatial and temporal scale dependence in PI illustrate that the classification of a predation experiment as small- or large-scale (movement or consumption controlled) depends both on the size of the arena and on the species-specific relationship between emigration and consumption rate. For example, the studies by Dahl (1998a, b), which used the smallest arenas (length 1.5 m) included in this analysis, involved some taxa for which migration rates were much higher than consumption rates, such as Ephemerella and Baetis (trout) in

Fig. 4. Relationship between arena length and area-specific migration and consumption rate for the experiments in Table 4. Migration and consumption rates are mean values for taxa analyzed in each experiment.

508

Fig. 5b. For these taxa predicted PIs at the scale of Dahl’s experiments are approximately equal to PIs predicted for smaller scales (Fig. 5a, b) and PI is predicted to reach an equilibrium within a few days (Fig. 6a, b). Thus, for these taxa the experiments should be classified as small-scale experiments with movement control. For other taxa with lower emigration rates, such as Orthocladiinae and Gammarus in Fig. 5a, the experiment was performed at the scale where the strongest scale dependence is predicted to occur and the predicted time to equilibrium is much longer ( \ 60 d). The experiment may thus be classified as intermediate-scale (mixed control) for these species. The experiments in Dahl (1998a, b) could not be classified as consumption controlled for any of the analyzed taxa. In contrast, the largest experiments examined here, Lancaster (1990) and Forrester (1994), (15 and 35 m, respectively), should be classified as large-scale, consumption-controlled experiments for most taxa. In these studies, predicted PIs remained constant at the experimental and larger scales (Fig. 5e, f) and the times to equilibria were predicted to be long (100 d or more) except for Eurylophella (Fig. 6e, f). Figure 5 also illustrates that the predicted shape and sign of the relationship between PI and experimental scale are highly variable among prey taxa. This is because the scale dependence in PI is a function of both predator consumption and prey movement rates and these rates vary substantially among prey taxa (Table 4). For example, an increase from negative to high positive PIs with increasing scale is predicted if prey have a freezing response (mc \ mp ) and experience significant predation (e.g., Orthocladiinae, Fig. 5c) and an increase from negative to small positive or zero PI is expected if prey with a freezing response experience little predation (e.g., Tanypodiinae, Fig. 5a and Ephemerella, Fig. 5c). Several of the mayfly taxa in Fig. 5f show the opposite pattern; large positive PIs are predicted for small-scale experiments because prey disperse out of predator arenas (mp \ mc ), whereas at larger scales very small PIs are expected because consumption rates are low. Little or no scale dependence in predator impacts are expected if the predator impacts caused by predator avoidance responses at small scales are approximately equal to the predator impacts owing to direct predator consumption at larger scales (e.g., Baetis, Fig. 5c, e). The sign of the relationship between spatial scale and PI can also be influenced by the duration of the experiment and, thus, by the absolute values of consumption and migration rates. For example, contrasting scale relationships are predicted for Baetis in Forrester (1994) and Baetis(bullhead) in Dahl (1998a), even though the relationships between consumption and migration rates were similar (Table 4). However, the absolute values of the rates are more than two orders of OIKOS 92:3 (2001)

magnitude larger in Dahl (1998a) and these have a strong positive effect on the PI predicted at large scales (given by qt) but no effect on the PI predicted for small scales because the small-scale value is given by the ratio between two rates, i.e., PI = ln(mp /mc ). The absolute values of the consumption and emigration rates also have some influence on the scale domain of an experiment; the predicted PI for Baetis(bullhead) levels off at an arena length of 0.1 m (Fig. 5b) whereas PI for Baetis in Forrester (1994) is strongly scale dependent at this same arena length (Fig. 5f).

Identifying the dominating scale domain of the experiments in the large data set The experiments listed in Table 4 were performed in arenas that were large (length 1.5– 35 m) compared to those used in the experiments in the large data set (73% in the range 0.1– 1 m). Inspection of Fig. 4 suggests that, in general, migration rates are much larger than consumption rates for arenas B1.5 m. This suggests that most stream experiments should be classified as belonging to the small scale domain with movement control for typical prey taxa. In small-scale experiments it is expected that PI quickly equilibrates (Table 3, Fig. 2a). To test this prediction we calculated the rate of change in PI for the period from the start of the experiment to the first sampling event and for the period from first sampling to the end of the experiment for the nine experiments reporting data from more than one time. We found that the rate of change of PI was significantly larger than zero for the initial period (median 14 d), but close to zero for the rest of the time (14– 28 d) for both epibenthic and chironomid prey (Fig. 7). The median length of these experiments was 0.5 m.

Discussion

Fig. 5. Predicted relationships between spatial scale and observed predator impacts for the taxa listed in Table 4. Predictions were calculated using eqs 7–9 and parameter values given in Table 4. Arrows denote the length of the experimental unit used in each experiment. In b) the large-scale asymptotes were PI= 10.8 for Tanypodiinae and PI = 13.0 for Baetis. OIKOS 92:3 (2001)

The conceptual model proposed by Wiens (1989) is based on the idea that the scale dependence of ecological responses is non-monotonic. Within some scale domains there is weak or monotonic scale dependence and between such domains there are zones of transition where scale dependence is strong or rapidly changing and where there is a switch in the controlling processes. Our mathematical model of the scale dependence of predator impact on prey densities proved to be a nice illustration of this general idea. Using the model we could identify two scale domains with little scale dependence and a transition zone where scale dependence can be very strong and where there is a switch in the control of prey densities from prey movements to predator consumption. 509

If ecological responses show this type of nonmonotonic scale dependence it should be useful to identify scale domains and transition zones. For exam-

Fig. 7. Mean 995% confidence intervals for the rate of change in predator impact (PI/d) for nine experiments where predator impact was measured at more than one time. The rate of change in PI was calculated for the period from the start of the experiment to the first sampling event (first period), and for the period from the first sampling event to the end of the experiment (last period). Data were derived from Bechara et al. (1992, 1993), Dudgeon (1991, 1993), Gilliam et al. (1989), Peckarsky (1991), Lancaster (1990), and Lancaster et al. (1990).

Fig. 6. Predicted predator impacts over time for the taxa listed in Table 4. Predictions were calculated using eqs 7 and 8 and parameter values given in Table 4. Arrows denote the duration of each experiment.

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ple, one would probably like to perform experiments in the same scale domain as the natural system of interest, because the results of experiments performed in other domains or in transition zones may reflect other processes than those controlling the natural system. Thus, we wanted to examine if this general idea was applicable to real experiments. As a first step we used data from stream predation experiments to test the two key assumptions that underlie our model, i.e., that area-specific prey migration rates decrease with increasing arena size and that predator consumption rates are scale independent. The first assumption was supported by data culled from the literature; the relationship between area-specific prey migration rate (estimated by per capita emigration rate) and arena length was consistent with the assumption that migration rate scales to the perimeter to area ratio of arenas. This pattern emerged even though there was variation in factors, such as arena shape, current velocity, and the mobility of the organisms, that is likely to introduce additional variability. We expected to see clearer relationships for more homogeneous groups of taxa, and indeed we found a stronger correlation between emigration rate and arena length when restricting the analysis to mayflies (rs = − 0.92, N= 11, pB 0.001, taxa within experiments as replicates). Other studies also have found that per capita emigration rates decrease with spatial scale, although the slope of the relationship sometimes deviates from predictions based on perimeter to area ratios (Pokki 1981, Bach 1984, Kareiva 1985, Sutcliffe et al. 1997, Englund and Hamba¨ck unpubl.). The assumption is likely to be valid in any system where the number of individuals that cross the perimeter is proportional to the density in the source area (Englund 1997). An exception to this patOIKOS 92:3 (2001)

tern may be situations where the scale of movements is much larger than the scale of the arena. If every movement is much longer than the length of the arena, every movement will result in emigration regardless of arena size and, thus, the area-specific migration rate will be independent of arena size (see discussion in Englund 1997). The second assumption was also supported by our data. Consumption rate per prey was not significantly related to arena length; however, we caution that the power of this test was low. Because we examined the relationship between scale and predator consumption rates for different types of predators and prey, variation in consumption rates independent of spatial scale were also quite high. Analyzing chironomids and non-chironomid taxa separately did not produce any significant relationships; however, we found a significant correlation between scale and consumption rates of predators feeding on mayflies (rs = −0.88, N=11, pB0.005). A linear regression of log(consumption rate) on log(arena length) yielded a weak and statistically non-significant slope (slope= −0.51, N=11, p=0.14). Cooper et al. (1998) reported predator impacts for closed experiments (i.e., those not allowing prey migration) performed at different scales in a stream system. These data can be used to study the scale dependence of consumption rates under the assumption that predator impacts in these closed systems primarily reflected consumption by the manipulated predator (the experiments were short, 3–8 d). PI for Baetis decreased with arena length, but the effect was weak (slope = −0.22) compared to the effect of arena length on migration rate found for the taxa in Table 4 (slopes = −1.13 for all taxa, −1.54 for mayflies, and −1.50 for Baetis, taxa within experiments as replicates). Thus, some data suggest that consumption rates in stream experiments might decrease with spatial scale in some cases, but that the scale dependence of consumption rates may be weaker than the scale dependence of area-specific migration rates. A simulation showed that PI, rather than reaching an asymptote, decreased with increasing scale at large scales when weak scale dependence in consumption rates (slope = −0.35) was included in our model (Fig. 8). The effects of scale-dependent consumption rates are negligible at small scales where movements dominate and across all scales when overall consumption rates are low (Ephemerella in Fig. 8). This shows that a negative relationship between consumption rate and arena size has only minor influence on our scale-dependent results. Thus, our data on the scale dependence of consumption and migration rates support the idea that there is a change from movement control to consumption control as arena size is increased. From this conclusion follows that there is also a transition zone where both processes control predator impact. At the transition we should OIKOS 92:3 (2001)

expect strong scale dependence if the PI predicted from prey movements alone differs from the PI predicted if only predator consumption influences prey densities. Next we made an attempt to classify the analyzed experiments as belonging to one of the scale domains. Direct comparison of the magnitude of consumption and migration rates has an intuitive appeal and this method accurately indicates the controlling process when the difference between the two rates is very large. However, this method does not work as well when both processes influence PI, in part because it does not take into account that the influence of consumption increases with increasing duration of the experiment. Simulating the scale dependence of PI proved to be a better method. By comparing the PI predicted for the observed parameter values and the PIs predicted with movements only or consumption only it is possible to obtain a quantitative estimate of the relative importance of the two processes. Such an analyses showed that for approximately 50% of the analyzed taxa predator impact was controlled by both migration and consumption, i.e., they should be classified as belonging to the transition zone. Although the primary objective of the simulations of scale dependence was to demonstrate a method to determine the scale domain of an experiment, they also provided other important insights. They illustrated that the scale domain of an experiment depends both on arena size and the mobility of the prey. This is why we have chosen to define large and small scales in terms of migration rates rather than arena size. The simulations also showed that the type of scale dependence predicted is highly variable depending on taxon-specific consumption rate, prey movement responses to predators, and rates of prey movement independent of predators. This

Fig. 8. Predicted relationships between spatial scale and observed predator impacts for two of the prey taxa in Table 4. Solid lines denote the predicted PI when area-specific migration rate (m) is inversely proportional to arena length (l), i.e. m 8l − 1, and consumption rate (q) is scale independent. Broken lines indicate predicted PIs when the scale dependence of consumption rate is weaker than the scale dependence of migration rate, i.e. m 8l − 1 and q 8l − 0.35. Predictions were calculated with eq. 7 and the parameter values given in Table 4.

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means that it is not possible to predict the type of scale dependence we should expect unless consumption and movement rates are known. It also should be noted that predicted scale dependence was very strong for some taxa. Bechara et al. (1993) used six meter long channels and decreasing the length of channels one order of magnitude (to 0.6 m) is predicted to change the observed PI on Orthocladiinae from strongly positive to strongly negative. So far the models predictions about the effect of arena size on predator impact have not been tested. Thus, a logical next step would be to manipulate predator densities at several spatial scales and measure prey densities and consumption and migration rates. We also made an attempt to categorize the scale domain for studies in the COMPLETE data set, which included a majority of published experiments addressing predator effects on the stream benthos. Extrapolation using the observed scale dependence of consumption and migration rates suggested that the majority of these experiments should be classified as small-scale and movement controlled for most prey taxa. Obviously, the extrapolation outside of the observed scale range is highly uncertain and should be viewed with great caution. To our knowledge no study has reported prey migration and predation rates for small-scale stream experiments ( B1 m) that could be used to support or refute our results; however, the conclusion that prey movements control prey densities in small-scale stream experiments is supported by other evidences. Peckarsky and Dodson (1980) manipulated densities of predatory stonefly larvae in small cages (length = 0.3 m) and found that the most important mechanism affecting prey densities was predator avoidance in five out of six experiments. Studies where both predator densities and the size of the mesh used to enclose predators were manipulated in small arenas show that mesh size affects PI for a range of prey taxa (Cooper et al. 1990: arena length = 0.1 m, Lancaster et al. 1991: arena length = 0.3 m, Dahl 1998c: arena length =1.5 m). As mesh size is expected to influence prey migration rates but not consumption rates these results support the finding that movements rather than consumption determine predator impacts on many prey taxa in small experimental arenas. This result has important implications for the interpretation of this large body of experiments. First, it suggests that most stream predation experiments cannot be extrapolated directly to larger scales. Extrapolation to smaller scales may be possible but such extrapolation likely will be uninteresting in most cases. Second, it suggests that we should expect that most variation in PI among published studies is caused by differences in the movement responses of prey to predators rather than by variation in consumption rates. For example, recent meta-analyses of stream predation experiments showed that benthic-feeding fish suppress 512

densities of benthic prey more than do drift feeding fish (Dahl and Greenberg 1996, Englund et al. 1999a, Englund and Evander 1999). Although Dahl and Greenberg (1996) focused their discussion on differences in consumption rates between the two types of predators, our result suggests that contrasting movement responses by chironomids to the two types of predatory fish influence this pattern. Third, when synthesizing the results from different studies that estimate the magnitude of effects of biological interactions it is desirable to find a metric that is not influenced by the duration of experiments (Osenberg et al. 1997, 1999). In small-scale stream predation experiments we expect PI to reach equilibrium quickly and PI may thus serve as a time-independent index of predator impacts. Using the change in predator impact per time unit (PI/t) as suggested by Osenberg et al. (1997) may be more appropriate in large-scale experiments, where we, at least initially, expect PI to increase linearly over time (Fig. 2c).

Conclusions Field ecologists need theoretical tools to explain why and predict when experimental results are scale dependent. For ecological systems where the scale dependence of process rates or patterns is non-linear we believe that the concepts of scale domains and transition zones proposed by Wiens (1989) could be useful tools. When scale domains can be identified it may be possible to perform experiments in the same domain as the system of interest, thereby avoiding the problem of extrapolating between domains. Knowledge about scale domains can also influence the interpretation of already published experiments as illustrated by our analysis of stream predation experiments. The most important implication of this analysis is that published experiments may tell us very little about the population-level effects of predators on their prey. We chose to apply our model of open predation experiments to studies performed in streams, mainly because this field seems to be the only one where migration and consumption rates have been estimated in experiments. We see no reason why the model should not be applicable to other systems where open experiments are used, e.g. marine soft bottoms (Hall et al. 1990), lakes (Butler 1989), and various terrestrial systems (Holmes et al. 1979, Skinner and Whittaker 1981, Heads and Lawton 1984). Elsewhere we have reviewed empirical and theoretical models of scale dependence for a wider range of experimental systems, including closed systems and other types of ecological responses (Englund and Cooper unpubl.). These models suggest scale domains and transition zones may be found in many types of experiments other than those examined here. OIKOS 92:3 (2001)

Acknowledgements – Stefan Henriksson helped extract data. Valuable comments on the manuscript were provided by Sebastian Diehl, Jill Lancaster, Jon Moen, Barbara Peckarsky, Andrew Sih and Dave Wooster. Financial support was provided by Swedish Council for Forestry and Agricultural Research, Swedish Natural Research Council (STINT) (post doctoral grant), La¨ngmanska kulturfonden, Ax:son Johnsons stipendiefond, and Anna-Greta and Holger Crafoords fond to GE. OS was supported by NSF grant DEB96-29473, and a post-doctoral fellowship from the National Center for Ecological Analysis and Synthesis, a center funded by NSF (DEB 94-21535), the Univ. of California, Santa Barbara, and the State of California. This research also was partially supported by NSF grant DEB-9407591 to SDC.

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