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Journal of Animal Blackwell Publishing, Ltd. 2005 The strength of trophic cascades across : 74, 1029–1038 predictions from and energetics

JONATHAN B. SHURIN*† and ERIC W. SEABLOOM* *Department of Zoology and Centre for Research, University of British Columbia, 6270 University Blvd, Vancouver, BC V6T 1Z4, Canada; and †Department of Zoology, Oregon State University, Corvallis, OR 97331–2914, USA

Summary 1. Top-down control of trophic structure is often highly variable both within and among ecosystems. We explored the roles of relative body sizes of predators and prey, their metabolic types, the production-to- ratio (P : B) of , and system pro- ductivity in determining the strength of the indirect effects of predators on plants. 2.We used a well-studied model with three trophic levels that is parameterized based on allometric relationships for rates of ingestion and metabolic efficiency. The model predicts that invertebrate and ectotherm predators and should propagate cascades to a greater degree than vertebrates and endotherms because of their higher metabolic efficiency. 3. Increasing the -to- body-size ratio strengthened the effects of cascades, while predator body-size was predicted to have no effect. Increasing system or plant P : B magnified cascades. Because herbivore : plant body size ratios and plant P : B are both generally greater in aquatic than terrestrial systems (especially those with uni- cellular producers), the model predicts stronger cascades in than on land. This predic- tion is supported by a recent cross-system comparison of experiments. 4. We discuss features of natural systems that are not incorporated into the model and their implications for the intensity of trophic cascades across ecosystems. Key-words: allometry, cross-system comparison, indirect effects, metabolic rate, popu- lation dynamic equations, production, top-down control, trophic structure. Journal of Animal Ecology (2005) 74, 1029–1038 doi: 10.1111/j.1365-2656.2005.00999.x

magnitudes of their effects are often quite variable both Introduction within and among systems (Wootton et al. 1996; The importance of trophic cascades in regulating the Leibold et al. 1997; Persson 1999; Polis 1999; Shurin of organisms has long been a controversial et al. 2002). Several hypotheses have been proposed to topic in ecology (Hairston et al. 1960; Murdoch 1966; explain variability in the strength of cascades based on McQueen et al. 1986; Polis 1991; Power 1992; Strong factors such as life history, organism size and edibility, 1992; Carpenter & Kitchell 1993; Polis & Strong 1996; productivity, adaptive behaviour, nutrient recycling and Holt 2000). Evidence for cascades has been found in non-equilibrium dynamics (Leibold 1989; DeAngelis systems as disparate as grasslands, , , 1992; Abrams 1993; Chase 1996, 1998, 1999; Polis 1999; beds, forests and marine pelagia (Power 1990; McClaren Peacor & Werner 2001). However, the processes that & Peterson 1994; Brett & Goldman 1996, 1997; Estes regulate the expression of trophic cascades in natural et al. 1998; Micheli 1999; Pace et al. 1999; Post et al. systems, and their variability among systems, remain 1999; Schmitz et al. 2000; Halaj & Wise 2001). Although largely unknown. trophic cascades may be prevalent in nature, the Four features of food webs that may be important for determining the strength of cascades are (1) the relative body sizes of consumers and their resources, (2) the Correspondence: Jonathan B. Shurin, Department of Zool- metabolic types of organisms (e.g. ectotherms vs. endo- ogy and Centre for Biodiversity Research, University of Brit- therms), (3) the turnover time (the biomass-to-production © 2005 British ish Columbia, 6270 University Blvd, Vancouver, BC V6T ratio) of the species and (4) system productivity. For Ecological Society 1Z4, Canada. E-mail: [email protected] instance, in the pelagic zones of lakes, body-size ratios

1030 between planktivorous fish and are generally of plants and herbivores) influence the strength of J. B. Shurin & on the order of 106, while the ratio between zoop- top-down control. E. W. Seabloom lankton and is around 105 (Cohen et al. Our modelling approach is clearly a great simplifica- 2003). herbivores in grasslands are generally tion of natural food webs. We ignore many important of similar size to their predators, and around 104 larger aspects of real communities, such as omnivory, inter- than the plants they consume. The majority of predator : specific heterogeneity in edibility within trophic levels, prey body-size ratios fall within the range of 101−103 nutrient fluxes and recycling, population structure and (Cohen et al. 1993). Relative plant and herbivore body adaptive behaviour (Polis 1991; Strong 1992; Abrams sizes are also a major feature distinguishing terrestrial 1993; Polis & Strong 1996; Leibold et al. 1997). Although and aquatic (especially planktonic) food webs (Hairston these factors clearly influence the expression of cascades, & Hairston 1993). Many aquatic systems are dominated it is not obvious that they differ systematically among by unicellular producers (phytoplankton and periphy- (e.g. aquatic vs. terrestrial; Chase 2000). The ton) that are much smaller than their grazers, whereas model we use can be considered the most simplified herbivores on land range from much larger (e.g. ) representation of food chains that captures many to much smaller (e.g. arthropods) than the plants factors known to vary among ecosystems. The model they consume. Differences in relative body size may be therefore provides a more appropriate basis for cross- important for regulating the intensity of trophic cascades system comparisons of the intensity of cascades than a because body size is related closely to metabolic, growth null-model that there will be no differences. Deviations and demographic rates (Peters 1983). In addition, physio- between the model predictions and empirical data may logical rates such as feeding and respiration are related point to other critical features that distinguish different to an organism’s metabolic type, with endotherms, system types. vertebrate ectotherms and invertebrates showing distinct scaling relations (Peters 1983; Yodzis & Innes Models 1992). The production-to-biomass ratio (P : B) of plants is also related to the fraction of primary produc-  - tion consumed by herbivores, and is greatest in aquatic systems and lowest with woody plants (Cyr & Pace The model of a three-species food chain presented by 1993; Nielsen et al. 1996; Cebrian 1999). Finally, pro- McCann & Yodzis (1994) was based on Yodzis & Innes’s ductivity may play a role in determining the number (1992) model of predators and prey. It can be written as of trophic levels and the strength of top-down con- follows: trol (Oksanen et al. 1981; McQueen et al. 1986; Power 1992; Sarnelle 1992; Wootton & Power 1993; Chase dR  R  HR 3 =−R 1 333− xy et al. 2000). 3   HH −+δ dt  K  fRRHH()()1 3 o Here we use a well-studied approach to modelling food chain dynamics to examine the influence of relative dH  R  PH 3 = xH y 3 −1 − xy 33 HH3  +  PP −+δ body sizes, metabolic types, plant P : B and system pro- dt  RR3 o  fHHPP()()1 3 o ductivity on the strength of trophic cascades. Yodzis & Innes (1992) developed a model of parameterized dP  H  3 = xP y 3 −1 eqn 1 PP3  +  - dynamic equations using empirical dt HH3 o relations among body size, metabolic type and rates

of ingestion, metabolism and biomass production (see where R3, H3 and P3 are the biomass densities of the also McCann & Yodzis 1994; Huxel & McCann 1998; resources (plants), herbivores and predators, respec- McCann et al. 1998; Post et al. 2000). The form of tively, in the three system, and K is the the model is the same as Rosenzweig & MacArthur’s of the plant trophic level. The con- (1963) equations; however, the parameter values are version efficiencies of the predator and herbivore are constrained based on empirical relations between described by δ, the fraction of ingested energy lost to metabolic rates and body size. The model is therefore egestion and excretion, and f, the proportion of prey biologically plausible; however, it retains the generality biomass consumed. The form of the model is somewhat that comes from having relatively few parameters. In unusual in that these terms appear in the denominator addition, it can be used to make predictions about the of the loss term for the prey, rather than the growth behaviour of systems with different types of organisms term for the consumer; however, the same model could because it converts all fluxes into units of biomass easily be written in the more traditional way without

rather than population density. We use Yodzis & Innes’ changing any of its properties. The parameter xi is the model to ask how predator impacts on the abundance mass specific metabolic rate of species i measured of their prey, and their indirect effects on the prey’s relative to the production-to-biomass ratio of the © 2005 British resources, vary with plant P : B and the body sizes and plant (for full derivation, see Yodzis & Innes 1992). Ecological Society, Journal of Animal metabolic types of organisms. Our goal is to ask how Note that the metabolic rate determines rate at which Ecology, 74, two features that distinguish aquatic and terrestrial both assimilate and lose biomass. This 1029–1038 food webs (plant turnover time and relative body sizes value depends on the metabolic type (i.e. vertebrate

1031 endotherm, vertebrate ectotherm, invertebrate) and The resulting equations are: Modelling trophic organisms’ body-sizes a:   dR R333HR cascades 3 =−R 1  − xy −⋅025 3   HH + am dt K RR3 o = TH H xH −⋅ fam025 RR R dH  R  PH eqn 2 3 = 3 − − 33 −⋅ xHHH3  y 1 xyPP eqn 4 am025 dt  RR+  HH+ = TP P 3 o 3 o xP −⋅ fam025 RR R dP  H  3 = xP y 3 −1 PP3  +  dt HH3 o where aTi is the allometric scaling coefficient for species i’s respiration rate, f is the fraction of primary produc- R Notice that we have eliminated the parameters relating tivity consumed by the herbivore, a is the production- R to the consumers losses due to egestion, excretion or to-biomass ratio (P : B) of the resource and m is the i messy eating, so that the densities of the predator and body-size of species i. Many differences among food herbivore are now expressed relative to δ and f (eqn 3, chains are embodied in the parameter x. For instance, McCann & Yodzis 1995). This is equivalent to assum- x increases when the body-size ratio between the resource ing that consumers ingest all of their prey and do not and consumer becomes larger, when the consumer eats lose energy to digestion. This transformation does not a smaller fraction of the biomass of each resource unit, alter any of the conclusions that follow. when the consumer respires at a greater rate (higher The above model shows a wide array of dynamical a ) or with a lower P : B of the resource (Table 1). The T behaviour, including stable equilibria, limit cycles, per- parameter x can be intuited as the metabolic demands sistent chaos and ‘paradox of enrichment’ type extinctions of each relative to the rate of biomass (McCann & Yodzis 1994; Huxel & McCann 1998). The production of the plant. dynamic stability of the system depends strongly on the The parameter y measures the ingestion rate per i functional responses of the predator and herbivore (H unit metabolic rate for species i (termed the ‘ecological o and R ), and the carrying capacity of the basal resource scope’) and depends only on the metabolic type. o (K). Low values of H and R and high values of K Maximum observed values of y found by Yodzis & o o i to instability. Here we focus on the equilibrial Innes (1992) for endotherms, vertebrate ectotherms abundances of plants, herbivores and , and and invertebrates are shown in Table 1. These values the dynamic stability of the two and three trophic-level reflect maximum physiologically constrained inges- systems. We constrained the values of x and y to plausible tion rates, not the realized rates when ecological or i i values based on allometric scaling relations, metabolic behavioural limitations are considered. The values efficiencies and predator : prey body-size ratios. of y and x are related, because both are constrained i i By solving the third equation in model 1 for H , we find by the metabolic type of the organism (Table 1). H 3 o that the equilibrial density of the herbivore is: and Ro are the half-saturation constants for the Type II functional responses of the predator and the H H* = o eqn 5 herbivore. 3 − yP 1 By following the transformations in McCann & Yodzis (1995) we can reduce the number of parameters in the indicating that consumer density depends only on its above model. We performed the transformations: half saturation constant (H0) and the metabolic type of the predator, but not the metabolic rates of any of the

H species. The equilibrial densities of the plant (R3) and H → − δ predator (P ) are: fHH(1 ) 3 eqn 3 P RyKyRKA +− −± P → R* = oP Po eqn 6 −−δδ 3 − ffHP(11 H )( P ) 21(yP )

Table 1. Definitions of the model parameters, their units and relationship with metabolic efficiency (x). The range of values for invertebrates, vertebrate ectotherms, and endotherms are taken from Table 2 in Yodzis & Innes (1992)

Vertebrate Parameter Definition Units Invertebrates ectotherms Endotherms

× 0·25 aT Scaling coefficient for respiration (kg year) kg*kg 0·5 2·3 54·9 a Production : biomass ratio (P : B) (kg × year) kg kg0·25 9·2 6·6 34·3 © 2005 British R * f Fraction of resource biomass eaten None 0·2 0·2 0·2 Ecological Society, R x Mass-specific metabolic rate None 0·05–0·15 0·31–0·98 1·4–4·5 Journal of Animal i y Maximum ingestion rate (scaled to metabolic rate) None 19·4 3·9 1·6 Ecology, 74, max mR/mH Prey : predator body-size ratio None 0·1–0·001 0·1–0·001 0·1–0·001 1029–1038

1032 KR(())()+−−+++21 xH y yR R R yR A P* = oHoH Po oo Po J. B. Shurin & 3 − 21xKyPP() E. W. Seabloom eqn 7 where

A≡

−−−22 −+ − ()()yKRKyRxyHyRPoPoHHoPo141 ()( ) eqn 8

Note that xP, the metabolic rate of the predator, does

not affect R3* or H3* . For all the parameter value

combinations we explored, the solution for R3* where A was negative in the numerator of eqn 4 produced

negative values of R3* . Because these solutions are not reasonable (i.e. they imply negative resources), we focused only on the positive solutions.

 -

The equivalent to the three trophic-level model with only an herbivore (H) and basal resource (R) present is:

dR  R  HR 2 =− R 1 222 − xy 2   HH + dt  K  RR2 o Fig. 1. The effects of the herbivore (a) and predator’s (b) eqn 9 metabolic rates on equilibrial abundances of all species in the dH  R  two and three trophic-level food chains. All equilibria were 2 = xH y 2 − 1 dt HH2  RR +  stable for the parameter values shown. The dotted line is the 2 o predator, the solid lines are the plant, and the dashed lines are Here, the equilibrial densities of the two species are the grazer. The heavy lines are the two trophic level case, and given by: the thin lines are with three trophic levels present. The values = = = = of the other parameter values are Ro Ho K yP 5, and = R yH 3·9. R* = o eqn 10 2 − yH 1 level (P) on the resource (R). This can be measured Ry(KKR −− ) H* = oH o eqn 11 as the ratio of the biomass of the resource with two 2 − 2 xKyHH( 1) trophic levels present to that with three trophic levels (R = R*/ R*), while the impact of the predator on the indicating that the density of the resource depends only eff 2 3 herbivore (H ) is given by H* /H* . H and R there- on the functional response and feeding rate (but not the eff 2 3 eff eff fore measure the increase in herbivore density and metabolic efficiency) of the herbivore. The equilibrial decrease in plants with removal of the predator. These density of the grazer depends on all of the parameters measures of effect sizes were used because they are in the model. directly analogous to the metrics of response sizes used Figure 1 illustrates the effects of predator and herbivore in experiments (Shurin et al. 2002). In this system of metabolic rates (and therefore body size and plant P : B) equations, the predator effect sizes are: on the equilibrial densities in the two and three trophic- level systems. Increasing herbivore metabolic rates (high RK( +− R yKy)( − 1) =− ooHP Heff 2 eqn 12 xH) relative to plant P : B (for instance, due to small − xKHyHoH( 1) herbivore-to-plant size ratios) to more predators (P*), fewer plants (R* ) and equivalent grazer densities 2R 3 3 R = o eqn 13 ()H* in the three-level system (Fig. 1a). When the predator eff   3 −−+A (yKRHo 1)  is absent, increasing xH reduces grazer biomass (H2* )  −  (yP 1) and has no effect on plants (R2* ). Increasing the

predator’s metabolic rate (xP) reduces predator By varying the parameters, we can make predictions abundance but has no effect on either herbivores or about the relative magnitude of trophic cascades among plants (Fig. 1b). different types of systems. For the analyses that follow, we took the log of the effect sizes so that R and H © 2005 British e eff eff would be distributed symmetrically around zero. We Ecological Society,   Journal of Animal evaluated the equilibrial values of Heff and Reff and the Ecology, 74, The strength of the trophic cascade (which we call Reff) stability of the equilibria in the two and three trophic-level 1029–1038 is the magnitude of the indirect effect of the top trophic systems by using numerical derivatives (Press et al. 1988). 1033 effects on herbivores than vertebrates or endotherms.

Modelling trophic This can also be seen from the fact that yP appears in cascades the numerator of the expression for Heff (eqn 10). Heff

increases with yP but is unaffected by variation in xP (eqn 12, Fig. 1). The model also predicts that predators should have slightly larger effects on invertebrate herbivores than on vertebrate ectotherms and endotherms (compare panels a, b and c in Fig. 2). This indicates that predators are most effective at reducing the biomass of herbivores with high feeding rates and low metabolic rates. The

slight variation in Heff across herbivore types results

from the interdependence between yH and xH. Equation −1 − −2 12 shows that Heff varies as xH and yH(yH 1) ; however,

invertebrates with high feeding rates (large yH) also

have low metabolic rates (low xH, Table 1). The reduc-

tion in Heff as we move from endotherms to inverte-

brates indicates that the effect of xH predominates over

yH. However, the effect of herbivore metabolic type (yH)

on Heff is small relative to the effect of predator type

(yP) or herbivore metabolic rate (xH). Heff decreases

because grazer density declines with increasing xH in the absence of predators, but remains constant in the Fig. 2. The effect of predator removal on consumer biomass three-level system (Fig. 1a). The effect of predator

(Heff) as a function of the predator’s ingestion rate (yP) for removal therefore declines with increasing xH.

herbivore feeding rates (yH) corresponding to endotherms (a), The strength of cascades (Reff) increases as the pred- vertebrate ectotherms, (b) and invertebrates (c). The four lines ator’s feeding rate becomes greater relative to its mass- on each panel indicate different values of x representing H specific metabolic rate, suggesting that cascades should plant:herbivore body-size ratios ranging from 10−1–10−3 (high be most prevalent in systems with invertebrate and xH represents high plant : herbivore body size ratio). The symbols indicate the stability of the equilibria as follows: () ectotherm predators. Figure 3 shows Reff as a func- both two and three trophic-level systems are stable, and ( ) tion of yP, yH and xH. Low values of Reff indicate large only the two-level system is stable. Note that there were no increases in plant density with the addition of the pre- cases where the three-level system was stable while the two- dator (eqn 13). Predators with high feeding rates (y ) level system was not. Maximum feeding rates are 1·6 for P endotherms, 3·9 for vertebrate ectotherms and 19·4 for such as invertebrates and ectotherms, as well as herbiv- invertebrates. The half saturation constants for the herbivore ores with high feeding rates (yH) and low metabolic = = = = and predator are Ro Ho 2 (a) and Ro Ho 5 (b and c). rates (xH), are most effective at transmitting cascades The other parameter values were the same on all three panels to the plant trophic level. Any factor that reduces x , or = = = H (K 5, fr 0·2 and xP 0·1). increases yH or yP should therefore promote cascades. All the conditions that generate large predator effects on herbivores (large H , Fig. 2) are therefore the same   eff as those for strong trophic cascades (large negative

The first question we ask is, ‘how does the strength of values of Reff, Fig. 3). However, the reason for the effects

the predator’s effects on herbivores and plants vary of xH on Reff are different from those on Heff. Plants

with metabolic type?’ Figure 2 shows the effects of the decline with xH only in the three-level system, while her-

predator’s ingestion rate (yP) on Heff for different her- bivores decline only in the two-level system (Fig. 1a).

bivore feeding rates (yH) and metabolic rates (xH). The The difference in density with and without predators is metabolic rates were chosen to be appropriate to the driven by plants in the three-level system and grazers in

type of herbivore (based on aT and ar, Table 1) and to the two-level system. span plant-to-herbivore body-size ratios ranging from 0·1 to 0·001. Making the plant larger than the herbiv- -  ore did not affect the qualitative results. The effect of

the predator on the grazer (Heff) increases as the pred- The model predicts that predator effects on herbivores ator’s ingestion rate becomes greater relative to its increase when the herbivore has a low metabolic rate

metabolic rate (yP, Fig. 1). This result makes intuitive relative to the production-to-biomass ratio of the plant sense, as predators that feed at high rates and have low (low x , Fig. 2). The effect of the predator on herbivore © 2005 British H metabolic demands are likely to sustain high popula- biomass (H ) varies as 1/x (eqn 12). Because the con- Ecological Society, eff H 1 Journal of Animal tion densities and have a greater impact on their prey sumer’s metabolic rate declines as the - /4 power of its Ecology, 74, (Oksanen & Oksanen 2000). Invertebrate and ecto- body size (eqn 2), a larger size ratio between the her- 1029–1038 therm predators are therefore expected to have greater bivore and plant leads to larger effects of the predator 1034 growth rate (high P : B). The carrying capacity of the J. B. Shurin & environment is represented by K in eqns 4 and 9.

E. W. Seabloom Increasing K leads to increases in P3, R3 and H2 (the classical case of alternating trophic level responses to enrichment, eqns 4, 5 and 9; see Oksanen et al. 1981; Chase et al. 2000). The differences in plant and her- bivore biomass between the two and three-trophic-level

models therefore increase with K (larger Heff and

smaller Reff).



The stability of the equilibria is affected by herbivore and predator metabolic and feeding rates, plant turnover, carrying capacity and the functional responses. For the parameter values shown in Fig. 1, all equilibria with endotherm herbivores were stable in both the two and three trophic-level systems. High herbivore feeding rates

and low metabolic rates (high yH and low xH) destabilize the three-species food chain (Figs 2, 3). Regardless of the

stability of the equilibria, Heff was predicted to increase

with increasing yH and yP, and decline with increasing

xH and xP. We performed numerical simulations to explore the effects of the parameters on system dynamics with unstable equilibria. Figure 4 shows the effects of Fig. 3. The effect of predator removal on plant biomass (R ) eff xH and yH on plant density in the two- and three-species as a function of the predator’s ingestion rate (y ) for different P food chains. Although the predator induces limit cycles, herbivore feeding rates (y , the three panels) and metabolic H the response of mean plant density is similar to that rates (xH). Notation is as in Fig. 2. Large negative values of Reff indicate large increases in plant biomass in the presence of the when the equilibria are stable. Increasing yH leads to

predator. The four lines on each panel indicate different values stronger cascades by decreasing R2* more than R3* , and of x representing plant : herbivore body-size ratios ranging H causes higher amplitude fluctuations in R3* (Fig. 4a). −1− −3 from 10 10 . The symbols indicate the stability of the Decreasing x leads to stronger cascades through higher equilibria as follows: () both two and three trophic-level H R* without affecting R* , and also causes instability systems are stable, and () only the two-level system is stable. 3 2 We found no cases where the three-level system was stable in the three-level case (Fig. 4b). The equilibrium solutions while the two-level system was not. shown in Figs 1–3 may not always reflect the mean densities of the organisms in unstable situations where species show limit cycles or chaos. However, the unstable

on the herbivore via lower xH. Large herbivore : plant cases we examined by simulation showed similar body-size ratios also lead to more negative values of responses to changing parameter values to the stable

Reff (Fig. 3), indicating that trophic cascades should solutions (Fig. 4). Our analyses of stable equilibria may be most pronounced when herbivores are larger than therefore have relevance for many cases when popula- the plants they consume. Interestingly, the size of the tions show intrinsic fluctuations.

predator (expressed as xP, which depends on the body size ratio between the predator and the plant) does Discussion not appear in Heff or Reff, indicating that the strength of predator effects on herbivores and plants does not depend The relationships among organismal size, metabolic on the size of the predator. type and rates of growth, ingestion and assimilation provide insights into the control of trophic structure in   :     different ecosystems. The parameterized food chain model predicts that trophic cascades should be most As with plant and herbivore body size, the P : B of the pronounced in food chains with invertebrate predators

plant appears in the expression for xH. Heff varies as 1/ and herbivores, and with ectotherms rather than endo-

xH, while xH enters into the equation for Reff through A therms. Small plant : herbivore body size ratios are

(eqn 8). Increasing plant P : B decreases xH and xP (i.e. predicted to promote cascades, as should plants with plant P : B appears in the denominator of x and x ) high production-to-biomass ratios (short turnover © 2005 British H P and thereby leads to stronger predator effects on both times). Because plant growth rates ( Cyr & Pace 1993; Ecological Society, Journal of Animal herbivores (large Heff, Fig. 2) and plants (larger nega- Nielsen et al. 1996; Cebrian 1999) and herbivore : plant Ecology, 74, tive values of Reff, Fig. 3). Strong cascades are therefore size ratios (Cohen et al. 1993, 2003) are greater in many 1029–1038 expected when the plant has a high mass-specific aquatic ecosystems than terrestrial habitats, the model 1035 concerning metabolic type may not be realized. However, Modelling trophic if physiological limitations on feeding and metabolism cascades play a major role in nature, then the model may accurately reflect differences in the strength of top-down forces. A meta-analysis of 114 trophic cascade experiments found that, across ecosystems, systems with invertebrate herbivores showed the strongest cascades (Borer et al. 2005). This result supports the predictions of our model. However, endothermic herbivores were also found to promote a strong top-down effect. Our model predicts

that the predator metabolic rate (xP) should have no effect on cascades (eqn 13). Thus, empirical evidence supports the model’s predictions regarding herbivore’s metabolic type but not the predator’s. The model also predicts that cascades should be strongest when the production-to-biomass ratio of

plants (ar) is greatest. Highly productive plants provide greater nutritional advantage to herbivores through a greater flux of energy and materials per unit standing stock. Herbivores are therefore able to reduce the biomass of productive plants to a greater extent while maintaining their own metabolic needs. The data of Halaj & Wise (2001) support this prediction. Halaj & Wise (2001) found the strongest cascades in agricul- tural systems and the weakest in woodlands and grass- lands, although the effects of predators on herbivores were similar among the three systems (Table 3 in Halaj & Wise 2001). In terrestrial systems, P : B is greatest in Fig. 4. The effects of predator removal on plant density with agricultural crops, lowest in woody plants and inter- unstable equilibria. The straight lines are stable equilibria in two-species food chains. (a) shows the effects of the herbivore’s mediate in grasslands (Cebrian 1999); however, differ- = = feeding rate, yH (black lines are yH 3·9, grey lines are yH 3·0). ences in plant- may also have played a

(b) shows the effects of the herbivore’s metabolic rate, xH (black role in generating these patterns. Plants in agricultural = = lines are xC 0·3, grey lines are xC 0·5. Other parameter fields may be more uniformly palatable to herbivores values are as in Fig. 2b,c (x = 0·3 in a, and y = 3·9 in b). C C than those in grasslands or forest. The results of Halaj & Wise (2001) support the model prediction of stronger provides two reasons for expecting stronger cascades cascades in systems where plants have greater P : B in water. A review of 102 trophic cascade experiments ratios. found that plants responded more strongly to predator Plant turnover time or P : B is also a major feature removal in aquatic systems (Shurin et al. 2002). The distinguishing aquatic and terrestrial ecosystems. model offers two potential explanations for this result Terrestrial plants allocate more of their biomass to based on factors known to distinguish different eco- structural and transport tissues (stems and trunks) and systems. We discuss each of these conclusions in detail less to photosynthetic tissues than aquatic producers. below, as well as the available empirical evidence for Phytoplankton, periphyton, macro- and macro- evaluating them. phytes acquire nutrients directly from the surrounding Ectothermic and invertebrate predators and her- water and are supported by their buoyancy in a fluid bivores should be more effective in transmitting trophic medium. Land plants, therefore, have consistently lower cascades to plants than endotherms or vertebrates growth rates than their aquatic counterparts (P : B ratios,

because of their relatively high ingestion rates (high yi) Nielsen et al. 1996; Cebrian 1999) because of greater

and low metabolic rates (low xi due to low mass-specific allocation to structure and transport at the expense of

respiration rate, aT, Table 1). Invertebrates and ectotherms growth (i.e. photosynthetic) tissues. The order of increas- are able to consume more prey and maintain greater ing turnover time (decreasing P : B) among systems biomass densities with lower metabolic expenditures reported by Cebrian (1999) is phytoplankton > benthic (Oksanen & Oksanen 2000). This prediction is based microalgae > macroalgae > > freshwater on the physiologically constrained maximum ingestion macrophytes > marshes > grasslands > > rate, and does not take into account ecological limita- forests and shrublands. Our model indicates that high © 2005 British tions on feeding such as predator avoidance or P : B should promote strong cascading effects of predators Ecological Society, Journal of Animal structure. If vertebrates and invertebrates are affected (Fig. 3). P : B appears in the model through xH and xP, Ecology, 74, differentially by ecological constraints (e.g. gape limita- the metabolic rates of the herbivore and predator

1029–1038 tion, thermal tolerance), then the model predictions relative to P : B for the plant. Increasing P : B decreases xH 1036 and thereby leads to greater cascading effects of pred- provide a firm basis for expecting more pronounced J. B. Shurin & ators (Fig. 3). Because aquatic primary producers cascades in aquatic systems. Differences in trophic E. W. Seabloom (both unicellular and multicellular) have higher P : B complexity between environments remain to be shown. ratios than terrestrial plants, the model predicts that In addition to their effects on the equilibrium abun- cascades should be most pronounced in aquatic systems. dances of plants and herbivores, predators influenced High producer growth rates may explain empirical the stability of the plant–herbivore interaction. Stable evidence for stronger trophic cascades in aquatic equilibria were possible for both the two and three ecosystems (Halaj & Wise 2001; Shurin et al. 2002). trophic-level models for each type of herbivore (Fig. 2). Enriching the system by increasing the plant carrying We found regions in which the three-level model was capacity (K) has similar effects of increasing top-down less stable than the two-level model; however, the inverse control, a prediction that is supported by several aquatic was never true. Thus we predict that predator removal examples (Sarnelle 1992; Wootton & Power 1993). will tend to stabilize food chains. This prediction is sup- The second important difference between aquatic ported by a meta-analysis (Halpern et al. 2005) that and terrestrial systems lies in the relative body sizes showed that predator removal stabilized herbivore of plants and herbivores. Aquatic ecosystems are often populations in 40 experiments in six different ecosys- dominated by unicellular algae that are many orders of tems. The stability of the system depended on produc-

magnitude smaller than their herbivores (Cohen et al. tivity (K), the functional responses (Ro and Ho), feeding 2003). Many terrestrial plants, by contrast, are large and metabolic rates of the herbivore and predator

relative to their consumers (Hairston & Hairston 1993). (xi and yi). The unstable conditions we encountered Although large herbivores and small plants occur in consisted of limit cycles rather than chaotic dynamics. many terrestrial systems (e.g. grasslands with grazing Regardless of dynamic stability, the equilibria showed ungulates), these same systems also contain many consistent patterns with respect to changes in the para- arthropod herbivores that are much smaller than plants. meter values (Figs 2 and 3). Our exploration of unstable In addition, the largest terrestrial herbivores are all equilibria through simulation indicated that average endotherms (mainly mammals) with high metabolic abundances over time showed similar patterns to those

rates that are predicted to have weak effects on plants of the stable equilibria. For instance, increasing yH or

(Fig. 3). Increasing the size ratio between herbivores decreasing xH increased the strength of cascades in cases

and plants decreases xH (eqn 2) and thereby increases with limit cycles, as well as affecting the amplitude of the impact of predator removal on the basal trophic the fluctuations (Fig. 4). Our results from analyses of level (Fig. 3). Thus, the model predicts that food chains stable equilibria may therefore have relevance for other with unicellular producers should show stronger cascades situations where species densities fluctuate over time. than those containing multicellular plants. Because Although much pertinent information about organ- unicellular dominate many aquatic environ- isms relating to trophic structure is incorporated by ments but are largely absent on land, the model predic- metabolic type, body size, P : B and productivity, the tions relating to plant and herbivore body size also model ignores many important aspects of the structure suggest stronger cascades in aquatic ecosystems. of real food webs. First, the model represents ‘species The model’s predictions agree well with meta-analyses cascades’ (sensu Polis et al. 2000) rather than ‘ of the published literature on trophic cascade experi- cascades’, where many heterogeneous species comprise ments showing stronger effects of predator removal on a trophic level. However, all community cascades must plant biomass in aquatic systems than terrestrial (Halaj at some level consist of one or more species cascades. & Wise 2001; Shurin et al. 2002; Borer et al. 2005). The Interspecific heterogeneity within trophic levels can model provides a mechanistic explanation for stronger substantially alter trophic structure and the strength of cascades in pelagic food webs based on P : B ratios top-down control (Leibold 1989; Power 1992; Abrams of plants and body-size differences between plants and 1993; Wootton et al. 1996; Leibold et al. 1997). Inedible herbivores. Although the model predicts stronger species can attenuate the indirect effects of predators cascades in water than on land, the reasons are different over resources and limit the flow of energy from resources from those envisioned by Strong (1992). Strong argued to consumers. Inedibility could be incorporated in the

that phytoplankton are more uniformly palatable than model by adjusting fR, the fraction of the resource con- terrestrial plants, and that terrestrial food webs have sumed, the P : B ratio of the plant or by adding species more omnivory and trophic complexity. Phytoplankton with different traits at any of the trophic levels. These vary from highly edible to completely defended forms would all have the effect of decreasing the herbivore’s

(Leibold 1989; Tessier & Woodruff 2002). In addition, efficiency (xH, Table 1), and thereby dampen the cascade omnivory is common in aquatic, as well as terrestrial, (Fig. 3). Variation in species traits within trophic levels environments (Diehl 1993; Vander Zanden & Rasmussen or along environmental gradients are likely to be import- 1996). It is unclear whether edibility of plants or degree for regulating the strength of cascades in nature, © 2005 British of trophic complexity differ systematically between but are not incorporated into the model. A second Ecological Society, Journal of Animal aquatic and terrestrial food webs (Chase 2000). Well- potential limitation to the application of the model to Ecology, 74, documented aquatic–terrestrial contrasts in plant P : B natural systems is that yi (species i’s feeding rate scaled 1029–1038 and the relative body sizes of herbivores and plants to its metabolic rate) is the physiologically constrained 1037 maximum and not the realized rate. If ecological Borer, E.T., Seabloom, E.W., Shurin, J.B., Anderson, K., Modelling trophic constraints vary systematically among organisms of Blanchette, C.A., Broitman, B., Cooper, S.D. & Halpern, cascades different metabolic types, or between systems, then the B.S. (2005) What determines the strength of a trophic cas- cade? Ecology, 86, 528–537. realized ys may differ substantially from ymax. However, Brett, M.T. & Goldman, C.R. (1996) A meta-analysis of the if physiological constraints on y are large relative to freshwater trophic cascade. Proceedings of the National ecological limitations, then realized feeding rates may Academy of Sciences USA, 93, 7723–7726. Brett, M.T. & Goldman, C.R. (1997) Consumer versus be correlated with ymax. This model is clearly a simplified representation of resource control in freshwater pelagic food webs. Science, 275, 384–386. a in that it does not incorporate many com- Carpenter, S.R. & Kitchell, J.F. (1993) The Trophic Cascade in plexities such as behaviour, edibility, omnivory, stage Lakes. Cambridge University Press, Cambridge, UK. structure or detrital energy pathways that influence Cebrian, J. (1999) Patterns in the fate of production in plant the expression of trophic cascades (Polis 1999). How- communities. American Naturalist, 154, 449–468. ever, many fundamental differences among ecosystems Chase, J.M. (1996) Abiotic controls of trophic cascades in a simple grassland food chain. Oikos, 77, 495–506. occur in terms of the metabolic types and body-sizes Chase, J.M. 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Polis, G.A., Sears, A.L.W., Huxel, G.R., Strong, D.R. & Received 16 September 2004; accepted 18 April 2005

© 2005 British Ecological Society, Journal of Animal Ecology, 74, 1029–1038