Jonathan B. Shurin, Department of Zoology and Centre for Biodiversity Research, University of British Columbia, 6270 University Blvd, Vancouver, BC V6T 1Z4, Canada. E-mail: email@example.com
1Top-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-biomass ratio (P : B) of plants, and system productivity in determining the strength of the indirect effects of predators on plants.
2We used a well-studied food chain 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 herbivores should propagate cascades to a greater degree than vertebrates and endotherms because of their higher metabolic efficiency.
3Increasing the herbivore-to-plant body-size ratio strengthened the effects of cascades, while predator body-size was predicted to have no effect. Increasing system productivity 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 unicellular producers), the model predicts stronger cascades in water than on land. This prediction is supported by a recent cross-system comparison of trophic cascade experiments.
4We discuss features of natural systems that are not incorporated into the model and their implications for the intensity of trophic cascades across ecosystems.
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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 metabolic types of organisms (e.g. ectotherms vs. endotherms), (3) the turnover time (the biomass-to-production ratio) of the species and (4) system productivity. For instance, in the pelagic zones of lakes, body-size ratios between planktivorous fish and zooplankton are generally on the order of 106, while the ratio between zooplankton and phytoplankton is around 105 (Cohen et al. 2003). Ungulate herbivores in grasslands are generally of similar size to their predators, and around 104 larger than the plants they consume. The majority of predator : prey body-size ratios fall within the range of 101−103 (Cohen et al. 1993). Relative plant and herbivore body sizes are also a major feature distinguishing terrestrial and aquatic (especially planktonic) food webs (Hairston & Hairston 1993). Many aquatic systems are dominated by unicellular producers (phytoplankton and periphyton) that are much smaller than their grazers, whereas herbivores on land range from much larger (e.g. ungulates) to much smaller (e.g. arthropods) than the plants they consume. Differences in relative body size may be important for regulating the intensity of trophic cascades because body size is related closely to metabolic, growth and demographic rates (Peters 1983). In addition, physiological rates such as feeding and respiration are related to an organism's metabolic type, with endotherms, vertebrate ectotherms and invertebrates showing distinct scaling relations (Peters 1983; Yodzis & Innes 1992). The production-to-biomass ratio (P : B) of plants is also related to the fraction of primary production consumed by herbivores, and is greatest in aquatic systems and lowest with woody plants (Cyr & Pace 1993; Nielsen et al. 1996; Cebrian 1999). Finally, productivity may play a role in determining the number of trophic levels and the strength of top-down control (Oksanen et al. 1981; McQueen et al. 1986; Power 1992; Sarnelle 1992; Wootton & Power 1993; Chase et al. 2000).
Here we use a well-studied approach to modelling food chain dynamics to examine the influence of relative body sizes, metabolic types, plant P : B and system productivity on the strength of trophic cascades. Yodzis & Innes (1992) developed a model of parameterized consumer-resource dynamic equations using empirical relations among body size, metabolic type and rates of ingestion, metabolism and biomass production (see also McCann & Yodzis 1994; Huxel & McCann 1998; McCann et al. 1998; Post et al. 2000). The form of the model is the same as Rosenzweig & MacArthur's (1963) equations; however, the parameter values are constrained based on empirical relations between metabolic rates and body size. The model is therefore biologically plausible; however, it retains the generality that comes from having relatively few parameters. In addition, it can be used to make predictions about the behaviour of systems with different types of organisms because it converts all fluxes into units of biomass rather than population density. We use Yodzis & Innes’ model to ask how predator impacts on the abundance of their prey, and their indirect effects on the prey's resources, vary with plant P : B and the body sizes and metabolic types of organisms. Our goal is to ask how two features that distinguish aquatic and terrestrial food webs (plant turnover time and relative body sizes of plants and herbivores) influence the strength of top-down control.
Our modelling approach is clearly a great simplification of natural food webs. We ignore many important aspects of real communities, such as omnivory, interspecific heterogeneity in edibility within trophic levels, nutrient fluxes and recycling, population structure and adaptive behaviour (Polis 1991; Strong 1992; Abrams 1993; Polis & Strong 1996; Leibold et al. 1997). Although these factors clearly influence the expression of cascades, it is not obvious that they differ systematically among habitats (e.g. aquatic vs. terrestrial; Chase 2000). The model we use can be considered the most simplified representation of food chains that captures many factors known to vary among ecosystems. The model therefore provides a more appropriate basis for cross-system comparisons of the intensity of cascades than a null-model that there will be no differences. Deviations between the model predictions and empirical data may point to other critical features that distinguish different system types.
where R3, H3 and P3 are the biomass densities of the resources (plants), herbivores and predators, respectively, in the three trophic level system, and K is the carrying capacity of the plant trophic level. The conversion efficiencies of the predator and herbivore are described by δ, the fraction of ingested energy lost to egestion and excretion, and f, the proportion of prey biomass consumed. The form of the model is somewhat unusual in that these terms appear in the denominator of the loss term for the prey, rather than the growth term for the consumer; however, the same model could easily be written in the more traditional way without changing any of its properties. The parameter xi is the mass specific metabolic rate of species i measured relative to the production-to-biomass ratio of the plant (for full derivation, see Yodzis & Innes 1992). Note that the metabolic rate determines rate at which heterotrophs both assimilate and lose biomass. This value depends on the metabolic type (i.e. vertebrate endotherm, vertebrate ectotherm, invertebrate) and organisms’ body-sizes a:
where aTi is the allometric scaling coefficient for species i's respiration rate, fR is the fraction of primary productivity consumed by the herbivore, aR is the production-to-biomass ratio (P : B) of the resource and mi is the body-size of species i. Many differences among food chains are embodied in the parameter x. For instance, x increases when the body-size ratio between the resource and consumer becomes larger, when the consumer eats a smaller fraction of the biomass of each resource unit, when the consumer respires at a greater rate (higher aT) or with a lower P : B of the resource (Table 1). The parameter x can be intuited as the metabolic demands of each heterotroph relative to the rate of biomass production of the plant.
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)
Scaling coefficient for respiration (kg × year)
Production : biomass ratio (P : B) (kg × year)
Fraction of resource biomass eaten
Mass-specific metabolic rate
Maximum ingestion rate (scaled to metabolic rate)
Prey : predator body-size ratio
The parameter yi measures the ingestion rate per unit metabolic rate for species i (termed the ‘ecological scope’) and depends only on the metabolic type. Maximum observed values of yi found by Yodzis & Innes (1992) for endotherms, vertebrate ectotherms and invertebrates are shown in Table 1. These values reflect maximum physiologically constrained ingestion rates, not the realized rates when ecological or behavioural limitations are considered. The values of yi and xi are related, because both are constrained by the metabolic type of the organism (Table 1). Ho and Ro are the half-saturation constants for the Type II functional responses of the predator and the herbivore.
By following the transformations in McCann & Yodzis (1995) we can reduce the number of parameters in the above model. We performed the transformations:
The resulting equations are:
Notice that we have eliminated the parameters relating to the consumers losses due to egestion, excretion or messy eating, so that the densities of the predator and herbivore are now expressed relative to δ and f (eqn 3, McCann & Yodzis 1995). This is equivalent to assuming that consumers ingest all of their prey and do not lose energy to digestion. This transformation does not alter any of the conclusions that follow.
The above model shows a wide array of dynamical behaviour, including stable equilibria, limit cycles, persistent chaos and ‘paradox of enrichment’ type extinctions (McCann & Yodzis 1994; Huxel & McCann 1998). The dynamic stability of the system depends strongly on the functional responses of the predator and herbivore (Ho and Ro), and the carrying capacity of the basal resource (K). Low values of Ho and Ro and high values of K lead to instability. Here we focus on the equilibrial abundances of plants, herbivores and carnivores, and the dynamic stability of the two and three trophic-level systems. We constrained the values of xi and yi to plausible values based on allometric scaling relations, metabolic efficiencies and predator : prey body-size ratios.
By solving the third equation in model 1 for H3, we find that the equilibrial density of the herbivore is:
indicating that consumer density depends only on its half saturation constant (H0) and the metabolic type of the predator, but not the metabolic rates of any of the species. The equilibrial densities of the plant (R3) and predator (P3) are:
Note that xP, the metabolic rate of the predator, does not affect or . For all the parameter value combinations we explored, the solution for where A was negative in the numerator of eqn 4 produced negative values of . 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:
Here, the equilibrial densities of the two species are given by:
indicating that the density of the resource depends only on the functional response and feeding rate (but not the metabolic efficiency) of the herbivore. The equilibrial density of the grazer depends on all of the parameters in the model.
Figure 1 illustrates the effects of predator and herbivore metabolic rates (and therefore body size and plant P : B) on the equilibrial densities in the two and three trophic-level systems. Increasing herbivore metabolic rates (high xH) relative to plant P : B (for instance, due to small herbivore-to-plant size ratios) leads to more predators (), fewer plants () and equivalent grazer densities () in the three-level system (Fig. 1a). When the predator is absent, increasing xH reduces grazer biomass () and has no effect on plants (). Increasing the predator's metabolic rate (xP) reduces predator abundance but has no effect on either herbivores or plants (Fig. 1b).
The strength of the trophic cascade (which we call Reff) is the magnitude of the indirect effect of the top trophic level (P) on the resource (R). This can be measured as the ratio of the biomass of the resource with two trophic levels present to that with three trophic levels (Reff = /), while the impact of the predator on the herbivore (Heff) is given by /. Heff and Reff therefore measure the increase in herbivore density and decrease in plants with removal of the predator. These measures of effect sizes were used because they are directly analogous to the metrics of response sizes used in experiments (Shurin et al. 2002). In this system of equations, the predator effect sizes are:
By varying the parameters, we can make predictions about the relative magnitude of trophic cascades among different types of systems. For the analyses that follow, we took the loge of the effect sizes so that Reff and Heff would be distributed symmetrically around zero. We evaluated the equilibrial values of Heff and Reff and the stability of the equilibria in the two and three trophic-level systems by using numerical derivatives (Press et al. 1988).
The first question we ask is, ‘how does the strength of the predator's effects on herbivores and plants vary with metabolic type?’Figure 2 shows the effects of the predator's ingestion rate (yP) on Heff for different herbivore feeding rates (yH) and metabolic rates (xH). The metabolic rates were chosen to be appropriate to the type of herbivore (based on aT and ar, Table 1) and to span plant-to-herbivore body-size ratios ranging from 0·1 to 0·001. Making the plant larger than the herbivore did not affect the qualitative results. The effect of the predator on the grazer (Heff) increases as the predator's ingestion rate becomes greater relative to its metabolic rate (yP, Fig. 1). This result makes intuitive sense, as predators that feed at high rates and have low metabolic demands are likely to sustain high population densities and have a greater impact on their prey (Oksanen & Oksanen 2000). Invertebrate and ectotherm predators are therefore expected to have greater effects on herbivores than vertebrates or endotherms. This can also be seen from the fact that yP appears in 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 12 shows that Heff varies as and yH(yH−1)−2; however, invertebrates with high feeding rates (large yH) also have low metabolic rates (low xH, Table 1). The reduction in Heff as we move from endotherms to invertebrates 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 three-level system (Fig. 1a). The effect of predator removal therefore declines with increasing xH.
The strength of cascades (Reff) increases as the predator's feeding rate becomes greater relative to its mass-specific metabolic rate, suggesting that cascades should be most prevalent in systems with invertebrate and ectotherm predators. Figure 3 shows Reff as a function of yP, yH and xH. Low values of Reff indicate large increases in plant density with the addition of the predator (eqn 13). Predators with high feeding rates (yP) such as invertebrates and ectotherms, as well as herbivores with high feeding rates (yH) and low metabolic rates (xH), are most effective at transmitting cascades to the plant trophic level. Any factor that reduces xH, or increases yH or yP should therefore promote cascades. All the conditions that generate large predator effects on herbivores (large Heff, Fig. 2) are therefore the same as those for strong trophic cascades (large negative values of Reff, Fig. 3). However, the reason for the effects of xH on Reff are different from those on Heff. Plants decline with xH only in the three-level system, while herbivores decline only in the two-level system (Fig. 1a). The difference in density with and without predators is driven by plants in the three-level system and grazers in the two-level system.
The model predicts that predator effects on herbivores increase when the herbivore has a low metabolic rate relative to the production-to-biomass ratio of the plant (low xH, Fig. 2). The effect of the predator on herbivore biomass (Heff) varies as 1/xH (eqn 12). Because the consumer's metabolic rate declines as the -1/4 power of its body size (eqn 2), a larger size ratio between the herbivore and plant leads to larger effects of the predator on the herbivore via lower xH. Large herbivore : plant body-size ratios also lead to more negative values of Reff (Fig. 3), indicating that trophic cascades should be most pronounced when herbivores are larger than the plants they consume. Interestingly, the size of the predator (expressed as xP, which depends on the body size ratio between the predator and the plant) does not appear in Heff or Reff, indicating that the strength of predator effects on herbivores and plants does not depend on the size of the predator.
plant p : b and carrying capacity
As with plant and herbivore body size, the P : B of the plant appears in the expression for xH. Heff varies as 1/xH, while xH enters into the equation for Reff through A (eqn 8). Increasing plant P : B decreases xH and xP (i.e. plant P : B appears in the denominator of xH and xP) and thereby leads to stronger predator effects on both herbivores (large Heff, Fig. 2) and plants (larger negative values of Reff, Fig. 3). Strong cascades are therefore expected when the plant has a high mass-specific growth rate (high P : B). The carrying capacity of the environment is represented by K in eqns 4 and 9. 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 herbivore 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 xH and yH on plant density in the two- and three-species food chains. Although the predator induces limit cycles, the response of mean plant density is similar to that when the equilibria are stable. Increasing yH leads to stronger cascades by decreasing more than , and causes higher amplitude fluctuations in (Fig. 4a). Decreasing xH leads to stronger cascades through higher without affecting , and also causes instability in the three-level case (Fig. 4b). The equilibrium solutions 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 cases we examined by simulation showed similar responses to changing parameter values to the stable solutions (Fig. 4). Our analyses of stable equilibria may therefore have relevance for many cases when populations show intrinsic fluctuations.
The relationships among organismal size, metabolic 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 pronounced in food chains with invertebrate predators and herbivores, and with ectotherms rather than endotherms. Small plant : herbivore body size ratios are predicted to promote cascades, as should plants with high production-to-biomass ratios (short turnover times). Because plant growth rates ( Cyr & Pace 1993; Nielsen et al. 1996; Cebrian 1999) and herbivore : plant size ratios (Cohen et al. 1993, 2003) are greater in many aquatic ecosystems than terrestrial habitats, the model provides two reasons for expecting stronger cascades in water. A review of 102 trophic cascade experiments found that plants responded more strongly to predator removal in aquatic systems (Shurin et al. 2002). The model offers two potential explanations for this result based on factors known to distinguish different ecosystems. We discuss each of these conclusions in detail below, as well as the available empirical evidence for evaluating them.
Ectothermic and invertebrate predators and herbivores should be more effective in transmitting trophic cascades to plants than endotherms or vertebrates because of their relatively high ingestion rates (high yi) and low metabolic rates (low xi due to low mass-specific respiration rate, aT, Table 1). Invertebrates and ectotherms are able to consume more prey and maintain greater biomass densities with lower metabolic expenditures (Oksanen & Oksanen 2000). This prediction is based on the physiologically constrained maximum ingestion rate, and does not take into account ecological limitations on feeding such as predator avoidance or habitat structure. If vertebrates and invertebrates are affected differentially by ecological constraints (e.g. gape limitation, thermal tolerance), then the model predictions concerning metabolic type may not be realized. However, if physiological limitations on feeding and metabolism 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 agricultural systems and the weakest in woodlands and grasslands, 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 agricultural crops, lowest in woody plants and intermediate in grasslands (Cebrian 1999); however, differences in plant-species diversity may also have played a role in generating these patterns. Plants in agricultural fields may be more uniformly palatable to herbivores than those in grasslands or forest. The results of Halaj & Wise (2001) support the model prediction of stronger cascades in systems where plants have greater P : B ratios.
Plant turnover time or P : B is also a major feature distinguishing aquatic and terrestrial ecosystems. Terrestrial plants allocate more of their biomass to structural and transport tissues (stems and trunks) and less to photosynthetic tissues than aquatic producers. Phytoplankton, periphyton, macro-algae and macrophytes acquire nutrients directly from the surrounding water and are supported by their buoyancy in a fluid medium. Land plants, therefore, have consistently lower growth rates than their aquatic counterparts (P : B ratios, Nielsen et al. 1996; Cebrian 1999) because of greater allocation to structure and transport at the expense of growth (i.e. photosynthetic) tissues. The order of increasing turnover time (decreasing P : B) among systems reported by Cebrian (1999) is phytoplankton > benthic microalgae > macroalgae > seagrass > freshwater macrophytes > marshes > grasslands > mangroves > forests and shrublands. Our model indicates that high P : B should promote strong cascading effects of predators (Fig. 3). P : B appears in the model through xH and xP, the metabolic rates of the herbivore and predator relative to P : B for the plant. Increasing P : B decreases xH and thereby leads to greater cascading effects of predators (Fig. 3). Because aquatic primary producers (both unicellular and multicellular) have higher P : B ratios than terrestrial plants, the model predicts that cascades should be most pronounced in aquatic systems. High producer growth rates may explain empirical evidence for stronger trophic cascades in aquatic ecosystems (Halaj & Wise 2001; Shurin et al. 2002). Enriching the system by increasing the plant carrying capacity (K) has similar effects of increasing top-down control, a prediction that is supported by several aquatic examples (Sarnelle 1992; Wootton & Power 1993).
The second important difference between aquatic and terrestrial systems lies in the relative body sizes of plants and herbivores. Aquatic ecosystems are often dominated by unicellular algae that are many orders of magnitude smaller than their herbivores (Cohen et al. 2003). Many terrestrial plants, by contrast, are large relative to their consumers (Hairston & Hairston 1993). Although large herbivores and small plants occur in many terrestrial systems (e.g. grasslands with grazing ungulates), these same systems also contain many arthropod herbivores that are much smaller than plants. In addition, the largest terrestrial herbivores are all endotherms (mainly mammals) with high metabolic rates that are predicted to have weak effects on plants (Fig. 3). Increasing the size ratio between herbivores and plants decreases xH (eqn 2) and thereby increases the impact of predator removal on the basal trophic level (Fig. 3). Thus, the model predicts that food chains with unicellular producers should show stronger cascades than those containing multicellular plants. Because unicellular autotrophs dominate many aquatic environments but are largely absent on land, the model predictions relating to plant and herbivore body size also suggest stronger cascades in aquatic ecosystems.
The model's predictions agree well with meta-analyses of the published literature on trophic cascade experiments showing stronger effects of predator removal on plant biomass in aquatic systems than terrestrial (Halaj & Wise 2001; Shurin et al. 2002; Borer et al. 2005). The model provides a mechanistic explanation for stronger cascades in pelagic food webs based on P : B ratios of plants and body-size differences between plants and herbivores. Although the model predicts stronger cascades in water than on land, the reasons are different from those envisioned by Strong (1992). Strong argued that phytoplankton are more uniformly palatable than terrestrial plants, and that terrestrial food webs have more omnivory and trophic complexity. Phytoplankton vary from highly edible to completely defended forms (Leibold 1989; Tessier & Woodruff 2002). In addition, omnivory is common in aquatic, as well as terrestrial, environments (Diehl 1993; Vander Zanden & Rasmussen 1996). It is unclear whether edibility of plants or degree of trophic complexity differ systematically between aquatic and terrestrial food webs (Chase 2000). Well-documented aquatic–terrestrial contrasts in plant P : B and the relative body sizes of herbivores and plants provide a firm basis for expecting more pronounced cascades in aquatic systems. Differences in trophic complexity between environments remain to be shown.
In addition to their effects on the equilibrium abundances of plants and herbivores, predators influenced the stability of the plant–herbivore interaction. Stable equilibria were possible for both the two and three trophic-level models for each type of herbivore (Fig. 2). We found regions in which the three-level model was less stable than the two-level model; however, the inverse was never true. Thus we predict that predator removal will tend to stabilize food chains. This prediction is supported by a meta-analysis (Halpern et al. 2005) that showed that predator removal stabilized herbivore populations in 40 experiments in six different ecosystems. The stability of the system depended on productivity (K), the functional responses (Ro and Ho), feeding and metabolic rates of the herbivore and predator (xi and yi). The unstable conditions we encountered consisted of limit cycles rather than chaotic dynamics. Regardless of dynamic stability, the equilibria showed consistent patterns with respect to changes in the parameter values (Figs 2 and 3). Our exploration of unstable equilibria through simulation indicated that average abundances over time showed similar patterns to those of the stable equilibria. For instance, increasing yH or decreasing xH increased the strength of cascades in cases with limit cycles, as well as affecting the amplitude of the fluctuations (Fig. 4). Our results from analyses of stable equilibria may therefore have relevance for other situations where species densities fluctuate over time.
Although much pertinent information about organisms relating to trophic structure is incorporated by metabolic type, body size, P : B and productivity, the model ignores many important aspects of the structure of real food webs. First, the model represents ‘species cascades’ (sensuPolis et al. 2000) rather than ‘community cascades’, where many heterogeneous species comprise a trophic level. However, all community cascades must at some level consist of one or more species cascades. Interspecific heterogeneity within trophic levels can substantially alter trophic structure and the strength of top-down control (Leibold 1989; Power 1992; Abrams 1993; Wootton et al. 1996; Leibold et al. 1997). Inedible species can attenuate the indirect effects of predators over resources and limit the flow of energy from resources to consumers. Inedibility could be incorporated in the model by adjusting fR, the fraction of the resource consumed, the P : B ratio of the plant or by adding species with different traits at any of the trophic levels. These would all have the effect of decreasing the herbivore's efficiency (xH, Table 1), and thereby dampen the cascade (Fig. 3). Variation in species traits within trophic levels or along environmental gradients are likely to be important for regulating the strength of cascades in nature, but are not incorporated into the model. A second potential limitation to the application of the model to natural systems is that yi (species i's feeding rate scaled to its metabolic rate) is the physiologically constrained maximum and not the realized rate. If ecological constraints vary systematically among organisms of different metabolic types, or between systems, then the realized ys may differ substantially from ymax. However, if physiological constraints on y are large relative to ecological limitations, then realized feeding rates may be correlated with ymax.
This model is clearly a simplified representation of a food web in that it does not incorporate many complexities such as behaviour, edibility, omnivory, stage structure or detrital energy pathways that influence the expression of trophic cascades (Polis 1999). However, many fundamental differences among ecosystems occur in terms of the metabolic types and body-sizes of organisms, plant P : B ratios and productivity. The model allows us to understand how these features influence the impact of top-down control in linear food chains. The importance of trophic cascades varies greatly within and among ecosystems and is generally greater in aquatic environments (Shurin et al. 2002). The model's predictions with regard to body size and plant P : B may explain these differences. Structural differences among food webs like heterogeneity within trophic levels, omnivory, non-equilibrium dynamics and individual behaviour are all likely to be important for determining the magnitude of cascades (Polis 1991; Power 1992; Strong 1992; Polis & Strong 1996; Peacor & Werner 2001). However, it is not clear that these factors show consistent differences among ecosystems (Chase 2000). By contrast, plant P : B and relative sizes of consumers and resources show marked, consistent variation among habitats (Cebrian 1999). Discrepancies between the model predictions and empirical data may point us to other salient features of food webs that differ among systems and influence the magnitude of top-down control over trophic structure. None the less, despite its simplified form, this model accurately predicted the effects predator removal on food-chain stability and the effects of herbivore metabolism, body size ratios and plant turnover rates on the strength of trophic cascades.
The work and the manuscript benefited from discussions with Kurt Anderson, Elizabeth Borer, Carol Blanchette, Bernardo Broitman, Jon Chase, Scott Cooper, Stephen Cox, Perry de Valpine, Ben Halpern, Michel Loreau, Kevin McCann, Lauri Oksanen, David Post, Shane Richards and two anonymous reviewers. Funding was provided by postdoctoral fellowships from the National Center for Ecological Analysis and Synthesis, a Center funded by NSF (grant no. DEB-0072909), the University of California and UC Santa Barbara.