A Mechanistic Approach for Modeling Temperature-Dependent Consumer-Resource Dynamics

vol. 166, no. 2 the american naturalist august 2005 ൴

David A. Vasseur1,2,* and Kevin S. McCann2,†

1. Biology Department, McGill University, 1205 Avenue Docteur Many of the characteristics that researchers use to describe Penfield, Montreál, Que´bec H3A 1B1, Canada; biological populations and communities, such as their 2. Department of Zoology, University of Guelph, Guelph, Ontario density, distribution, diversity, and dynamics, rely on en- N1G 2W1, Canada vironmental temperature. Recent climate projections sug- Submitted September 8, 2004; Accepted April 13, 2005; gest that unprecedented warming will occur in many of Electronically published May 17, 2005 the earth’s environments during the current century (Houghton et al. 2001) with uncertain impacts on bio- Online enhancement: appendix. logical populations and communities. Predictive models often consider that populations persist within a range of climatic conditions known as their climate envelope. Cur- rent climate projections indicate that envelopes and their abstract: Paramount to our ability to manage and protect bio- inhabitant populations will move to higher latitudes in logical communities from impending changes in the environment is response to changing conditions because of local specia- an understanding of how communities will respond. General math- tion and extinction events at the range extrema or changes ematical models of community dynamics are often too simplistic to in migratory routes (Parmesan et al. 1999). Although evi- accurately describe this response, partly to retain mathematical trac- dence for such shifts in nature is accumulating (Parmesan tability and partly for the lack of biologically pleasing functions rep- et al. 1999; Humphries et al. 2002; Root et al. 2003), they resenting the model/environment interface. We address these prob- lems of tractability and plausibility in community/environment may not occur as predicted by the population envelope models by incorporating the Boltzmann factor (temperature depen- approach since the envelopes of interacting species may dence) in a bioenergetic consumer-resource framework. Our analysis not change symmetrically (Davis et al. 1998), the temporal leads to three predictions for the response of consumer-resource associations of species may become disrupted (Harrington systems to increasing mean temperature (warming). First, mathe- et al. 1999), and, for populations with fixed ranges, the matical extinctions do not occur with warming; however, stable sys- strength of the interactions themselves may vary with tem- tems may transition into an unstable (cycling) state. Second, there perature (Post et al. 1999; Jiang and Morin 2004). This is a decrease in the biomass density of resources with warming. The biomass density of consumers may increase or decrease depending suggests we should invest in models treating communities on their proximity to the feasibility (extinction) boundary. Third, rather than populations as the basic unit with which to consumer biomass density is more sensitive to warming than resource assess the importance of environmental change (e.g., Ives biomass density (with some exceptions). These predictions are in 1995; Ripa et al. 1998; Ripa and Ives 2003). line with many current observations and experiments. The model Known physiological constraints and responses of or- presented and analyzed here provides an advancement in the testing ganismal and within organism processes have been par- framework for global change scenarios and hypotheses of latitudinal amount in formulating predictions for population pro- and elevational species distributions. cesses. Such approaches use a bioenergetic framework to estimate the flux in measurable individual and population Keywords: predator-prey, allometry, global change, environmental variability, temperature, mathematical model. parameters associated with changes in body size (e.g., Pe- ters 1983) and temperature (e.g., Brown et al. 2004). These relationships have then been used to explain variability in metabolic rate (Gillooly et al. 2001), developmental rate * E-mail: [email protected]. (Gillooly et al. 2002), population growth rate (Savage et † E-mail: [email protected]. al. 2004), and global trends in species density (Enquist and Am. Nat. 2005. Vol. 166, pp. 184–198. ᭧ 2005 by The University of Chicago. Niklas 2001); to predict temperature-induced changes in 0003-0147/2005/16602-40613$15.00. All rights reserved. population ranges (Humphries et al. 2002) and energy Temperature-Dependent Consumer-Resource Dynamics 185 usage (Ernest et al. 2003); and to describe gradients of Consumers ingest resources according to a Monod func- global biodiversity (Allen et al. 2002). The extension of tion defined by the maximum ingestion rate J and the bioenergetic models to the community level has been lim- resource density required to achieve half-saturation of this ited to the analysis of body size in consumer-resource rate, R0 (Yodzis and Innes [1992] used a more general dynamics (Yodzis and Innes 1992). Extending this bio- form of the functional response capable of producing a energetic framework to include community-based tem- sigmoidal saturation function). The parameter d is the perature dependence will arguably provide a useful tool fraction of biomass lost during ingestion and digestion, for the description and prediction of communities in and fe is the fraction of biomass removed from the resource changing environments. population that is actually ingested by the consumer. The Although more realistic, community models require ad- parameter M represents the amount of energy lost to con- ditional parameter and function specification often com- sumer metabolism. This basic model structure can be eas- ing at the cost of generality, and when the interactions ily iterated to include more diverse trophic structures and among populations are themselves dependent on environ- levels, although particular attention must be paid to the mental conditions, scaling up models to the level of com- implementation of multispecies functional responses munities and ecosystems leads to an explosion in the num- (Koen Alonso and Yodzis, forthcoming). All parameters ber of required parameters. Researchers are left with a used in the model are nonnegative. trade-off between model simplicity, which allows mathe- Yodzis and Innes (1992) recognized the need to reduce matical tractability but sacrifices model comprehensive- the number of parameters in this model in order to obtain ness. We address this problem with a general but biolog- a general theory for the dynamics of consumer-resource ically plausible model for temperature dependence in systems. They suggested that the per unit time rates r could consumer-resource systems. We combine the bioenergetic- be scaled with resource body mass (mR) and J and M with allometric framework provided by Yodzis and Innes (1992) consumer body mass (mC) according to empirically de- with a recent advancement in temperature scaling theory rived power laws and assuming a scaling exponent of 0.75. (Gillooly et al. 2001; Brown et al. 2004) to scale the per The “true” value of such scaling exponents remains a con- unit time parameters of a consumer-resource model. We tentious point, but Brown et al. (2000) present strong show that the mathematical analysis of this system remains derivations for quarter-power scaling laws in biology. Stan- tractable while incorporating environmental dependence dardizing these relationships by body mass generates a in a biologically pleasing manner. We make three main mass-specific scaling exponent Ϫ0.25, which can then be predictions for the dynamics of consumer-resource sys- incorporated in the energetic model framework using tems undergoing long-term temperature change and show power functions. that this framework has great potential for future analyses Our argument for the inclusion of temperature as a of global change scenarios and ecological species gradients. parameter in consumer-resource models originates from its ability to explain variability in rates of growth, ingestion, and metabolism in the empirical literature; second The Model to body mass, temperature explains the largest amount of Yodzis and Innes (1992) provided the following framework residual variation in biological rates (Peters 1983). The for a system of consumers and resources based on the wide range of temperature experienced by many species model first proposed by Rosenzweig and MacArthur influences their rates of growth, foraging, reproduction, (1963): and metabolism (among others) via a direct influence on enzyme kinetics. This influence is often described using the empirically derived coefficient Q as the change in rate dR R JC R 10 p rR 1 ϪϪ , associated with a 10Њ change in temperature. This constant dt()( K f R ϩ R ) e0 has been measured for a suite of biological and ecological rates across a broad taxonomic scale (e.g., Peters 1983). dC R However, recent advances in metabolic theory have shown p C ϪM ϩ (1 Ϫ d)J ,(1) dt[()] R ϩ R that scaling biological rates in this manner can lead to as 0 much as 15% error over the range of biologically plausible Њ Њ where the state variables R and C describe the amount of temperatures from 0 to 40 C since the Q10 is itself de- energy in the resource and consumer populations in units pendent on temperature (Gillooly et al. 2001). Rather, it of biomass density (mass per unit area). In the absence is more precise to scale rates according to the Boltzmann of consumers (C p 0 ), resources increase according to the factoreϪE/kT , where T is temperature in Kelvin, E is the Verhulst equation, which is described by the production/ activation energy, and k is Boltzmann’s constant (Gillooly biomass ratio r and the population carrying capacity K. et al. 2001). This factor originates from first principles; 186 The American Naturalist

Ϫ1 0.25 Table 1: Empirically derived intercepts (ai) of the allometric relationships from Yodzis and Innes (1992) in kg (kg year) kg

Parameter ai(T0) Details Source

Metabolism aM (T0): Vertebrate ectotherms 2.3a Average ﬁeld metabolic rate for iguanid lizards and Brett and Glass 1973; Nagy 1982 active metabolic rate in salmon Invertebrates .51 (20) Average oxygen consumption for 77 species of marine Ikeda 1977 zooplankton

Ingestion aJ (T0): b Vertebrate ectotherms 6.4 Derived from maximal O2 consumption in lizards and Brett 1971; Bennett and Dawson in feeding experiments in salmon 1976 Invertebrates 9.7 (20) Average feeding rates of eight marine planktonic Ikeda 1977 copepods

Production ar (T0): Unicells .386 (20) Average maximal division rate in 14 species of salt Williams 1964 marsh pennate diatoms

Note: We have added information on the temperatures at which the allometric intercepts were measured,T0. a Field metabolic rate for iguanid lizards was measured during the active season, which excludes hot, dry periods and cold periods (Nagy 1982). Brett and Glass (1973) measured active metabolic rates at 5Њ,15Њ, and 20ЊC. We take 20ЊC as the intercept for this coefﬁcient. b ˙ Њ Њ Bennet and Dawson (1976) supplied a summary ofVo2max during activity by reptiles at 30 C. Brett (1971) measured food intake at 15 C. Our corrected Њ Ϫ1 0.25 p coefﬁcient for vertebrate ectotherm ingestion at 20 C is 6.4 kg (kg year) kg (assumingEJ 0.7 eV). higher kinetic energy leads to a larger fraction of molecular well describe the dependence of community processes on collisions with enough energy to activate a reaction. We ambient temperature. For ectotherms, the correlation be- write the biological rate functions including body mass tween ambient and body temperature can be weaker be- and temperature scaling: cause of behavioral thermoregulation; however, ambient temperature still largely determines body temperature for Ϫ Ϫ p 0.25 Er(T T00)/kTT r farr(T0)meR , these organisms. For endotherms, maintenance of a con-

Ϫ Ϫ stant body temperature invokes complicated rate sensitiv- Ϫ p 0.25 EJ(T T00)/kTT (1 d)J faJJ(T0)meC ,(2)ities to temperature, requiring model extensions outside Ϫ0.25 E (TϪT )/kTT the limits of our general framework (e.g., Humphries et M p a (T )meM 00, M 0 C al. 2002, 2004). where the E are the rate-specific activation energies. The In a survey of published literature, Yodzis and Innes i (1992) provide a set of allometric intercepts for scaling the intercepts of the allometric relationships (ai(T0)) are empirically derived constants representing the maximum sus- three biological rates r, J, and M to consumer and resource tainable rates (physiological maxima) measured at tem- body mass. We incorporate their intercepts in the size and temperature scaling functions (eqq. [2]) by resolving the perature T0, and the fi are the fractions of those rates temperatures at which they were measured, T0 (table 1). realized in nature. The intercepts (ai) are conservative within each of four metabolic classifications: unicellular In addition, we sought studies in the literature describing poikilotherms, invertebrate poikilotherms, vertebrate ec- the effect of temperature on the rates r, J, and M. Gillooly totherms, and vertebrate endotherms (Robinson et al. et al. (2001) suggest that the range of activation energies 1983). However, the accuracy of these relationships is di- for population rates ought to be within the range of experimentally derived activation energies for all chemical minished when the average body masses (mC , mR) represent a large, skewed, and/or platy- or leptokurtic dis- reactions (0.2–1.2 eV), and they provide activation ener- tribution of body mass (Savage 2004). gies for metabolic rates of a variety of functional groups. Incorporating temperature dependence into a con- More commonly, temperature effects have been quantified using Q coefficients rather than activation energies. Pro- sumer-resource energetic model not only introduces math- 10 vided that the range of temperatures used to estimate the ematical complexity but also complicates the assumptions Q is known, an approximation of the activation energy of the model; the methods by which certain metabolic 10 classes cope with heat loss to a cooler ambient environ- (E) can be determined using the relationship ment or heat gain from a warmer ambient environment p 2 become an important consideration. For poikilotherms, E 0.1(kT010)lnQ , (3) ambient temperature is closely tracked by body temperature, and thus the expressions in equations (2) should where T0 is the median of the range over which Q10 was Temperature-Dependent Consumer-Resource Dynamics 187 measured and k is Boltzmann’s constant (Dixon and Webb dence of ingestion and production. We sought to include 1964; Gillooly et al. 2001). It is important to consider that those studies reviewing a large range of taxonomy to pro- our model formulation precludes the need for estimates vide “average” values for each metabolic functional group. of M and J for populations of unicellular organisms since In one of the larger summaries, Robinson et al. (1983) they are herein always assigned to the resource level. Pop- described the body size and temperature dependence of ulations of heterotrophic organisms, which may under metabolic rate in 109 species of active fish and reptiles normal circumstances occupy a “consumer” role, can be (vertebrate ectotherms) and 729 species of invertebrate incorporated into the model as a “resource” given that poikilotherms, obtaining Q10 coefficients of 1.44 and 1.7 their production per unit biomass (r) is some fraction (fr) for these groups, respectively. These coefficients are smaller of their maximum capacity for ingestion less metabolic than those reported by many studies; however, Clarke and costs: Johnston (1999) noted that Q10 coefficients for diverse taxonomic assemblages tended to be smaller than those p Ϫ Ϫ r fr[(1 d)J M](4)of single-species groups. They found Q10 coefficients of

1.83 for 69 species of teleost fish, while within species, Q10 (from Yodzis and Innes 1992). As we show later in this coefficients averaged 2.40. This suggests that some form article, choice of the function representing maximal pro- of functional compensation occurs in response to tem- duction and the realized fraction of this amount has no perature in the diverse assemblage, highlighting the need effect on the qualitative behavior of the system. for parameter specificity when formulating system-specific

We present a short survey of experimentally and em- predictions. Hansen et al. (1997) provide Q10 coefficients pirically derived values of the Q10 coefficients and corre- for respiration and ingestion rates across a variety of zoo- sponding activation energies for each model parameter plankton functional groups, with coefficients averaging and metabolic classification in table 2. While data for me- 2.51 and 2.97, respectively. The Q10 coefficients for the tabolism/temperature relationships abound, there has been ingestion rates of vertebrate ectotherms have been mea- relatively little work describing the temperature depen- sured for juvenile cod using digestion velocity (3.41; Knut-

Table 2: Empirically derived Q10 temperature coefﬁcients and their corresponding activation energies

Parameter Q10 Ei (eV) Details Source

Metabolism (EM): Vertebrate ectotherms .433 Fish Gillooly et al. 2001 .500 Amphibians Gillooly et al. 2001 .757 Reptiles Gillooly et al. 2001 1.44 .274 Standard metabolism of 109 species of active Robinson et al. 1983 (5Њ–40ЊC) ﬁsh and reptiles 1.83 .432 Standard metabolism of 69 species of teleost Clarke and Johnston 1999 (0Њ–30ЊC) ﬁsh Invertebrates .790 Multicellular invertebrates Gillooly et al. 2001 1.7 .393 Respiration rates of 729 species of Robinson et al. 1983 (0Њ–40ЊC) poikilotherms Respiration rates of seven functional groups Hansen et al. 1997 061. ע 652. 61. ע 2.51 including ciliates, meroplankton larvae, and copepods

Ingestion (EJ): Vertebrate ectotherms 3.41 .671 Digestion velocity in juvenile cod (Gadus Knutsen and Salvanes 1999 (6Њ–13ЊC) morhua) 3.0 .748 Gastric evacuation rate in cod (G. morhua) Temming and Herrmann 2003 (1Њ–15ЊC) Maximal ingestion rates of 11 functional Hansen et al. 1997 045. ע 772. 16. ע Invertebrates 2.97 groups including ﬂagellates, ciliates, meroplankton larvae, and copepods 1.9 .462 Flow velocity through the burrow of the ﬁlter Julian et al. 2001 (10Њ–22ЊC) feeding Urechis caupo

Production (Er): Unicells 1.88 .467 Maximum expected growth rate for marine Eppley 1972 (0Њ–40ЊC) and freshwater photoautotrophic algae 2.8 .788 Net photosynthesis in lake cyanobacteria mats Wieland and Ku¨hl 2000 (20Њ–30ЊC) Note: The Boltzmann constantk p 8.618 # 10Ϫ5 eV KϪ1. 188 The American Naturalist

Ϫ0.25 sen and Salvanes 1999) and gut evacuation rate (3.0; Tem- MamMC Ϫ Ϫ Ϫ x pp # e (ErME )(T T00)/kTT ,(6) ming and Herrmann 2003). Eppley (1972) summarized Ϫ0.25 rfamrr R data for phytoplankton growth in marine and freshwater Ϫ (1 d)JfaJJ Ϫ Ϫ environments and found that growth rates fell between 0 pp# (EJME )(T T00)/kTT p Ϫ y e .(7) and m doublings per day, where log10 m 0.0275T MaM 0.070 and T is measured in degrees Celsius. This rela- p tionship gives aQ10 1.88 for the rate of maximal pri- Parameter x describes the rate of consumer metabolism mary production in unicellular organisms. The inefficiency normalized to the production biomass ratio of the resource of C3 photosynthesis at high temperatures due to pho- population, and parameter y is the consumer’s realized torespiration may explain the relatively weaker dependence rate of ingestion per unit metabolism (Yodzis and Innes of primary production on temperature when compared 1992). The two parameters x and y contain the temper- with heterotrophic metabolism (Ricklefs and Miller 2000). ature dependence of the model, and these quantities are Last, the assimilation efficiency (1 Ϫ d ), which is inde- in turn governed by the activation energy differences Ϫ Ϫ Ϫ pendent of both body size and temperature (Peters 1983), ErMEEand JMEE . The difference rME represents is related to the food type: 0.45 for herbivorous consumers the potential thermal efficiency (PTE) of the consumer- and 0.85 for carnivorous consumers (see Yodzis and Innes resource system. For positive values of PTE, the system 1992). has an increasing potential to retain energy with warming; increases in production at the resource level outpace increases in energy metabolism at the consumer level. This change in system energy is a potential change, since the Model Analysis amount of energy lost to trophic transfer can change dis- In the analysis of differential equation models, it is useful proportionately. The impact of the consumer on its re- to define both the qualitative and quantitative response of sources is determined by the parameter y, whose response Ϫ the model to changes in parameter values. Here we focus to temperature is determined by the differenceEJME . on the model’s response to temperature. We refer to the For positive values, warming-induced increases in the changes in the stability of the equilibrium (stable/unstable) maximum ingestion rate outpace metabolic demands, and the character of the equilibrium (node, focus, limit leading to a larger impact (per unit mass) on the resource cycle) as qualitative changes. The response of the equilib- population. We call this difference the consumer thermal rium densities and their resilience are herein described as impact (CTI). quantitative changes. For simplicity, we describe our re- Yodzis and Innes (1992) analyzed the qualitative changes sults only in response to increasing temperature (herein in the equilibrium in response to changes in y and, in called warming) since the effect of decreasing temperature doing so, cataloged all possible outcomes for variation in is easily inferred. the parameters influencing y. We reiterate their derivations in order to clearly describe how temperature (T) and the

activation energies (Ei) inﬂuence the model inside this range of previously cataloged behavior. Qualitative Response to Temperature Change The model has three possible equilibrium states, and we categorize them according to the existence or nonex- For the purpose of determining the qualitative behavior istence of each species: of the model equilibrium, it is useful to deﬁne the nondimensionalized model (see app. A for derivation): p Eq1ee : trivial, R , C 0;

p p Eq2ee : resource only, R K, C 0; (8) dRp ϪϪ R xy RC R 1 , RR(1 Ϫ )f dt() K (1 Ϫ d)fR (ϩ R ) p 0eep Ϫ d e0 Eq : coexistence, R , C 1 R 3eey Ϫ 1 ()[]Kx e. dC R p xC Ϫ1 ϩ y ,(5)For plausible parameter combinations, Eq1 is always un- dt[()] R ϩ R 0 stable, while one of Eq2 or Eq3 can be stable depending on parameter values. To determine the response of the where the units of time have been chosen such that r p qualitative behavior of the model to changes in temper-

1. Correcting all allometric relationships such that the ai ature, it is useful to employ a bifurcation analysis, which p Њ deﬁne the rate atT0 20 C (293 K), we can represent tracks equilibria and their inherent characteristics across the new parameters x and y as a range of parameter space. Points in parameter space Temperature-Dependent Consumer-Resource Dynamics 189 where equilibria undergo qualitative changes in stability two possible metabolic classiﬁcations for the consumer are called bifurcation points. population considered here. Figure 1 shows the Hopf bi-

The exchange of stability between Eq2 to Eq3 denotes a furcation surface for (a) an invertebrate poikilotherm con- transcritical bifurcation, which we herein refer to as the sumer and (b) a vertebrate ectotherm consumer. When “feasibility boundary” for the system. A feasible two- CTI 1 0, the Hopf bifurcation locus increases with warm- ! ! species equilibrium (Eq3) exists when0 R e K , and ing, reaching an asymptotic value of 1 in theR 0 /K plane. Ϫ 1 thereforeK(y 1) R 0 deﬁnes the feasibility boundary This leads to a larger incidence of limit cycles in R 0 /K (transcritical bifurcation) for the two-species system. In parameter space since limit cycles occur below the Hopf ! the feasible parameter space of Eq3, the equilibrium has bifurcation. Alternatively, whereCTI 0 , the Hopf bifur- two stable states: a stable node, whereby communities at cation locus decreases with warming, and the incidence of nonequilibrium densities approach the equilibrium mon- limit cycles inR 0 /K parameter space lessens. When otonically, and a stable focus, whereby the approach is CTI p 0, the Hopf bifurcation is independent of tem- characterized by decaying oscillations around the equilibrium point. The location of the focus/node boundary is dependent on both model parameters x and y, and its proximity to the feasibility boundary is controlled mainly by the body mass ratiomCR/m and PTE (for a full description, see Yodzis and Innes 1992). Also in the feasible parameter space, the equilibrium Eq3 can be an unstable focus. In this case, it is surrounded by a stable limit cycle, which is approached from outside (inside) by decaying (expanding) oscillations. The Hopf bifurcation denotes the change in stability of Eq3 from a stable focus to an unstable focus and marks the emergence of a stable limit cycle. The state Eq3 is locally stable when (see app. A)

K(y Ϫ 1) ! ϩ 1, (9) R 0(y 1) and it is locally unstable, but surrounded by a limit cycle, when condition (9) is not satisﬁed. From the above equations, it is apparent that the qualitative behavior of the model is completely determined by the parameters R0, K, and y. To further reduce the di- mensionality of the analysis, Yodzis and Innes (1992) de-

ﬁned the ratioR 0 /K as an inverse measure of resource abundance as perceived by the consumer population; as

R0 decreases (K increases), consumers perceive a higher density (or enriched) resource supply, resulting in a larger flux of biomass from the resource to the consumer population. We make the assumption that this “inverse enrichment ratio” is independent of temperature, and we discuss the full implications of this assumption later in this article. We analyze changes in the loci of the feasibility boundary and Hopf bifurcation across a range of biologically Figure 1: Hopf bifurcation surface across the independent parameters plausible temperatures (0ЊC ! T ! 40ЊC ) and across a p temperature and CTI for (a) an invertebrate consumer (aJM/a 19 ) and Ϫ p ע range ( 1) in the CTI (EJME ). This encompasses a (b) a vertebrate ectotherm consumer (aJM/a 2.8 ). Above the Hopf satisfactory range in CTI given that measured activation bifurcation, the equilibrium is stable, and below the Hopf bifurcation, the equilibrium is unstable. Along the two independent axes, the Hopf energies of chemical reactions range from 0.2 to 1.2 eV p bifurcation surface approachesR0/K 1 according to a Monod function. (Gillooly et al. 2001). Since the value of the parameter y The two axes of invariance occur atCTI p 0 eV, where temperature depends only on the consumer population, it is sufficient change has no effect on the Hopf bifurcation, and atT p 20ЊC , where to perform two bifurcation analyses corresponding to the the intercepts of the temperature-dependent functions are defined. 190 The American Naturalist perature. In both panels, the response of the Hopf bifurcation is similar in nature, but saturation of the curve to p R 0 /K 1 occurs much quicker for invertebrate consumers (a) because of the relative differences in the ratio of the interceptsaJM/a (see table 1). Similar results are observed for the feasibility boundary of invertebrate consumers and vertebrate ectotherm consumers (fig. 2), with the exception that the feasibility boundary is not a saturating function, but rather it grows/ decays exponentially with warming. WhenCTI 1 0 , the locus of the feasibility boundary grows exponentially with warming, and whileCTI ! 0 , the locus of the feasibility boundary decays exponentially with warming. Below the feasibility boundary, the system is persistent, and where p the curve crosses theR 0 /K 0 plane, the consumer cannot persist regardless of its capacity to ingest resources. Empirical estimates of the activation energies influencing the bifurcation loci Ei (table 2) suggest that the CTI for invertebrate consumers falls in the range 0.1–0.38 and for vertebrate ectotherm consumers 0.24–0.47. Figure 3 shows the two-dimensional slices from figures 1 and 2 corresponding to empirical estimates of the CTI from table 2. In both examples, there is an increase in the parameter space occupied by unstable equilibria (limit cycles) with warming (although very slight for invertebrate consumers). However, the proportion of unstable parameter space within the domain of feasible solutions decreases with warming, since the feasibility boundary increases in

R 0 /K. This leads to two predictions for the qualitative stability of consumer-resource systems in response to warming. First, warming may lead to qualitative instability; for ! any value ofR 0 /K 1 , a stable system may cross the Hopf bifurcation, resulting in an unstable (cycling) state. Sec- Figure 2: Locus of the feasibility boundary (transcritical bifurcation) across the independent parameters temperature and CTI for (a)anin- ond, warming does not lead to mathematical extinctions p vertebrate consumer (aJM/a 19 ) and (b) a vertebrate ectotherm con- (where mathematical extinction corresponds to crossing p sumer (aJM/a 2.8 ). Above the feasibility boundary, the consumer can- the feasibility boundary) but rather may facilitate the per- not persist. Below the feasibility boundary, both the resource and sistence of consumer-resource systems because of the in- consumer persist, and in this region, the system stability is determined by the Hopf bifurcation (ﬁg. 1). In both a and b, the feasibility boundary crease in feasibleR 0 /K parameter space. varies exponentially along the two independent axes. The two axes of invariance occur atCTI p 0 eV, where temperature change has no effect on the feasibility boundary, and atT p 20ЊC , where the intercepts of the temperature-dependent functions are deﬁned. Quantitative Response to Temperature

It is useful to determine the quantitative behavior of Eq3 in response to warming since the density of resource and dR ϪRy(E Ϫ E ) consumer populations can be highly variable within any e0p JM.(10) dT (y Ϫ 1) 22kT particular region of qualitatively similar parameter space.

We determine the response of Re and Ce to temperature This generates a rather simple relationship governing the using the derivatesdR ee/dT and dC /dT and using a pa- directional response of equilibrium resource density to rameterized example. temperature change; where theCTI 1 0 , resource density ! The derivative of equilibrium resource density Re with decreases with warming. Alternatively, if theCTI 0 , re- respect to ambient temperature is a dampened exponential source density increases with warming. Where the function whose sign is completely determined by the CTI: CTI p 0, resource density is constant across temperature. Temperature-Dependent Consumer-Resource Dynamics 191

Figure 3: Two-dimensional slices of the bifurcation loci given the CTI values estimated from table 2. For invertebrate consumers (a), we estimated p p CTI from the coefﬁcientsEJM0.77 andE 0.65 measured by Hansen et al. (1997). For vertebrate ectotherm consumers (b), we estimated CTI p p from the average values ofEJM0.71 andE 0.38 for ﬁsh (excluding reptiles and amphibians).

! Empirical estimates in table 2 suggest that, for the majority PTE 0,dCe /dT reaches a local maximum in the range 1 ! ! ! of systems, CTI should be 0 and thus thatdR e /dT 0 . K/2 R e K (recall that Re is decreasing with warming) ≤ The consumer population displays a more complex re- and is always negative beyondR e K/2 . Alternatively, 1 ≤ sponse to temperature: whenPTE 0 ,dCe /dT is positive in the range K/2 ! ! R eeKRand reaches a local maximum whenK/2 . It is important to consider that these relationships describe dC (1 Ϫ d)f (E Ϫ E )(1 Ϫ R /K)RdR 2R eep rM eeϩ e1 Ϫ e, the quantitative behavior of equilibrium resource density dT x[()] kT2 dT K in the range of stable parameter space; however, they do (11) not provide any information about the response of equilibrium densities to temperature within unstable (cyclic) where the sign ofdCe /dT relies on the PTE, the relative parameter space. In this region, it is more informative to equilibrium resource densityR e /K , and the response of perform numerical simulation to obtain the attributes of equilibrium resource density to temperature (dR e /dT ). the limit cycle (see parameterized example below). We With the potential for two real roots, the consumer density therefore proceed in our analysis of the quantitative be- may reach local extrema in response to varying temper- havior of the system using numerical simulation of an ature; however, these local extrema can lie outside the exemplary consumer-resource system. range of biologically relevant conditions. The analysis of Using the coefﬁcients EM and EJ derived from Hansen et equation (11) proceeds more easily under the assumption al. (1997), Er derived from Eppley (1972), and the intercepts ! thatdR e /dT 0 , and the results below adhere to this as- listed in table 1, we parameterize a model representing the sumption. Near the feasibility boundary, Re is only slightly interaction between a herbivorous invertebrate consumer 1 p Ϫ p less than K, and thusdCe /dT 0 . This is a rather obvious and a unicellular resource (PTE 0.19 and CTI p conclusion given thatCe 0 at the feasibility boundary 0.12). Figure 4 shows the equilibrium response of this and increases to positive density with warming. When system to warming given three values of the enrichment 192 The American Naturalist

Figure 4: Equilibrium densities of resources (a) and consumers (b) across a gradient of temperature parameterized to represent the interaction between an invertebrate consumer and a unicellular resource. Resource biomass density is normalized by the carrying capacity K; consumer biomass 0.25 is normalized by the resource carrying capacity K and by the body mass ratio(mCR/m ) . Three ratios of the inverse enrichment ratioR0/K are p p shown to highlight three potential behaviors:R0/K 13 (dashed line) shows the equilibrium response near the feasibility boundary,R0/K 6 (solid p line) shows the equilibrium response in the stable region, andR0/K 0.9 (dotted line) shows the equilibrium response as it crosses the Hopf p p bifurcation into unstable (cyclic) space. Coefﬁcients ai are from table 1,ErJ0.467 (table 2; Eppley 1972),E ,M 0.772 and 0.652, respectively p Ϫ p (table 2; Hansen et al. 1997),frJ, f , fe 1(1 , andd) 0.45 .

ratioR 0 /K : a low-enrichment community that crosses the enrichment are decreasing functions for the range of tem- p ! feasibility boundary (R 0 /K 13 ), a medium-enrichment peratures shown, sinceR e K/2 over the entire range. On community that remains in the stable state space the logarithmic scale of figure 4, it is evident that the p (R 0 /K 6 ), and a high-enrichment community that equilibrium consumer density is more sensitive than the p crosses the Hopf bifurcation (R 0 /K 0.9 ). Resource den- equilibrium resource density given their relative slopes at p sity is a decreasing function of temperature for all values medium- and high-enrichment levels (R 0 /K 6 , 0.9; ex- p of the enrichment ratio within the range of stable state cludingR 0 /K 13 since it is proximate to the feasibility space (fig. 4). In the cyclic domain, the amplitude of cycles boundary). We have factored out the body mass ratio 0.25 generally increases with warming, and the lower bound of (mCR/m ) from figure 4b to show the characteristic re- the cycle drops to low densities. This occurs in concor- sponse of consumer biomass density. dance with the “paradox of enrichment,” which suggests Given the coefficients listed in table 2, we expect CTI that consumer-resource systems become unstable and of- to be 10 for the majority of systems. This leads to a robust ten nonpersistent because of large amplitude cycles after prediction for the response of equilibrium resource bio- crossing the Hopf bifurcation (Rosenzweig 1971). The re- mass density: in the domain of stable solutions, equilib- sponse of consumer density is more complicated because rium resource density declines with warming because of of the potential for a local maximum within the range of the increasing capacity for ingestion in the consumer pop- biologically relevant temperatures. In the low-enrichment ulation. Similarly, the median density of resources in the community, the consumer density increases with warming unstable domain decreases with warming, with a tendency as it crosses the feasibility boundary and reaches a local for high-amplitude cycles at high temperatures. For the maximum thereafter. The curves for medium and high consumer population, the response of equilibrium density Temperature-Dependent Consumer-Resource Dynamics 193 to warming relies also on the PTE. In our example, PTE mass cannot be scaled out. We chose body sizes represen- is !0, resulting in a decreasing equilibrium consumer den- tative of the interaction between zooplankton and uni- sity in response to warming when the system is sufficiently cellular algae that fit the activation energies used in figure far from the feasibility boundary. Systems that are prox- 4 (1 # 10Ϫ7 kg and 6 # 10Ϫ12 kg, respectively). For the imate to the feasibility boundary will initially increase with low-enrichment community, the feasibility boundary pro- warming and reach a local maximum. For invertebrate duces a vertical asymptote at low temperatures where the consumers, the influence of the feasibility boundary is dominant eigenvalue passes through the origin. Similarly, limited to a small range of the stable parameter space. for the high-enrichment community, there is a vertical However, for vertebrate ectotherm consumers whose fea- asymptote at the Hopf bifurcation where the system be- sibility boundary is potentially crossed for nearly all fea- comes unstable (at both bifurcation points, the real por- sible values ofR 0 /K , this effect will be more prominent tion of the dominant eigenvalue is equal to 0, and the (see fig. 3). return time is thus infinite). For each of the low- and In addition to changes in the density, it is important to medium-enrichment communities, return time decreases consider changes in the “attractiveness” or resilience of with warming since they are bounded well away from the the system. Resilience of the system can be defined as the Hopf bifurcation. The decrease in return time results from rate at which population density returns to equilibrium a larger throughput of energy at warmer temperatures, after a perturbation (Pimm 1991), and the negative re- which allows the system to equilibrate more quickly fol- ciprocal of the dominant eigenvalue of the Jacobian matrix lowing perturbation (see DeAngelis 1992). It is important provides a quantitative measure of the amount of time to consider that larger body sizes of both resources and required for the system to reach eϪ1 (or approximately consumers increase the return time of the system because 37%) of the initial perturbation displacement. This mea- of their slower per unit mass rates of growth, ingestion, sure is an asymptotically unbiased estimate of the actual and metabolism but do not qualitatively affect the results. value; error in the estimate increases with the perturbation displacement. It is important to note that the eigenvalues Discussion determined in appendix A are those of the nondimensional system and require back transformation to properly esti- In response to an increase in mean temperature, three mate the return time. general predictions emerge from the analysis of our Figure 5 shows the dependence of return time on tem- temperature-dependent model, given the empirical param- perature for each of the three low-, medium-, and high- eter estimates in table 1. First, mathematical extinctions enrichment communities from figure 4. Since the return do not occur with warming; however, stable systems may time is estimated from the dominant eigenvalue of the transition into an unstable (cyclic) state. Vertebrate ec- Jacobian matrix, the effects of consumer and resource body totherm consumers are likelier than invertebrate consum-

Figure 5: Return times of the consumer-resource system parameterized to represent the interaction between an invertebrate consumer and a unicellular resource at the low-, medium-, and high-enrichment levels shown in figure 4. Return times are estimated as the reciprocal of the dominant eigenvalue of the Jacobian matrix; they represent the time required for the system to reach eϪ1 (or approximately 37%) of the initial perturbation displacement. The abrupt change in the low- and medium-enrichment curves is caused by the qualitative change in stability from stable node to stable focus. 194 The American Naturalist ers to drive the system into an unstable state because of are predicted to decline with warming, increasing their their relatively larger metabolic requirement per unit mass. susceptibility to extinction through demographic stochas- Second, the biomass density of resources always decreases, ticity, genetic bottlenecks, and Allee effects. In addition, and the biomass density of consumers will decrease with we predict that systems may destabilize with warming, warming provided that their existence is not proximate to leading to limit cycles with relatively low minimum den- the feasibility boundary (e.g., consumers are not on the sities of both resources and consumers. Short of the pres- verge of extinction because of an inability to acquire ad- ent date, climate change has only been implicated in one equate resources). Practical extinctions may occur with species-level extinction (Pounds et al. 1999), although warming, given that both resources and consumers can some studies predict high rates of extinction in the near reach low densities. Third, given our exemplary param- future (Thomas et al. 2004). In a study of population eters, consumer biomass density is more sensitive to densities of trees and terrestrial ectotherms (amphibians, warming than resource biomass density. This prediction reptiles, and invertebrates) the biomass density of both depends on the proximity to the feasibility boundary and groups decreased at warmer temperatures (Allen et al. on the relative magnitudes of CTI and PTE. In the cyclic 2002). Petchey et al. (1999) found that 30%–40% of spe- state, the median biomass density of consumers is more cies of aquatic microbes in a microcosm were extinct after sensitive than that of resources. However, the change in 7 weeks of a ϩ2ЊC weekϪ1 change in temperature (ap- cyclic amplitude is larger for resource biomass density, and proximately 0.1Њ–0.2ЊC per generation). In their experi- since the majority of the cycle period can be spent at low ment, producer (resource) biomass increased with warm- densities, the mean biomass density of resources may show ing while consumer biomass decreased, and only increased sensitivity. This distinction may be important consumers showed increased rates of extinction. Our when considering the sensitivity of cyclic systems in nature. model mimics the observed increase in resource biomass Our predictions for the response of population biomass only whenCTI ! 0 , suggesting that despite our empirical densities, sensitivities, and community stability and resil- estimates for CTI (which are 10), there may be instances ience rely on empirical estimates of the intercepts and where the capacity for ingestion by consumers increases parameters that govern the scaling functions. The inter- more slowly with warming than metabolism or where the cepts for production and ingestion provided by Yodzis and capacity for ingestion decreases with warming. There is Innes (1992; table 1) represent the physiological maxima evidence supporting the existence of thermal optima for that can be attained under ideal conditions. Realized rates ingestion in a variety of species and similarly for critical of these maxima may be only small fractions when species thermal limits in production and metabolism, which could K are “ecologically limited” (fr,fJ 1 ) rather than “physi- account for discrepancies between our model and empir- ≈ ologically limited” (fr,fJ 1 ). Our examples for inverte- ical data. Like any scaling relationship based on diverse brate and vertebrate ectotherm consumers represent the assemblages, our model is best suited to explain large-scale behavior of the model at the physiologically limited extent. patterns since species-specific responses to temperature are For ecologically limited consumers, the loci of the feasi- obscured within the general scaling framework. bility boundary and Hopf bifurcations are more sensitive Voigt et al. (2003) compiled plant, herbivore, and car- to temperature (e.g., they show more curvature); for ex- nivore abundances from two grassland communities to ample, an ecologically limited invertebrate consumer may determine whether the climate sensitivity of functional have bifurcation curves similar to those of a physiologically groups differed between trophic levels. Their analysis limited vertebrate ectotherm consumer. Additionally, the showed that the temporal variance in abundance, attrib- equilibrium densities depend on the extent to which con- utable to climatic variability, increased with trophic level. sumers and resources are ecologically or physiologically Similarly, Allen et al. (2002) show that the density of ter- limited. Despite these dependencies, the trends describing restrial ectotherms is more sensitive than the density of the response of equilibrium densities to warming are in- plants to temperature. Pianka (1981) suggested that dif- dependent of the extent of ecological or physiological lim- ferences in the metabolic requirements of carnivores, her- itation; ecologically limited species are more prone to un- bivores, and autotrophs could account for differences in dergo the qualitative changes of the first prediction and their climatic sensitivity. However, the response of auto- more likely to behave as systems proximate to the feasi- trophs additionally causes the resource availability to vary bility boundary when at low temperatures for the second for herbivores and can lead to amplified sensitivity at and third predictions. higher trophic levels (Voigt et al. 2003). In our example Although our model does not suggest mathematical ex- system (fig. 4), both mechanisms account for the increased tinctions will occur at warmer temperatures (mathematical sensitivity of higher trophic levels: consumers ingest more extinction corresponds to crossing the feasibility bound- energy per unit mass at higher temperatures, causing re- ary), biomass densities of both consumers and resources source density to decline. Fewer resources combined with Temperature-Dependent Consumer-Resource Dynamics 195 an increased consumer ingestion rate lead to a relatively rather than consumer grazing, provides a reasonable ex- larger decline in consumer density. The generality of this ample for which data are available. Enquist and Niklas property and mechanism for a broader class of consumer- (2001) show that tree biomass density is constant across resource systems remains to be determined. latitudes ranging from Ϫ40Њ to 60Њ, which is at least sug- Our amalgamation of the Boltzmann factor into the gestive that carrying capacity may be independent of tem- Yodzis and Innes (1992) consumer-resource model pro- perature. Much work remains to determine the scaling of vides a general framework for temperature-dependent carrying capacity with temperature and, as importantly, to models of interacting populations. General theories for the determine whether its implicit scaling in population mod- inclusion of further environmental variables in allometric els is analogous to the results produced by our community models may be limited, since body mass and temperature approach. tend to be the only variables accounting for significant The half-saturation density of the functional response amounts of variability in broadscale regressions (e.g., Pe- R0, in unison with the maximum capacity for ingestion J, ters 1983). The framework used to model consumer- controls the realized rate at which resources are consumed. resource dynamics in this study neglects to include the The effects of temperature on the half-saturation density direct effects of consumer mortality in the model. The itself are not well documented. However, there has been relationship between mortality, temperature, and body size some research on the temperature dependence of the at- follows the same relationship as metabolism (Savage et al. tack rate in the traditional Holling Type II functional re- 2004), and evidence suggests that the two are governed by sponse model (Holling 1965). This model defines the func- p ϩ the same scaling parameters (Ei for fish mortality is 0.45 tional response asf(R) aRC/(1 atRh ) , where a is the eV [Savage et al. 2004] and 0.43 eV for fish metabolism attack rate, th is the amount of time required to handle a [table 2]). If, indeed, the scaling parameters are conser- prey item, and R and C are resource and consumer den- vative, then directly including consumer mortality in the sities. The parameters of our Monod model can be ex- model would further limit any consumer from existing at pressed as those of the Holling Type II, where the maxi- { its physiologically limited extent (see app. B in the online mum capacity for ingestionJ 1/t h and the { edition of the American Naturalist). There is still much to half-saturation densityR 0h1/at . In our study, we as- be learned about the influence of temperature on alter- sume R0 is independent of temperature, and given that native model formulations, structures, and, with direct handling timescales as1/J (where1/J ∝ e EJ/kT ), this requires impact for our results, the potential temperature depen- that the Holling attack rate (a) scales according to a ∝ ϪEJ/kT dence of the per unit area parametersR 0 and K. e . Holling (1965) considers attack rate to be a function The temperature dependence of carrying capacity K has of the reactive distance of the predator, speeds of move- received surprisingly little attention in the literature (Sav- ment of the predator and prey, and their capture success. age et al. 2004). In Verhulst population models, K defines Thompson (1978) showed that the attack coefficient of the maximum attainable density of a population, and un- damselfly larvae increased with temperature according to der steady state conditions, it is the equilibrium density. a sigmoidal function over the range 5Њ–27.5ЊC. However, Verhulst growth is included in our model framework to inspection of Thompson’s curve suggests that exponential define density-dependent growth of resources and assumes growth well explains the changes in the attack rate below that factors such as nutrients and light are static. Popu- approximately 20ЊC. Marchand et al. (2002) observed a lation models that account for the metabolic costs offset- positive and nonlinear relationship between the attack rate ting growth but neglect community interactions suggest of juvenile brook charr and temperature below their upper that where resource supply is constant, K must decrease avoidance temperature of 22ЊC. The existence of an upper with temperature to balance the effects of increasing met- threshold for this relationship is not surprising, given that abolic costs (Savage et al. 2004). Savage et al. (2004) argue species-specific thermal tolerances are often well bounded that the findings of Allen et al. (2002; decreased ectotherm within our range of “biologically plausible” temperatures. density with temperature) support a negative relationship These results are at least suggestive of a positive expo- between temperature and carrying capacity. However, our nential relationship among attack rate and temperature, model framework similarly explains the decline in ecto- and they thus indicate that temperature invariance of R0 therm density using a community framework, scaling only may indeed be a reasonable suggestion. In practice, any the per unit time elements of the model. Our model differs scaling relationship that exists between R0 and temperature from that of Savage et al. (2004) in that resource density will most likely be influenced by environmental correlates is not influenced by temperature in the absence of pred- of temperature that alter the efficiency of consumers. For ators (since K is temperature invariant). Populations that example, an increase in the density of emergent macro- exist truly in the absence of predators are rare, but tree phytes in an aquatic system may reduce the effectiveness biomass density, which is arguably limited by competition of sensory cues in zooplanktivorous fish. 196 The American Naturalist

Our results suggest that long term-changes in temper- Solving fordR/dt p dC/dt p 0 , we obtain the nontrivial ature can influence the equilibrium dynamics of species equilibrium conditions: in predictable ways, given the dependence of their rates of growth, ingestion, and metabolism on temperature. Fur- R (1 Ϫ )f p Ϫ eed ther work addressing short-term periodic variability, such Cee1 R , as seasonal and El Ninõ oscillations, and stochastic vari- ()[]Kx ability will provide additional insight into the influence of R R p 0 .(A3) environmental conditions on short-term, nonequilibrium e y Ϫ 1 (equilibrium transient) dynamics. For those communities whose range is limited by the spatial extent of their habitat 1 Since all model parameters are nonnegative andCe 0 for (e.g., lake communities), we expect that our model pro- ! ! ! Ϫ all0 R e0KR , we derive/K (y 1) as the upper limit vides a testable and applicable framework for formulating for feasibility (transcritical bifurcation). SinceR 0 /K is al- predictions. For communities whose range is flexible, in- ways positive, the ratio(1 Ϫ d)J/M must be 11 for a two- tegrating this bioenergetic community approach into species equilibrium to exist. This provides a derivation of climate-envelope predictions will provide a stronger basis the energetic constraints of the model; energy ingestion for the ecological consequences of global change scenarios. must be greater than energy metabolism in order to sustain consumers in the model. Acknowledgments To determine the stability of the equilibrium densities We would like to thank M. Drever, M. Koen-Alonso, J. (eqq. [A3]), we define the Jacobian matrix at the nontrivial Rip, N. Rooney, J. Umbanhowar, C. Vasseur, and P. Yodzis equilibrium point Eq3, for comments on an early draft of the manuscript and E. Ѩ Ѩ Novak for assistance with the data collection. Two anon- R R ymous reviewers provided indispensable feedback on the ѨR ѨC JacF p model and manuscript. Partial funding for this project was Ѩ Ѩ Eq3 C C provided by a Natural Sciences and Engineering Research ѨR ѨC Council postgraduate scholarship to D.A.V. Eq3 Ϫ Ϫ Ϫ Ϫ Ky Ry00K R x APPENDIX A Ϫ Ϫ y(y 1)K (1 d)fe p ,(A4) Bifurcation and Sensitivity Analysis Ϫ Ϫ Ϫ fe0(1 d)(Ky K R ) 0 of the Model System yK

Starting with the model (eqq. [1]) from Yodzis and Innes and the solution of its eigenvalues, (1992), we deﬁne the nondimensionalized model by se- p (ͱ(ѨR/ѨR)2 Ϫ 4(ϪѨR/ѨC # ѨC/ѨR ע lecting units of time such thatr 1 . Normalizing the ѨR/ѨR remaining rates in the model by r gives l p .(A5) 1,2 2

dR R JC R The model equilibrium (A3) is globally stable when p R 1 ϪϪ , dt()( K rf R ϩ R ) both eigenvalues have negative real parts, and since e0 Ѩ Ѩ # Ѩ Ѩ ! ! Ϫ R/ C C/ R 0 for allR 0 /K y 1 , the discriminant Ѩ Ѩ dC M (1 Ϫ d)JR in equation (A5) will always be less thanR/ R . Therefore, p C Ϫϩ ,(A1) Ѩ Ѩ ! dt[()] r r R ϩ R we must satisfy only the conditionR/ R 0 . This con- 0 dition is satisﬁed when and given thatx p M/ry andp (1 Ϫ d)J/M , the model RyϪ 1 reduces to 0 1 . (A6) Kyϩ 1 dR R xy RC p R 1 ϪϪ , The condition (A6) deﬁnes the Hopf bifurcation locus in dt() K (1 Ϫ d)fR (ϩ R ) e0 theR 0 /K plane. Below this stability threshold, the equilibrium becomes locally unstable but is surrounded by a dC R stable limit cycle. Within the realm of locally stable equi- p xC Ϫ1 ϩ y .(A2) dt[()] R ϩ R libria, the approach to equilibrium can be monotonic 0 Temperature-Dependent Consumer-Resource Dynamics 197

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Appendix B from D. A. Vasseur and K. S. McCann, “A Mechanistic Approach for Modeling Temperature-Dependent Consumer-Resource Dynamics” (Am. Nat., vol. 166, no. 2, p. 184)

Including Consumer Natural Mortality in the Model System The Yodzis and Innes (1992) bioenergetic-allometric framework for consumer-resource interactions that forms our model system neglects to include the direct effects of a consumer’s natural mortality. In the two trophic level system described and analyzed here, energy escapes the model only through inefficient energy assimilation (the fraction d of ingested biomass) and through metabolism. For some populations, particularly those inhabiting the uppermost trophic level(s), natural mortality may be a significant energy sink and thus may influence the dynamics of the system. We derive the scaling properties of natural mortality (from Brown et al. 2004) and describe its influence in our model system. A consumer population (C) will experience a complete turnover of its biomass, due solely to natural mortality, in the period of time corresponding to the average life span of an individual in the population (tL). Thus, the instantaneous rate of change of consumer biomass density due to natural mortality is ϪCN, where the rate of natural mortality (N) is equal to1/tL . Like many biological rates, life span scales with body size and temperature ∝ 0.25 E/kT according to the relationshiptL me , where m is the average individual body mass, E is the activation energy, k is Boltzmann’s constant, and T is temperature in Kelvin (Brown et al. 2004; Savage et al. 2004). Ϫ Ϫ Natural mortality then scales asN ∝ me0.25 EN/kT . Converting this relationship to scale from an intercept value at a known temperature (aN (T0 ) ; see, e.g., Gillooly et al. 2001), we have

Ϫ Ϫ p 0.25 EN(T T00)/kTT N aN (T0 )meC .(B1)

This relationship takes the exact form as that for all biological rates in our article (cf. eqq. [2]).

The activation energy EN, which controls the rate of natural mortality as temperature varies, should be nearly equal to the activation energy for metabolism (EM) for any given consumer since life span scales inversely with ∝ metabolism (tL 1/M ; Savage et al. 2004). For example, Savage et al. (2004) predicted an activation energy for mass-corrected instantaneous mortality in fish of 0.45 eV, which is very close to the activation energy used for fish metabolic rate in our model (0.43 eV; derived in Gillooly et al. 2001). Provided that metabolism and natural mortality indeed scale similarly with mass and temperature, we can rewrite the two scaling functions as a single function, defining the new parameter MJ as the joint effect of metabolism and natural mortality:

Ϫ0.25 E (TϪT )/kTT p ϩ MJ 00 MJM[a (T0 ) aN (T0 )]meC . (B2)

To include natural mortality in our model, we replace metabolism M with the joint effects of metabolism and mortality MJ. The leading effect of this change is an increase in the per unit energy demands of the consumer 1 population, sinceMJ M at all temperatures. This transformation manifests as an increase in the parameter x (x p M/ry ) and a decrease in the parameter y (p J/M ), ultimately inﬂuencing the qualitative and quantitative dynamics of the model. Recalling that the parameters x and y are also the subject of variation in the physiological/ecological extent of limitation through the fractions fr and fJ (which denote the realized fractions of the biological rates r and J), it is evident that increasing M by adding natural mortality has the same dynamical consequences as reducing the fractions fr and fJ . Thus, the inclusion of natural mortality is analogous to shifting the originally speciﬁed system toward a more ecologically limited state. The consequences for the dynamics of our consumer-resource system under ecological limitation are discussed in the main text. We reiterate here that

1 App. B from D. A. Vasseur and K. S. McCann, “Temperature-Dependent Consumer-Resource Dynamics” ecologically limited systems are more prone to undergo bifurcations with temperature change and have reduced densities at all temperatures when compared with their physiologically limited counterparts. The extent of change in the model owing to this modification is dependent on the relative contributions of metabolism and natural mortality to the overall energy expenditure of the consumer. At any temperature and for ϩ any consumer body mass, the fraction of energy expenditure due to metabolism alone (M/MaJM ) is (T0 )/[aM (T0 ) Њ aN (T0 )]. We estimated this fraction to be 0.76 for vertebrate ectotherms (fish) using the 20 C intercept value for natural mortality from Savage et al. (2004; 0.73 individuals [individual year]Ϫ1 kg0.25) and metabolism data from table 1 (2.3 kg [kg year]Ϫ1 kg0.25). In practice, the relative contributions of metabolism and natural mortality to overall consumer energy expenditure may depend on a variety of factors. Further research defining the scaling attributes of natural mortality will be vital to determining its importance in modeling exercises.