Universal temperature and body-mass scaling of feeding rates

(a). The anatomy of the functional response

The functional response describes the relationship between the per capita feeding rate of a consumer i, F_ij (individuals_i s⁻¹) and the density N_j (individuals_i m^{−2 or −3}), of the resource j being fed on. Density is measured in units of per metre squared or cubed, depending on whether the consumer is foraging on a surface or in a volume. The parameters of a functional response model (sensu Holling [31]) are attack rate, a_ij (m^{2 or 3} s⁻¹), and handling time, h_ij (s). Attack rates describe the initial increase in a functional response (figure 1a, black arrows), whereas handling time limits the maximum feeding rate (figure 1b, horizontal asymptotic lines), which yields a hyperbolic Holling type II curve [31]:

1.1

Biologically, handling time describes the time a consumer needs to fight, subdue, ingest [31] and digest a resource item [32]. The attack rate is the rate of successful attacks (also termed capture rate [33], instantaneous rate of successful search [31], maximum filtration rate [34] or maximum interaction strength [4]) and includes processes such as searching (e.g. movement), detection, encounter and success of the attack [27]. Some functional responses in our database exhibited a sigmoid shape (type III, not shown), where the attack rate increases with resource density, a ∝ N^q, with a nonlinearity described by q [21,35,36]. We followed Englund et al. [30] and used the attack rate at the half saturation density to include type III functional responses in our analyses.

(b). Body-mass dependence

Since the nineteenth century, biologists have reported biological rates to scale nonlinearly with organism body mass (i.e. allometric scaling) [37]. Allometric scaling of metabolic rates is perhaps the most widely studied [38–40], though other quantities have received considerable attention, e.g. growth rates, death rates and population abundances [38,39]. The size dependence of functional responses has been investigated for different specific taxonomic groups [36,41,42], but there is a lack of a unifying framework, and limited data collation over ecosystem types and physiological blueprints. In the following, we develop a null hypothesis for the allometric relationships for handling time and attack rate.

Handling times: We recast the functional response following Koen-Alonso [33] to the form of

1.2

where F_ij,max = 1/h_ij (figure 1b, asymptotic lines) is the maximum feeding rate, and f_ij = a_ijN_j/(F_ij,max + a_ijN_j) is a non-dimensional feeding level taking values between 0 and 1 that describes the proximity to maximum feeding capacity of the consumer [43]. Handling time as used here is a phenomenological description, including both physical handling (i.e. attacking, overwhelming and chewing a resource) and digestion [32,44]. Metabolic theory suggests that metabolic rate scales with a 3/4 power law of body mass [38]. Therefore, to persist, a predator at least needs to balance its food intake rate (via F_ij) to the metabolic rate, meaning that F_ij,max should share scaling relationship with metabolic rate, which subsequently means that handling time should scale as

1.3

Besides the Metabolic Theory of Ecology, other frameworks invoke more flexible theories that predict exponents from 0.66 to 1 for metabolic rates [45,46], leading to a continuum of expectations for the handling time scaling in the range from −0.66 to −1.

Resource size naturally also influences handling time, as it takes longer to handle larger prey. Classically, many bioenergetic models [19,20,22,23] and older functional response studies [34] assumed that resource mass is linearly proportional to digestion time leading to an isometric scaling. However, there is a more complex interplay between the degree to which time is devoted to ingestion and the subsequent digestion of the resulting resource pieces [32]. In the extremes, assuming that resource mass is negligible for the consumer, handling time does not depend on resource mass. Overall, this means that the resource-scaling exponent is expected to be in the range from 0 to 1:

1.4

Attack rates describe the feeding rate at low resource density where handling time is negligible (F_ij → a_ijN_j for N_j → 0 [4]). As a null model, we assume that the possibility of encounter of two individuals (consumer and resource) is proportional to the area or the volume that is searched per unit time while ignoring any movement of the resource (zero resource mass scaling) [47]. Following McGill & Mittelbach [47], we assume the speed of movement to scale with a 1/4 exponent [39]. The area or volume that is searched per unit time is given by the locomotive rate (distance per time, see exponent for movement) and (i) the visual search width for surface animals (assumed proportional to body length: 1/3 exponent) or (ii) the visual search area by animals hunting in volumes (assumed proportional to squared reaction distance: two-third exponent) [47]. Subsequently, the expected scaling exponents for consumer mass become 0.58 (1/4 + 1/3) and 0.92 (1/4 + 2/3) for consumers in two- and three-dimensional environments, respectively [47]. The visibility of the resource increases with resource body mass with a 1/3 to a 2/3 power law in dependence if the environment is two- or three-dimensional [47]. Moreover, typically environments are not purely two- or three-dimensional due to habitat structure [47] leading to a broad continuum of possible slopes:

1.5

Analogies similar to speed of movement and reaction distance/area apply for animals with different feeding strategies (e.g. filtrators, sit-and-wait, nocturnal or olfactorial consumers); different exponents may naturally be achieved. Furthermore, this simple null hypothesis neglects all behavioural aspects.

(c). Temperature dependence

Biological rates depend not only on body masses but also, as all chemical reactions, on temperature. The Metabolic Theory of Ecology [38] extends the allometric descriptions of metabolic rate, I, with the addition of the Arrhenius temperature dependence [48–50] such that: where E_I (eV) is the activation energy describing the exponential increase in the metabolism with temperature, k (eV K⁻¹) the Boltzmann constant, T (K) is the absolute temperature and T₀ sets the intercept of the temperature relationship at T₀, rather than at zero Kelvin [11,29]. By applying the same idea to the handling time and attack rate relationships, equations (1.4) and (1.5) yield

1.6

and

1.7

where E_h and E_a refer to the activation energy of handling time and attack rates, respectively. Gillooly et al. [50] argue that for metabolic rates the expected range of activation energies fall in the range of 0.6–0.7 eV, as this is the average of all biochemical reactions. For simplicity, many studies normalize or test their data using the value of 0.65 (see Brown et al. [38] for an example). Other theories predict a larger range of possible activation energies ranging from 0.46 to 0.96 [51]. As the maximum feeding rate is the inverse of the handling time (see above), we expect an activation energy of from −0.96 to −0.46 for handling time. As there is no conceptional background for the temperature dependence of attack rate or movement, we use as a null expectation the average biological activation energy of 0.65 eV (ranging from 0.46 to 0.96 [51]) for the attack rate.

(d). Deviations from the global models: going to the extremes

Besides the log-linear ‘global’ models, many case studies document hump-shaped relationships for attack rates with consumer size or body-mass ratio [36,41,52–55] (figure 1c, humps). Also, humps with increasing temperature were reported for attack rates [56] and maximum feeding rates [57] as summarized by Englund et al. [30]. Especially, for attack rates in dependency of the body-mass ratio, humps are often reported [36,41,52–54], whereas they appear only in approximately 40 per cent of all studies for temperature [30]. Additionally, the parameters describing the humps such as optimal foraging mass ratio vary across consumer–resource pairs [52] and the shape of the humps with increasing temperature (also known as thermal windows [58,59]) even varies with ontogenetic stage. Moreover, these humps are hidden within a large scatter of data points (figure 1c) and can vary in width (figure 1d). Here, analyses of hump-shaped relationships were constrained to studies that were designed to investigate scaling with body mass or temperature.

(e). Expectations

We compiled and analysed the largest database of functional responses currently available. We investigated how handling times and attack rates depend on consumer mass, resource mass and temperature. Moreover, we collected data on the metabolic group of the consumers (endotherm vertebrates, ectotherm vertebrates, invertebrates and unicells), and the ecosystem types (freshwater, marine and terrestrial ecosystems), following Brose et al. [60]. With this dataset, we test the scaling relationships of attack rates and handling time with temperature, consumer and resource masses as described earlier. These analyses are carried out at global (pooling all data) and local resolutions (more fine-grained analyses).

(a). Data collection

We searched ISI Web of Science and Google Scholar for the keywords ‘functional response’, ‘feeding rates’ with ‘prey/resource density’ and ‘non linear interaction strength’. Additionally, we searched bibliographies of found publications for further literature. We included studies that provided information only about (i) temperature; (ii) consumer and resource identity; and (iii) the spatial size of the experimental units. If information on body masses was not provided, we searched for secondary literature to find average length or body masses of the species. If only length was given, we used allometric equations to calculate the mass of the organisms according to Peters [39]. We used studies that provided attack rates and handling times of type II and type III functional responses estimated via (i) Holling's functions [31], (ii) Rogers random equation [61,62], or (iii) time-series analyses (mainly occurring in Hansen et al. [34]). If a study included data that were not statistically analysed to provide values for attack rates or handling times, we refitted the data with Rogers random equation or Holling's function. The final functional response database included 648 functional responses from 86 studies including unicells (n = 22), invertebrates (n = 472), ectotherm vertebrates (n = 143) and endotherm vertebrates (n = 11) from marine (n = 86), freshwater (n = 273) and terrestrial (n = 288) ecosystems.

(b). Statistical procedures

We used the log-linear version of (1.6) and (1.7) yielding

2.1

and

2.2

to fit a linear model to the data. Units were K for T, mg for m, s for h_ij, m^{2 or 3} for a_ij and we set T₀ to 293.15 K. We applied a ‘random intercept model’ [63] using mixed effect models with a maximum-likelihood estimator (function ‘lme’ with ‘method = ML’ within the ‘nlme’ package [64] of the statistics program R [65]). To correct for differences between studies such as length of the experiment, moisture, pH, experimental procedure and the foraging dimension (two- versus three-dimensional), we used the study identity as a random factor in the handling-time model and a nested formulation of dimension/study identity as random factors in the attack-rate model. However, we could not apply a random intercept and slope model, because none of the studies included all three continuous variables (body mass of resource, consumer and temperature), and the majority of studies only provided few data points.

We fitted (2.1) and (2.2) starting with a model including free allometric slopes and a free activation energy and a full model including interactions of metabolic ecosystem type (including all possible combinations of ecosystem types with species metabolic types) with allometric slopes and activation energy. Then, we used the automated ‘stepAIC’ function [66]. This function adds or deletes automatically interactions or parameters from or to a model until the lowest AIC (Akaike information criterion) is reached, in order to find the most parsimonious model. We started with both models described earlier to avoid a local minimum.

Subsequently, we tested for hump-shaped deviations from the global model in allometry (figure 1c) and temperature (figure 1d) dependence. These deviations are assumed to occur at a narrow taxonomic range in allometry [52] and (at least for metabolism and growth) for temperature [58,59]. Moreover, these humps may be overlaid by the general global intra-taxonomic trend (figure 1c). Hence, we used only studies that were designed to investigate allometric or temperature scaling of functional response parameters. Hence, we saved residuals of the most parsimonious models and built (i) subsets only including studies that measured functional responses across a body mass gradient of a close taxonomic group with at least five measurements, and (ii) a subset including studies measuring functional responses on a temperature gradient with at least four functional responses. We fitted polynomial models, including a linear and a quadratic term to these subsets:

2.3

and

2.4

where ɛ_m and ɛ_T are the residuals from the body mass studies and from the temperature models, respectively, R is the consumer–resource body-mass ratio and T_C is the temperature in degree Celsius. We used the body-mass ratios instead of consumer or resource masses to normalize the independent variable. To correct for different linear deviations from the global intraspecific model while focusing on hump deviations, we applied a mixed effects model to the residual data. We used a ‘random intercept and slope model’ ([63], with ln(R) or T_C as random slope) using mixed effect models with a maximum-likelihood estimator (function ‘lme’ with ‘method = ML’ within the nlme package [64] of the statistics program R [65]) and the identity of the study as the random effect influencing the intercept and the linear slope of the model.

(a). Global relationships

Across the entire dataset, higher temperatures were associated with higher attack rates and lower handling times (figure 2 and table 1). There was no evidence that the temperature scaling varied among metabolic and ecosystem types. Interestingly, attack rates increased less steeply than expected (figure 3a and table 1) and handling times decreased less than expected (figure 3d and table 1).

Also across the entire dataset, increases in consumer mass were associated with increased attack rates and decreased handling times (figures 2 and 3b,e). The scaling exponents, however, differed across metabolic and ecosystem types (figure 3b,e and table 1). The mass scaling exponents of freshwater ectotherms and unicells and terrestrial invertebrates deviated from expectations (0.58 and 0.92; figure 3b). Moreover, scaling exponents for consumer mass and handling time were lower than expected for freshwater and marine ectotherms and terrestrial invertebrates (figure 3e).

The scaling of attack rates with resource mass was most often shallower than expected (figures 2c and 3c), marine unicells exhibited significantly negative exponents. The exponent of terrestrial endotherms was most consistent with the prediction of 0.33 to 0.66, and the exponent distributions of the four other groups overlap the predicted range (figure 3c). All exponents estimated for the dependence of handling time on resource mass were lower than unity, and, as expected, we found a continuum of possible slopes ranging from 0 to 1 (figure 3f and table 1).

Together, these results suggest some general trends: (i) temperature influences functional response parameters less strongly than expected by metabolic theory, (ii) attack rates exhibited exponents deviating form theoretical predictions, and (iii) handling times decreased less with consumer mass and increased less with resource mass than expected.

(b). Local relationships

Attack rates followed hump-shaped relationships with increasing consumer–resource body-mass ratios (figure 4a, see legend for statistical outputs), whereas we found no significant deviation from the global model for the temperature dependence of attack rates (figure 4b). Handling times followed a negative hump-shaped relationship with increasing consumer–resource body-mass ratios (figure 4c) and temperature (figure 4d).

(a). Temperature dependence

Consistent with prior functional response studies [28,30], we found that the activation energies of attack rates (0.42) and handling times (−0.30) were lower than those found for metabolism (0.38–0.80 for invertebrates [67], 0.43–0.79 for unicells to mammals [50]). This suggests that feeding increases less strongly with temperature than metabolism, which supports prior conclusions that consumer biomass should decrease with warming [10]. Interestingly, this implies that warming should decrease interaction strengths at per capita (lower individual consumption rates relative to metabolism) and at population levels (lower consumer population densities). While these weaker interaction strengths may imply a higher stability of populations and more persistent communities [3,4,68], they may also cause consumer starvation [10,69].

Consistent with previous studies [30,56,57], our results suggest that the temperature dependence of functional response parameters is more complex than simple Arrhenius terms. While the positive quadratic deviation we found in the residual analyses for handling time supports these prior conclusions, we found no hump-shaped relationship of attack rates with increasing temperature. These and similar additional complexities found for metabolic scaling and growth rates [67,70] imply that temperature effects on organismic biology may go beyond simple thermodynamics [58,59]. Ultimately, more complex models accounting for biological processes under the influence of temperature need to be developed.

(b). The allometry

We found systematic scaling relationships of handling times and attack rates with consumer and resource masses. However, these empirical relationships differed significantly from predictions by simple concepts based on the allometry of metabolism and from more complex models based on visibility and movement. Focusing on the three groups that had the highest number of data points (n>100, terrestrial invertebrates, freshwater invertebrates and ectotherm vertebrates) reveals that all three groups exhibited exponents generally lower than expected by theory (0.58 for two-dimensional searchers to 0.92 for three-dimensional searchers; table 1 and figure 3 [47]). Also, the exponents for resource mass were lower than expected, only the exponent for terrestrial invertebrates was not different from 0.33. These two observations suggest that the scaling of attack rates with consumer and resource masses implies more than simple changes in movement speed, visual search width or area and visibility. One potential explanation for the disparity between theoretical predictions and empirical data could be that many of the studies included here added habitat structure and this may lead to the breakdown of the model predictions [47]. In addition to encounter, realized attack rates also depend on the success rates [27], and the interplay of these two components can yield hump-shaped relationships between attack rates and consumer–resource body-mass ratios [36,41,52]. Strikingly, our residual analyses confirmed these hump-shaped deviations from the allometric scaling models [36,41,52].

The handling times were expected to follow to −1 power-law relationships with consumer mass [38,45,46]. We found that only half of the groups tested fall into this range, whereas the others were systematically higher. Interestingly, the groups represented by most data (n > 100, freshwater ectotherm vertebrates, invertebrates and terrestrial invertebrates; table 1) showed smaller exponents. We found a variety of exponents for the relationship between handling time and resource mass ranging from zero to unity. Traditionally, assuming two resource items each of 1 g will take the same time to handle as one resource of 2 g led to linear scaling models in functional response studies and theoretical models [19,20,22,23,34]. However, assuming digestion processes to be dependent (i) on the degree the resource is broken down into pieces or (ii) that digestion takes place only on the surface of the resource item, this relationship can be dampened. Moreover, handling not only includes digestion of the resource but also aspects of subduing that additionally could influence the exponent (see Jeschke et al. [32] for a detailed discussion on the components of handling time). Assuming that the time to subdue a resource is constant and independent of resource mass implies that it takes longer to handle two small resources than one large, because the time to subdue the resource is needed only once. Moreover, the large variation in the resource–mass slopes across the different combinations of ecosystem and metabolic types might be caused by substantial variation in morphological traits (e.g. external versus internal digestion, swallowers versus chewers) of the resource that is independent of allometry.