In order to estimate the scaling of carnivore energy budgets, we develop a simple model that incorporates both the scaling of costs associated with body mass and the differences in time and energy budgets associated with hunting strategy. In our model, predatory carnivores are divided into two basic hunting groups (following [

3]): those that feed on small prey (invertebrates <10 g and small vertebrates <2 kg) and those that feed on large prey (large vertebrates, typically >10 kg). We calculate daily energy expenditure (DEE) (kilojoules/day) as

*E*_{r}T_{r} +

*E*_{h}T_{h}, where

*E*_{h} and

*E*_{r} are the energetic expenditure rates while hunting and resting (kilojoules/hour) and

*T*_{h} and

*T*_{r} are time hunting and resting, respectively (hours/day). The scaling of the energetic costs of resting and hunting (in watts) was estimated from the allometric equations [

2,

20]

*E*_{r} = 5.5

*M*^{0.75} and

*E*_{h} = 10.7

*M*^{0.684} *v* + 6.03

*M*^{0.697}, where

*M* is body mass in kilograms and

*v* is the average speed (meters/second) during hunting [

20] (see

Materials and Methods). These equations allow us to estimate DEE, taking into account the effects of body mass and travel speed while hunting. We expect to see an increase in resting and hunting costs associated with an increase in size, but additionally, we expect a sharp rise in hunting costs as carnivores switch from feeding on small prey to large prey (see

Materials and Methods).

The model predictions for DEE (using parameter values in Materials and Methods) are compared against estimates of DEE for 14 species of free-ranging wild carnivores (

Text S1) (A). Given the higher travel speeds of the large-prey feeders, the model estimates an abrupt 2.3-fold increase in DEE with the switch to large-prey feeding. Consistent with this prediction, we find that a piecewise regression [

21] provides a significantly better fit to the observed values than a simple linear regression (). This analysis, however, is limited by the number of direct estimates of DEE obtained in the field using doubly labeled water.

| **Table 1**Three Alternative Models Were Fitted to Both DEE and DEI Values *(E)* in Relation to Species Body Mass *(M)* |

To provide a more robust test of the predicted increase in energy expenditure, we compared our model predictions against a surrogate for DEE, daily energy intake (DEI, kilojoules/day)—a measure that is more readily obtained in the field, and for which we found estimates for 32 species (see

Materials and Methods,

Text S1). As carnivores switch to hunting large prey, with a more than doubling of DEE, we expect a corresponding increase in DEI. We predicted DEI (kilojoules/day) from the model predictions of DEE, using the equation DEI = DEE/0.66. The constant was derived from an estimate of proportional assimilation efficiency for the value of animal food [

13] (B). Again, the observed pattern in DEI is consistent with the model predictions of DEI and we find a step change in observed DEI at the point where carnivores switch to large-prey feeding, with the piecewise regression model and a sigmoid regression providing significantly better fits than a simple linear regression of log intake rate against log body mass (see

Materials and Methods) (B).

Across all carnivores, the scaling exponents of DEE (0.74 ± 0.10, 95% confidence interval) and DEI (0.79 ± 0.09, 95% confidence interval) () are similar to a three-fourths scaling expected based on metabolic theory [

22] and the observed scaling exponent of FMR (used interchangeably with DEE) of 0.77 based on 58 eutherian mammals [

12]. However, within small- and large-prey dietary groups, we see a secondary scaling pattern with DEE and DEI exhibiting a shallower scaling (0.58 ± 0.11 and 0.6 ± 0.09, respectively). In order to illustrate these primary and secondary scaling patterns in our surrogate for expenditure, DEI, we re-plot the data from B with the best-fitting regression, the sigmoid model, together with trend lines based on the scaling exponent of 0.77 from FMR (converted to DEI) (C). These trend lines illustrate upper and lower boundaries of 1.4 and 0.4 times estimated DEI, respectively, estimated from extremes in observed FMR in carnivores [

2,

13,

23,

24].

Within each dietary group, relatively small carnivores exhibit costly strategies (falling near the upper boundary); while relatively large carnivores exhibit energy conserving strategies (near the lower boundary). This suggests that within each dietary group, expenditure and intake rates impose increasing constraints as size within groups increases, leading to the shallower exponents within groups. The sigmoid curve fit of observed values of DEI intersects the upper and lower boundary of predicted DEI at masses near the size limits of the two dietary groups. At the upper size range of the large-vertebrate-prey feeders, the sigmoid curve intersects the lower boundary just above 1,100 kg, near the maximum mass estimated for some extinct carnivores (see below). At the lower end of the small-prey feeders, the sigmoid curve intersects at around 100 grams near the mass of the smallest carnivore species. The qualitative trend suggests that these upper and lower limits to DEE and DEI represent constraints on behavioral and metabolic adjustments, which then impose constraints on carnivore size.

In order to estimate net rate of gain (kilojoules/day) for these two dietary groups, the model predictions of DEE were combined with observed estimates of intake rate. Net gain

*G* (kilojoules/day) was calculated as 0.66

*IT*_{h} −

*E*_{r}T_{r} −

*E*_{h}T_{h}, where

*I* is intake rate per hour hunting (kilojoules/hour) and 0.66 is the assimilation coefficient (described above). We estimated the maximum sustainable mass for each dietary group by approximating the mass where

*G* approached zero. Maximum intake rates for small-prey feeders were estimated as 746 and 1,496 kJ/h for invertebrates and small vertebrates, respectively (see

Materials and Methods). For large-vertebrate-prey feeders, prey size increases with predator size [

3,

4,

25], so we assumed that intake rate (kilojoules/hour) scaled with body mass

^{0.6}, with an intercept of 1,010 kJ/h at 1 kg body mass based on the observed scaling of DEI (converted to an hourly rate, ) estimated from the above piecewise regression. We also used a maximum intercept of 3,132 kJ/h based on two well-studied large vertebrate hunting species (African wild dog

Lycaon pictus and gray wolf

Canis lupus) with very high rates of intake (see

Materials and Methods). The model predictions in represent the upper limits of carnivore mass for the two dietary groups, and this is shown against observed estimates of net gain for several carnivore species for which we found independent estimates of DEE and DEI ().

Using the above calculations, we predict a maximum carnivore mass of 18 or 45 kg (for invertebrate and small-vertebrate-prey feeders, respectively, ). Using estimates of intake rate for large-vertebrate prey, we predict a maximum carnivore mass of approximately 700 kg or 1,100 kg (for the average and maximum intercepts, respectively). These size limits are shown against the estimated masses of some of the largest known extinct mammalian carnivores. While we estimate that small prey can sustain carnivores up to 18–45 kg, above 14.5 kg carnivores feeding on large prey achieve a higher net gain.