## Abstract

The latitudinal gradient of increasing biodiversity from poles to equator is one of the most prominent but least understood features of life on Earth. Here we show that species diversity can be predicted from the biochemical kinetics of metabolism. We first demonstrate that the average energy flux of populations is temperature invariant. We then derive a model that quantitatively predicts how species diversity increases with environmental temperature. Predictions are supported by data for terrestrial, freshwater, and marine taxa along latitudinal and elevational gradients. These results establish a thermodynamic basis for the regulation of species diversity and the organization of ecological communities.

Global gradients in biodiversity exist for all major groups of terrestrial (1), freshwater (2), and marine taxa (3), but the general principles underlying their origin and maintenance remain unclear (4, 5). Here we present a theoretical framework that explains gradients of species diversity in terms of energetics. Our model is derived by extending the well-established “energetic-equivalence rule” (6) to include temperature. In its original form, the energetic-equivalence rule states that the total energy flux of a population per unit area,*B*
_{T}, is invariant with respect to body size. Species of different size have similar values of*B*
_{T} because individual metabolic rates,*B*
_{i}, increase with body size,*M*
_{i}, as *B*
_{i} ∝*M*
_{i}
^{3/4}, whereas population densities per unit area, *N*
_{i}, decrease with body size as*N*
_{i} ∝*M*
_{i}
^{−3/4}(*B*
_{T} =*N*
_{i}
*B*
_{i} ∝*M*
_{i}
^{−3/4}
*M*
_{i}
^{3/4}= *M*
^{0}). This inverse relation between abundance and body size is observed for plants and for endothermic and ectothermic animals; it reflects mechanistic connections between individual metabolic rates, rates of energy flux by populations, and the partitioning of available energy among species in a community (6, 7).

We can extend the energetic-equivalence rule to include temperature by incorporating the biochemical kinetics of metabolism. Recent work has shown that whole-organism metabolic rate varies with body size and temperature as *B* =*b*
_{o}
*M*
^{3/4}
*e*
^{−E/kT}(8), where *b*
_{o} is a normalization constant independent of size and temperature (∼2.65 × 10^{10} W g^{−3/4}) (9). The Boltzmann factor, *e*
^{−E/kT}, describes the temperature dependence of metabolic rate, where *E* is the activation energy of metabolism (∼0.78 eV or ∼1.25 × 10^{−19} J) (9), *k* is Boltzmann's constant (8.62 × 10^{−5} eV K^{−1}), and*T* is absolute temperature (K). The total energy flux of a population is therefore *B*
_{T} =*N*
_{i}
*B*
_{i} =*N*
_{i}
*b*
_{o}
*M*
_{i}
^{3/4}
*e*
^{−E/kT}, which yields(1)where constancy of *C*
_{0} = ln(*B*
_{T}/*b*
_{o}) with respect to temperature follows from our extension of the energetic-equivalence rule. Temperatures of ectotherms are approximately equal to ambient environmental temperatures, *T*
_{env}, whereas temperatures of endotherms are ∼40°C. Equation 1 therefore leads to three predictions for the relation between population density and temperature: (i) for ectotherms, the natural logarithm of mass-corrected population density should be a linear function of 1000/*T*
_{env}; (ii) the slope of this linear relation should be *E*/1000*k* ≈ 9.0 K for both plants and animals because the two groups share similar activation energies for metabolism (8); and (iii) for endotherms, mass-corrected population density should be independent of*T*
_{env}.

Abundance data compiled on a variety of plant and animal species provide strong support for all three predictions. First, the natural logarithm of mass-corrected population density for tree species throughout the world shows a positive, linear relation to inverse absolute temperature (Fig. 1A). Second, the 95% confidence interval (CI) for the slope of this relation includes the predicted value of 9.0 K (*x̄*; 95% CI, 7.66 to 10.17). Data on mass-corrected population density for terrestrial vertebrate- and invertebrate-ectotherms also support the first and second predictions (slope *x̄*; 95% CI, 6.61 to 16.88) (Fig. 1B). Finally, in accordance with the third prediction, mass-corrected population density shows no significant relation to environmental temperature for endothermic mammals (slope*x̄*; 95% CI, −0.82 to 2.73) (Fig. 1C).

To control for body temperature, we multiply the population densities in Fig. 1, B and C, by the Boltzmann factor,*e*
^{−E/kT}, assuming a temperature of 40°C for mammals and temperatures equal to*T*
_{env} for ectotherms. When temperature-corrected population density is plotted against body size on a natural logarithmic scale, endotherms and ectotherms fall on approximately the same line (Fig. 2). Moreover, the slope of this relation is close to the predicted value of −3/4 (*x̄* = −0.78; 95% CI, −0.82 to −0.74). Given that the intercept in Fig. 2 provides an estimate of *C*
_{0} = ln(*B*
_{T}/*b*
_{o}), that*e*
^{C0} =*e*
^{−}
^{19.63}= *B*
_{T}/*b*
_{o} = 2.95 × 10^{−9} km^{−2} g^{3/4}, and that*b*
_{o} ≈ 2.65 × 10^{10} W g^{−3/4} (9), we can estimate*B*
_{T} to be ∼80 W km^{−2} regardless of taxon, body size, temperature, or geographic location.

Having established temperature invariance for*B*
_{T}, we can now use the energetic-equivalence rule to predict changes in the diversity of ectotherms along temperature gradients. The average population density in a community composed of *J* individuals and *S* species isN̅ = *J/AS*, where *A* is the area of the plot delimiting the community and (10). The average metabolic rate of an ectotherm isB̅ =B̅o̅
*e*
^{−E/kTenv}, where B̅o̅ =*b*
_{o}
M̅3̅/̅4̅and is calculated on the basis of the frequency distribution of body sizes for species constituting the community of interest. Holding*A* constant across community samples,B̅T̅ = *N̄ B̅
* = (J/AS)B̅o̅
*e*
^{−E/kTenv}and(2)where *C*
_{1} = ln [(B̅o̅/B̅T̅)(J/A)] is assumed to be independent of temperature. This, in turn, requires two assumptions: temperature invariance for abundance, *J/A*, and temperature invariance for the average derived from the body size distribution, M̅3̅/̅4̅, which affects B̅o̅. These assumptions are supported by the approximate invariance of plant size and abundance across latitudes (11). Still, the model is relatively robust to departures from these two assumptions because richness is predicted to vary exponentially with temperature but less than linearly with both average body size, due to its ¾-power scaling exponent, and total abundance, due to sampling properties of species-abundance distributions (12). Thus, for example, the predicted 50-fold increase in tree diversity moving from boreal (−5°C) to tropical (30°C) forests as a consequence of biochemical kinetics (*e*
^{−E/k(273 + 30)}/*e*
^{−E/k(273 − 5)}) = 50) should overwhelm any effects attributable to changes in average tree size or total tree abundance (<threefold) (11). We take these two assumptions as working hypotheses for the groups of ectotherms considered here but note that they are not expected to hold true for groups that are narrowly defined (e.g., pine trees) or that are strongly regulated in abundance and/or body size by temperature (e.g., reptiles). The model can be extended to explicitly account for temperature effects on other variables.

Equation 2 yields three predictions for the relation of species diversity to temperature: (i) the natural logarithm of species richness should be a linear function of 1000/*T*_{env} for ectotherms; (ii) the slope should be −*E*/1000*k* ≈ −9.0 K along both latitudinal and elevational gradients in temperature for terrestrial taxa; and (iii) the slope should also be −9.0 K for aquatic taxa because they share a similar activation energy for metabolism (8).

We find strong support for predictions (i) and (ii) by using two independent data sets on tree diversity along gradients of latitude in North America and elevation in Costa Rica (Fig. 3, A and B). For both data sets, the relation between the natural logarithm of species richness and inverse absolute temperature is approximately linear, and the slopes are both close to the predicted value of −9.0 K (Table 1). Data on amphibian richness along latitudinal and elevational gradients also support these two predictions with slopes statistically indistinguishable from each other and from those observed for the trees (Fig. 3, C and D). Finally, latitudinal data on riverine fish, marine gastropods, and even the number of fish ectoparasite species per host support all three predictions of the model (Fig. 4). Overall, we see that the slopes are consistently close to the predicted value of –9.0 K (*x̄* = −9.21; range, −7.17 to −10.81), although the confidence intervals do not always include this value.

We do not mean to imply that temperature is the only variable that affects biodiversity. The significant residual variation about the relations in Figs. 3 and 4 emphasizes the importance of other variables including biogeographic history (13), habitat heterogeneity (14), area (15), and geometric constraints on species distributions (16). Indeed, we only predict the slopes of the diversity-temperature plots. The intercepts may vary by taxon, habitat, and sampling method. In particular, N̄, and thereforeB̅T̅, will vary as functions of plot size *A*, even if *J*/*A* is held constant, because *S* increases nonlinearly with *J* as a consequence of sampling, turnover in species composition through space (12), and the fractal-like distribution of habitat (10, 14).

Nevertheless, our model accounts for much of the variation in biodiversity (Table 1). Of more importance, it yields testable, quantitative predictions based on first principles of biochemical kinetics and provides a theoretical framework for understanding how temperature and productivity regulate biodiversity. The species-energy hypothesis proposes that biodiversity is positively correlated with productivity because more productive environments contain more individuals and can therefore support more species populations above some minimum size required for persistence (1,17). Data for endotherms support this hypothesis. The average population densities of mammals are temperature invariant. This implies that the observed increase in mammal diversity toward the tropics (1) results from an increase in total density,*J*/*A*, for mammals. By contrast, average population densities of trees and other ectotherms show an inverse Boltzmann relation to temperature (N̅ ∝*e ^{E/kT}
*). This result, combined with the observed Boltzmann relation of diversity to temperature for independent data collected on a variety of ectothermic taxa (

*S*∝

*e*

^{−E/kT}), supports our model assumption that total ectotherm abundance is approximately independent of temperature (J/A = N̅

*S*∝

*e*

^{0/kT}).

Temperature influences the diversity of terrestrial and aquatic ectotherms primarily through its effects on the biochemical kinetics of metabolism. Metabolic rates, in turn, dictate resource requirements at the level of the individual and rates of resource supply required of the ecosystem to maintain communities composed of multiple individuals. Evolutionary rates are ultimately constrained by generation times of individuals and mutation rates. Both of these rates are correlated with metabolic rates and show the same Boltzmann relation to temperature (9, 18). Our results therefore support the hypothesis that elevated temperatures increase the standing stock of species by accelerating the biochemical reactions that control speciation rates (5).

↵* To whom correspondence should be addressed. E-mail: drewa{at}unm.edu