An equation of state unifies diversity, productivity, abundance and biomass

To derive the relationship among macro-level ecological variables, which would constitute an ecological analog of the thermodynamic equation of state, we introduce a fourth state variable, B, the total biomass in the community. The ecological analog of the thermodynamic equation of state, an expression for biomass, B, in terms of S, N, and E, arises if we combine METE with a scaling result from the metabolic theory of ecology (MTE)^18,21. In particular, we assume the MTE scaling relationship between the metabolic rate, (varepsilon ,) of an individual organism and its mass, m: (varepsilon sim {m}^{3/4}). Without loss of generality²², units are normalized such that the smallest mass and the smallest metabolic rate within a censused plot are each assigned a value of 1. With this units convention, the proportionality constant in this scaling relationship can be assigned a value of 1. From the definition of the structure-function, it follows²³ that averaging the biomass of individuals times the abundance of species, nε^4/3, over the distribution R and multiplying by the number of species gives the total ecosystem biomass as a function of S, N, and E. Explicitly:

Both the sum and integral in the above equation can be calculated numerically, and Python code to do so for a given set of state variables S, N, and E, is available at github.com/micbru/equation of_ state/.

We can also approximate the solution to Eq. 1 analytically (Supplementary Note 2) to reveal the predicted functional relationship among the four state variables. If E >> N >> S >> 1:

where (capprox (7/2)Gamma (7/3)) ≈ 4.17 and (beta) = ({lambda }_{1}+{lambda }_{2}) is estimated^13,22 from the relationship (beta {{{{{rm{ln}}}}}}(1/beta )approx S/N). Equation 2 approximates the numerical result to within 10% for 5 of the 42 datasets analyzed here, corresponding to N/S greater than ~100 and E/N greater than ~25. Multiplying the right-hand side of Eq. 2 by (1-1.16{beta }^{1/3}) approximates the numerical result to within 10% for 33 of the 42 datasets analyzed here, corresponding to N/S greater than ~3 and E/N greater than ~5. The inequality requirements are not necessary for the numerical solution of Eq. 1, which is what is used below to test the prediction.

Empirical values of E and B can be estimated from the same data. In particular, if measured metabolic rates of the individuals are denoted by ({varepsilon }_{i},) where i runs from 1 to N, then E is given by the sum over the ({varepsilon }_{i}) and B is given by the sum over the ({{varepsilon }_{i}}^{4/3}.) Similarly, if the mass, m_i, of each individual is measured, then B is the sum over the m_i and E is the sum over the m_i^3/4. In practice, for animal data, metabolic rate is often estimated by measuring mass and then using metabolic scaling, while for tree data, metabolic rate is estimated from measurements of individual tree basal areas, which are estimators⁵ of the ({varepsilon }_{i}).

With E and B estimated from the same measurements, the question naturally arises as to whether a simple mathematical relationship holds between them, such as E = B^3/4. If all the measured m’s, are identical, then all the calculated individual (varepsilon {{hbox{‘}}}s) are identical, and with our units convention we would have E = B. More generally, with variation in masses and metabolic rates, the only purely mathematical relationship we can write is inequality between E and B^3/4: (E=sum {varepsilon }_{i}ge (sum {{{varepsilon }_{i}}^{4/3}})^{3/4}={B}^{3/4}). Our derived equation of state (Eq. 2) can be interpreted as expressing the theoretical prediction for the quantitative degree of inequality between E and B^3/4 as a function of S and N.

A test of Eq. 1 that compares observed and predicted values of biomass with data from 42 censused plots across a variety of habitats, spatial scales, and taxa is shown in Fig. 1. The 42 plots are listed and described in Table S2 and Supplementary Note 3. The communities censused include arthropods and plants, the habitats include both temperate and tropical, and the census plots range in area from 0.0064 to 50 ha. As seen in the figure, 99.4% of the variance in the observed values of B is explained by the predicted values of B.

Figure 2 addresses the possible concern that the success of Eq. 1 shown in Fig. 1 might simply reflect an approximate constancy, across all the datasets, of the ratio of E to B^3/4. If that ratio were constant, then S and N would play no effective role in the equation of state. Equation 1 predicts that variation in the ratio depends on S and N in the approximate combination S^1/4ln^3/4(1/(beta (N/S))). In Fig. 2, the observed and predicted values of E/B^3/4 calculated from Eq. 1, are compared, showing a nearly fourfold variation in that ratio across the datasets. The equation of state predicts 60% of the variance in the ratio.

Figure 3 shows the dependence on S and N of the predicted ratio E/B^3/4 over empirically observed values of S, N, and E. We examined the case in which S is varied for two different fixed values of each of N and E (Fig. 3a) and N is varied for two different fixed values of S and E (Fig. 3b). The value of E does not have a large impact on the predicted ratio, particularly when E >> N. On the other hand, the predicted ratio depends more strongly on N and S.

The total productivity of an ecological community is a focus of interest in ecology¹, as a possible predictor of species diversity²⁴ and more generally as a measure of ecosystem functioning²⁵. By combining the METE and MTE frameworks, we can now generate explicit predictions for certain debated ecological relationships, including one between productivity and diversity. Interpreting total metabolic rate E in our theory as gross productivity, then in the limit 1 << S << N << E, we can rewrite Eq. 2 to highlight the role of diversity and the other state variables in determining this quantity:

here, as shown in Eq. 2, c is a universal constant (~ 4.17). From Eq. 3, and the fact that up to a logarithmic correction (beta) varies as S/N, biomass exerts the strongest influence on E. With biomass fixed, productivity varies approximately as S^1/4, with a logarithmic correction. With biomass and species richness fixed, the dependence of productivity on abundance is logarithmic. Thus, the dependence of productivity on abundance is weakest, on biomass is strongest, and on species richness is of intermediate strength. Testing these predictions requires finding communities with the same values of pairs of state variables, and thus it is most feasible to compile further tests of Eq. 1 by allowing all four variables to vary as in Figs. 1 and 2. If N >> S, then (beta) is a function of N/S and we can re-express Eq. 3 entirely in terms of three ratios, E/S, B/S, and N/S, rather than the four state variables, and obtain:

To date, empirical cross-ecosystem surveys of relationships among community metabolism, biomass, species richness, and abundance have largely focused on uncontrolled pairwise relationships as, for example¹, between productivity and biomass but without controlling for species richness and abundance, and found considerably more scatter in the relationship than is shown in Fig. 1. In Supplementary Note 1, pairwise comparisons among state variables are shown for the datasets used here, illustrating the absence of strong relationships among pairs of state variables when the remaining pair of state variables is unconstrained. The largest R² value in all comparisons is 0.987, between ln(B) and ln(E), which is expected as they are computed from the same data. Noting ((1-0.994)/(1-0.987)approx 0.46,) the unexplained variance in the observed ln(B) using the equation of state is less than half that of the second-best predictor, ln(E). All other tested relationships shown in Supplementary Note 1 were significantly worse.

Source: Ecology - nature.com