State Variables at and Near Steady State.
Using Eqs. 21-23, with the transition functions specified in Eqs. 35-37
and R given by Eq. 3, we obtain (see SI-C for derivations):
\(\frac{\text{dN}}{\text{dt}}=\ \left[b_{0}-d_{0}\frac{(E}{E_{c})}\right][\frac{1.21N^{\frac{4}{3}}\operatorname{}{\left(\frac{1}{\beta}\right)\left(1+\frac{4N\ln(\frac{1}{\beta)}}{3E}\right)}}{E^{\frac{1}{3}}}-\frac{3N^{2}ln(\frac{1}{\beta)}}{2E}]+m_{0}\)(38)
and
\(\frac{\text{dE}}{\text{dt}}={[w}_{0}-d_{0}\frac{(E}{E_{c})}][\frac{2.42E^{\frac{2}{3}}N^{\frac{1}{3}}}{\ln^{\frac{2}{3}}\left(\frac{1}{\beta}\right)}-\frac{2.26E^{\frac{2}{3}}S^{\frac{1}{3}}}{\ln\left(\frac{1}{\beta}\right)}]\ -\frac{w_{10}E}{\operatorname{}\left(\frac{1}{\beta}\right)}+\ m_{0}.\)(39)
For the general case, with immigration and both speciation mechanisms
operating, along with extinction, we have
\(\frac{\text{dS}}{\text{dt}}=m_{0}e^{-\mu S-\gamma}+\sigma_{1}\frac{\text{KS}}{K+S}+\sigma_{2}b_{0}\left(\frac{1.21N^{\frac{4}{3}}\operatorname{}\left(\frac{1}{\beta}\right)}{E^{\frac{1}{3}}}\left(1+\frac{4N\ln\left(\frac{1}{\beta}\right)}{3E}\right)-\frac{1.5N^{2}\ln\left(\frac{1}{\beta}\right)}{E}\right)-\frac{{1.35\ d}_{0}}{E_{c}}\frac{S^{4/3}E^{2/3}}{ln(1/\beta)}.\)(40)
Steady States. Setting the time derivatives in Eqs. 38-40 to
zero, we obtain the following relationships among the static values of
the state variables and the parameters that describe the dynamics. From
Eq. 38:
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E=E_{c}\frac{b_{0}}{d_{0}}(1+\delta_{E})\)(41)
and from Eq. 39:
\(N=[\frac{0.41w_{10}}{w_{0}-d_{0}\frac{(E}{E_{c})}}]^{3}E\left(1-\delta_{N}\right)\text{.\ \ \ \ }\)(42)
The correction terms, \(\delta_{E}\ \)and \(\delta_{N},\ \)are of order\(\frac{{(m}_{0}}{b_{0})\left(\frac{E^{\frac{1}{3}}}{N^{\frac{4}{3}}}\right)}\)and \({(S/E)}^{\frac{1}{3}}\), respectively, which will generally be
<< 1.
From Eq. 40, for the immigration-only case
(\(\sigma_{1}=\ \sigma_{2}\)= 0), we have
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S}e^{3\mu\frac{S}{4}}=[\frac{0.41m_{0}ln(\frac{1}{\beta)}}{d_{0}(\frac{E}{E_{c})}}]^{\frac{3}{4}}E^{\frac{1}{4}}\text{\ \ }\)(43)
For the case \(m_{0}=\ \sigma_{2}=0,\) and S>> K ,
\(S=[\frac{{1.35\sigma}_{1}\ln(\frac{1}{\beta)}}{d_{0}(\frac{E}{E_{c})}}]^{3/4}E^{\frac{1}{4}}\)(44)
If S << K , then
\(S=[\frac{{1.35\sigma}_{1}\ln(\frac{1}{\beta)}}{d_{0}(\frac{E}{E_{c})}}]^{3}E\)(45)
Finally, for the case \(m_{0}=\sigma_{1}=0:\)
\(S=[\frac{{0.9\sigma}_{2}b_{0}}{d_{0}(\frac{E}{E_{c})}}]^{3/4}\ln(\frac{1}{\beta)}E\)(46)
The Steady-State Species-Area Relationship (SAR). We can derive
nested SARs from Eqs 43-46 because in a nested design,Ec , E , and N scale linearly with
area. In the immigration-only model, taking the logarithm of Eq. 43
gives:
\(\ S=\ \frac{\ln\left(E\right)}{3\mu}+\frac{\ln\left(m_{0}\right)}{\mu}-\frac{\operatorname{4ln}\left(S\right)}{3\mu}+\frac{\ln\left(\ln\left(\frac{1}{\beta}\right)\right)}{\mu}+\frac{\ln\left(\frac{0.41E_{c}}{d_{0}E}\right)}{\mu}.\)(47)
In a nested SAR design, Ec , E , andN scale linearly with area, and in general, E>> m 0. Hence Sscales logarithmically with area. There are ln(ln(area)) corrections
arise from the third and fourth terms (see Eq. 6) on the right-hand side
of Eq. 47 and the fifth term contributes a constant. Ifm 0 scales as a power of area, then the second
term also gives a ln(area), which is generally smaller than the first
term.
For sufficiently small S in Eq. 43, the exponential term can be
ignored, and then the first term on the right hand side of Eq. 47 can be
set equal to the third term. This results in S ~\(m_{0}^{\frac{3}{4}}E_{0}^{\frac{1}{4}}\ \sim\ \text{area}\ \)ifm 0 scales linearly with area, and S
~ area1/4 if m 0is scale-independent. This limit applies when S < 1/\(\mu\),
which for a typical 50 ha tropical forest plot is approximately S< 50 (see SI-E).
The METE SAR was derived (Harte et al. 2009) from the MaxEnt-predicted
species- abundance and the species-level spatial-occupancy
distributions. In contrast, the DynaMETE prediction does not depend on a
spatial distribution function, but it does depend on the choices made
for the mechanisms that govern the transition functions, f ,h , and q and that led to Eq. 41. Nevertheless, METE and
the static limit of DynaMETE in the migration-only model predict the
same approximate functional form for the SAR: S ~
ln(area) with ln(ln(area)) corrections. Numerical simulations for a
variety of choices of state variables show that, beyond this functional
similarity, they predict nearly overlapping SARs (see for example Fig.
2).
In the first speciation model, from Eq. 44 with S>> K we get quarter-power scaling of
species richness with E (up to a \(\ln\left(1/\beta\right)\)correction) and therefore with area. In that model, with S
<< K, and for any S in the second speciation
model, S scales linearly with area, up to a\(\ln\left(1/\beta\right)\) ~
ln(N )-ln(S ) correction. The first model, with a saturation
term K that is small compared to the steady state species
richness, thus yields a more realistic species-area relationship.
A Productivity-biomass-diversity-abundance relationship at
steady state. Metabolic scaling informs us that individual mass
(m ) is related to individual metabolism by\(m(\varepsilon)=\ m(1)\varepsilon^{4/3}\). Summing over the
steady-state structure function, total biomass, B , is then
(following the methods in SI-C):
B \(=m(1)S\sum_{n,\varepsilon}{\varepsilon^{\frac{4}{3}}nR(n,\varepsilon}\left|S,N,E\right)=\ m(1)\frac{3.57E^{\frac{4}{3}}}{S^{\frac{1}{3}}ln(\frac{1}{\beta)}}\).
(48)
This community mass-metabolism relationship thus involves species
richness, and also total abundance via the ln(1/\(\beta)\ \)term.
Interpreting the state variable E as total net productivityP of the community, Eq. 48 provides a relationship among
productivity, biomass, species richness and abundance:
\({{P=0.385\ B}^{\frac{3}{4}}S}^{\frac{1}{4}}\ln^{\frac{3}{4}}(\frac{1}{\beta)}\)(49)
where B and P are measured in units such that
m(\(\varepsilon=1\))=1. Eq. 49 does not depend on whether migration,
speciation or a combination of both contributes to diversification
because Eq. 40 was not used in its derivation. Nor does it depend on the
forms of the transition functions and the rate constants, which will
differ from habitat to habitat.
Noting the different scaling exponents in the contribution of biomass
and species richness to P , and that ln(1/\(\beta)\ \) varies
approximately as ln(N ) – ln(S ), the influence of biomass
on productivity is considerably stronger than that of species richness,
which in turn is stronger than that of abundance, which only enters via
the ln(1/\(\beta)\) term. Empirical surveys of the productivity-biomass
relationship (Ghedini et al. 2018; Jenkins 2015; Niklas 2007) are
qualitatively consistent with these results but extensive analysis will
be required to test Eq. 49.
Eq. 49 is an “ideal biodiversity law”, an analog of the ideal gas law
that relates thermodynamic state variables. Because this equation was
derived using the steady-state structure function in Eq. 3, it will
likely no longer hold under non-steady-state conditions. Following
disturbance, the full structure function (Eq. 19) is needed to derive
the productivity-biomass-abundance-species richness relationship, and it
will then depend on the details of the disturbance mechanism.
State variable dynamics near steady state. Time trajectories of
the state variables near steady state follow from Eqs. 38-40, which were
derived from the static structure function. We examine both the first
order responses of the state variables to several kinds of disturbance,
expressed by altered transition rate parameters, and the recovery to
steady state from a depleted state.
For these dynamical simulations we need to specify the transition rate
parameters. We choose a forest that resembles the 50 ha BCI tropical
forest plot (Condit et al., 2000; Condit, 1998; Condit et al., 2019;
Hubbell et al., 1999). Approximate transition rate constants for this
site are given in Table 2 and the rationale for these values is given in
SI-E.
Figures 3a–3d illustrate responses of the state variables, over 100
years, to different disturbances represented by changing the values of
the rate parameters in the transition functions. Independent of the
magnitude of the changes in these parameters, certain general patterns
emerge. A decrease in the immigration rate constant, for example under
habitat fragmentation that isolates an ecosystem from its
meta-community, results in a linear decrease in S at smallt , as well as a weak, damped oscillatory response of N ,
and nearly undetectable change in E (Fig. 3a). An increase in the
death rate (Fig. 3b) generates a slight decrease in S , an initial
large decline in N and a weaker decline in E , followed by
damped oscillatory behavior. A decrease in the ontogenic growth rate
(Fig. 3c) generates a nearly indiscernible increase in S , a large
damped oscillatory initial rise in N and weak damped oscillatory
initial decrease in E.
Figs. 3d shows the effect on state variable trajectories of combining
perturbations in migration, death and growth rates. We do not attempt
here a detailed comparison to real data because of the first order
approximation used to obtain these theoretical curves, but it is
encouraging that the time trajectories of the BCI state variables over
the period 1985-2015 (inset in Fig. 3d) also exhibit a steady decline inS , decline and then partial recovery in N , and weak
variability in E .
If we fix the transition rate parameters at their undisturbed values
(Table 2) in the immigration-only model, and initially reduce the state
variables 20% from their steady state values, their return to steady
state is shown in Fig. 4. Noteworthy is the monotonic recovery ofS , the large overshoot and then decline to steady state ofN , and a much weaker overshoot and decline of E .
If speciation is the driver of diversification, the pattern of recovery
of species richness is markedly different. In the first speciation
model, with K >> S , recovery ofS is sigmoidal and extremely slow, with 90% recovery taking
approximately 104 years. In the second speciation
model, and in the first with K << S ,S recovers to 90% of steady state in approximately 4000 years,
and at an ever-slowing, rather than sigmoidal, rate. The recovery
trajectories of N and E are nearly the same in both
speciation models and very similar to that in the immigration-only case
(Fig. 4).