The transition functions
Up until this point, we have described a very general theoretical
framework that in principle could be applied to a wide variety of
dynamical systems. To proceed we make specific model assumptions about
the mechanistic transition functions. There is no agreed-upon set of
processes governing the rates of change of n and\(\varepsilon,\ \)nor agreed-upon mathematical representations of
selected processes. As with all mechanistic modeling in ecology,
plausible mathematical representations of very complex and imperfectly
understood processes will necessarily, be simplifications.
Our choices are guided by conformity with results from metabolic theory
(Brown et al. 2004) and by scale-consistency requirements as described
below. The reasoning behind our choices of functional forms for the
transition functions are presented here, with some additional
mathematical details about diversification via immigration in SI-D.
Alternative forms for the transition functions can readily be
substituted. For notational simplicity, we suppress in this subsection
the time index for the state variables and rate constants that the
transition functions depend upon.
Contribution of birth and death tof(n, \(\mathbf{\ \varepsilon)}\). Species-level
demographics is constrained by metabolic scaling theory, so that
per-capita birth and death rate scale inversely with the 1/3 power of
the average metabolic rate of the individuals in the species (Niklas
2007; Marba et al. 2007):
\(f_{\text{birth}}=b_{0}\frac{n}{\varepsilon^{1/3}}\) (24)
Here, and in the other transitions that follow, a correlation betweenn and \(\varepsilon\) in the form of an energy equivalence
relationship (Brown et al., 2004) arises when expressions such as\(\frac{n}{\varepsilon^{1/3}}\) are averaged over the structure functionR (n , \(\varepsilon\)) (Harte et al., 2008).
The contribution of death to dn /dt also depends on\(\frac{n}{\varepsilon^{1/3}}\ \)but with a weak zero-sum constraint
which operates at the community level and arises from the value ofE , not N or n . Thus, we multiply\(\frac{n}{\varepsilon^{1/3}}\ \)by E /E c,
where E c is a soft metabolic limit:
\(f_{\text{death}}=-\frac{d_{0}\text{n\ }}{\varepsilon^{\frac{1}{3}}}\frac{E}{E_{c}}\)(25)
When we examine the spatial scaling properties of DynaMETE, the
parameter E c will also serve as a fundamental
scale parameter, with E c proportional to plot
area within a given habitat.
Contribution of death toh(n, \(\mathbf{\ \varepsilon).\ }\ \)Multiplying the death
rate in Eq. 25 by ε , we get the following contribution toh (n ,\(\ \varepsilon):\)
\(h_{\text{death}}={-d}_{0}n\varepsilon^{\frac{2}{3}}\frac{E}{E_{c}}\)(26)
We ignore the contribution of birth to dE/dt because birth
primarily partitions, rather than adds to, metabolism.
Contribution of immigration tof(n, \(\mathbf{\ \varepsilon)}\) andh(n, \(\mathbf{\ \varepsilon).}\) For a vegetation
community, we denote by m 0 the immigration rate
of seeds that result in germinants with \(\varepsilon\)= 1. The
probability that the immigrant is in an existing species with abundancen is assumed to be n /N , so:
\(f_{\text{immigration}}=m_{0}\frac{n}{N}\) (27)
Because the metabolic rate of these immigrants is 1:
\(h_{\text{immigration}}=m_{0}\frac{n}{N}\) . (28)
For an animal community, Eq. 28 would be multiplied by the metabolic
rate of individuals dispersing into the plot.
Contribution of ontogenic growth toh(n, \(\mathbf{\ \varepsilon)}\). An expression
for the ontogenic growth rate of an individual is also constrained by
metabolic scaling theory (West et al. 2001):
\(g_{\text{ontogenic\ growth}}=w_{0}\varepsilon^{2/3}-w_{1}\varepsilon\)(29)
This expression for g (n ,\(\ \varepsilon)\) (see Eq. 12) is
multiplied by n to give the contribution of ontogenic growth toh (n ,\(\ \varepsilon):\)
\(h_{\text{ontogenic\ growth}}=w_{0}n\varepsilon^{2/3}-w_{1}\text{nε}\)(30)
For reasons of scale consistency, the parameterw 1 equals a scale-independent parameter,w 10, divided by
ln2/3(1/\(\beta)\). If, instead, we assumed thatw 1 and not w 10 is scale
independent, then in the resulting equation for dE /dt, one of the
terms, -w 1E , scales isometrically with
area while the other scales as
area/\(\operatorname{}\left(\frac{1}{\beta}\right)\) or roughly
area/ln2/3(area).
Contribution of local extinction toq(n, \(\mathbf{\varepsilon}\)). We assume that
the local extinction of a species in an ecosystem occurs when, within
the local community, the last individual in that species dies. The
extinction contribution to q is then:
\(q_{\text{extinction}}=S\frac{\text{dn}}{\text{dt}}|_{death,n=1}=-\frac{Sd_{0}\frac{E}{E_{c}\delta_{n,1}}}{\varepsilon^{\frac{1}{3}}}\)(31)
where \(\delta_{n,1}\)= 1 if n = 1 and 0 otherwise.
Contribution of immigration toq(n, \(\mathbf{\varepsilon}\)). Most immigrants
will be from species in the meta-community already present in the local
community. We assume new species arriving in the local community
originate from the relatively rare species in the meta-community and
that the metacommunity is static.
The metacommunity species richness,S meta,, and total abundance,N meta, together determine \(\beta_{\text{meta}}\)(Eq. 6). From this, the fraction of immigrants that are new species
entering the \(\varepsilon\) = 1 cohort can be estimated. Using a
log-series meta-community SAD, and summing over the high rank (i.e. low
abundance) species from the highest ranked to a rank equal toS meta – S , which by our assumption are
the species not already present in the local community, we derive (see
SI-D for derivation):
\(q_{\text{immigration}}=m_{0}e^{-\mu S-\gamma}\) (32)
where\({\mu=ln(\frac{1}{\beta_{\text{meta}}\frac{)}{S_{\text{meta}},}}\text{\ β}}_{\text{meta}}\)is calculated from Eq. 6 using the values ofS meta and N meta which in
turn are estimated SI-E, and \(\gamma\) is Euler’s constant,
~0.577.
Contribution of speciation toq(n, \(\mathbf{\varepsilon}\)). Absent a more
complete understanding of how the speciation rate depends on community
variables, we examine two expressions for this rate, both of which are
certainly simplifications (see, for example, Smith et al. 2014). In one,
the speciation rate is proportional to the total number of species
(Rabosky, 2013). Motivation for this comes from the fossil record.
Following major extinction events, recovery of a diverse biota generally
begins slowly and then accelerates (Kirchner & Weil 1995). The
acceleration of diversification at small S is expected if the speciation
rate is approximately proportional to species richness. Hence in
speciation model 1 we have;
\(q_{speciation\_1}\) \(=\sigma_{1}\frac{\text{KS}}{K+S}\) (33)
where K is a saturation term.
In an alternative speciation model, each species would have a speciation
rate that is proportional to its population size divided by the turnover
time of individuals in the species (see, for example, Weiser et al.
2018). In other words, each birth has an equal chance of resulting in a
new species, and so species with large n and small\(\varepsilon\), and thus higher birth rate, have the highest speciation
rates:
\(q_{speciation\_2}=\ \frac{{\sigma_{2}b}_{0}\text{nS}}{\varepsilon^{\frac{1}{3}}}\)(34)
Summary of transition functions. From Eqs. 24-34, we have
\(f\left(n,\varepsilon\right)={{(b}_{0}-d}_{0}\frac{E}{E_{c}})\frac{n}{\varepsilon^{\frac{1}{3}}}+m_{0}\frac{n}{N}\),
(35)
\(h\left(n,\varepsilon\right)=w_{0}\text{nε}^{\frac{2}{3}}-\frac{w_{10}}{\ln^{\frac{2}{3}}(\frac{1}{\beta)}}\text{nε}\ {\ d}_{0}n\varepsilon^{\frac{2}{3}}\frac{E}{E_{c}}+\frac{m_{0}n}{N},\)(36)
\(q\left(n,\varepsilon\right)=m_{0}e^{-\mu S-\gamma}+\frac{\sigma_{1}\text{KS}}{K+S}+\frac{{\sigma_{2}b}_{0}\text{nS}}{\varepsilon^{\frac{1}{3}}}-S\delta_{n,1}\frac{d_{0}\frac{E}{E_{c}}}{\varepsilon^{\frac{1}{3}}}.\)(37)
In Eq. 37, both speciation models and immigration are included for
generality. With the transition functions specified, the iteration
procedure described by Eqs. 14-23 can be carried out and the time
trajectories of the state variables, the structure function, and the
metrics of macroecology that derive from the structure function can be
calculated.