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 metaS , 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.