DynaMETE.
DynaMETE is a hybrid theory based upon both the logic of the MaxEnt procedure and explicit mechanistic assumptions about the drivers of change. The mechanistic parent of DynaMETE is contained within a set of transition functions that describe demographic and ontogenic-growth rates. They derive from analyses of individuals and populations, not directly from the dynamics of the state variables. To up-scale these processes from the micro-level variables, n and \(\varepsilon,\)to the macro-level state variables, S , N , and E , we average the transition functions over the perturbed structure function. That function, in turn, is determined in an iterative procedure, from the constraints imposed by the perturbed state variables and their time derivatives using MaxEnt. Fig. 1 illustrates the recursive architecture of DynaMETE.
Importantly, our distinction between “steady-state” and “dynamic” applies to the state variables only; even in steady state, individuals can be growing or dying and abundances of species can be increasing or decreasing at any moment in time, provided the state variables are constant.
At any moment in time, the constraints on the dynamic structure function, R , include the state variables and their first time derivatives. Two Lagrange multipliers, \(\lambda_{1}\ \)and\(\lambda_{2}\), correspond to the constraints provided by ratios of state variables, N /S and E /S , just as in METE. Three additional Lagrange multipliers,\(\lambda_{3},\ \lambda_{4}\ \text{and}\ \lambda_{5},\ \)correspond to the constraints of (1/S )dN /dt , (1/S )dE /dt , and dS /dt .
Consider an ecosystem that has been in steady state up until timet = 0; the state variables have been constant as a consequence of a balance among the processes that increase and decrease their values, and the structure function is given by Eq. 3. For S , those processes might include extinction, speciation, and immigration; forN , they might include birth, death, and immigration; and forE , they might include ontogenic growth, death of individuals, and immigration. At t = 0, a disturbance is imposed. It could be a reduction in the immigration rate because of habitat fragmentation, or a change in the ontogenic growth rate or the per capita death rate because of climate change.
DynaMETE describes the time evolution of the system in the aftermath of that disturbance as an iteration of a sequence of steps (A-G) that are made more explicit in Table 1.
To express Table 1 in equation form, we introduce transition functions. We denote the rate of change of the population of an arbitrary species by:
dn/dt = f(n,\(\varepsilon\),{X},{c}) (11)
Here {X } refers to the set of state variables and {c } refers to the set of parameters such as migration rate or per-capita birth and death rates that govern changes in the abundances of the species and thus in N . We assume that f and the other transition functions do not depend on the {dX /dt }.
Similarly, the rate of change of the metabolic rate of an individual is:
d\(\varepsilon\)/dt = g (n ,\(\text{ε\ },\ \{X\}\),{c }). (12)
From Eqs. 11 and 12:
\(\frac{d\left(\text{nε}\right)}{\text{dt}}=ng(n,\varepsilon,\left\{X\right\},\ \{c\})+\varepsilon f(n,\varepsilon,\left\{X\right\},\{c\})\equiv h(n,\varepsilon,\left\{X\right\},\{c\})\)(13)
The function h describes processes that contribute to changes in the total metabolic rate of the species and thus in E .
In addition to \(f\) and \(h\), the transition function\(q\left(n,\varepsilon,\left\{X\right\},\{c\}\right)\) describes processes governing changes in species richness, including for example extinction, immigration, and speciation. Multiplying these transition functions by the time-evolving structure function and summing overn and \(\varepsilon\), yields the time-evolving time derivatives of N and E and S .
To describe the iterative process we introduce a discrete time-step index, i , and designate the time-dependent state variables, their time derivatives, the Lagrange multipliers, the transition rate parameters, and the structure function as {Xi }, {dXi /dt }, \(\lambda_{j,i},\){ci } and Rirespectively. The index, j, designating the Lagrange multipliers, runs from 1 to 5.
Again consider a system that in the past (i < 0) was in steady state, with static state variables,\(\ \)static structure function given by Eq. 3, and Lagrange multipliers given by Eqs. 5 and 6. For i < 0,\(\lambda_{3,i},\ \lambda_{4,i},\ \lambda_{5,i}\ \)and the time derivatives of the state variables vanish. At i = 0 a disturbance is imposed that is expressible as a change in one or more of the parameters, c , in the transition functions. It can be a time-varying or a fixed, one-time parameter change.
The iterative process cycles through three groups of equations using the perturbed transition functions. First, at any moment in time i , there are the constraint equations that determine the structure function from the instantaneous values of the state variables and their time derivatives:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N}_{i}=S_{i}\sum_{n,\varepsilon}{nR_{i}(n,\varepsilon,\left\{X_{i}\right\},\ \{c_{i}\})}\)(14)
\(E_{i}=S_{i}\sum_{n,\varepsilon}{\text{nε}R_{i}(n,\varepsilon,\left\{X_{i}\right\},\ \{c_{i}\})}\)(15)
\begin{equation} \nonumber \\ \end{equation}
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{dN_{i}}{\text{dt}}=S_{i}\sum_{n,\varepsilon}{f\left(n,\varepsilon,{\{X}_{i}\},\{c_{i}\}\right)R_{i}(n,\varepsilon,\left\{X_{i}\right\},\ \{c_{i}\})}\)(16)
\(\text{\ \ \ \ \ \ \ \ \ }\frac{dE_{i}}{\text{dt}}=S_{i}\sum_{n,\varepsilon}{h\left(n,\varepsilon,{\{X}_{i}\},\{c_{i}\}\right)R_{i}(n,\varepsilon,\left\{X_{i}\right\},\ \{c_{i}\})}\)(17)
\(\text{\ \ \ \ \ \ \ \ \ }\frac{dS_{i}}{\text{dt}}=\sum_{n,\varepsilon}{q\left(n,\varepsilon,{\{X}_{i}\},\{c_{i}\}\right)R_{i}(n,\varepsilon,\left\{X_{i}\right\},\ \{c_{i}\})}\)(18)
Eqs. 14 and 15 impose the same constraints as do Eqs. 1 and 2, whereas Eqs. 16-18 impose new constraints. For notational simplicity, conditionality of the structure function on the time derivatives of the state variables is implicit in Eqs 14-18, whereas we have made explicit the dependence of the structure function on the state variables as a consequence of their appearance in the constraints. Eqs. 14-18 implement step E in Fig. 1. From these equations, application of the MaxEnt inference procedure results in the following ecological structure function:
\(R_{i}(n,\varepsilon,\left\{X_{i}\right\},\{c_{i}\})={Z_{i}^{-1}e}^{-\lambda_{1,i}n}e^{{-\lambda}_{2,i}\text{nε}}e^{{-\lambda}_{3,i}f(n,\varepsilon,\{X_{i}\})}e^{{-\lambda}_{4,i}h(n,\varepsilon,\{X_{i}\})}e^{{-\lambda}_{5,i}q(n,\varepsilon,\{X_{i}\})}\)(19)
where Zi is a normalization factor that depends on the \(\lambda_{j,i}.\)
To iterate the structure function we need to update the state variables:
\({\{X}_{i+1}\}={\{X}_{i}\}+\{\frac{dX_{i}}{dt}\}\Delta t\)(\(\Delta t=1\ \text{with\ integer\ index\ }i)\) (20)
This is step B in Fig. 1.
Eq. 20 allows us to directly update the transition rate functions by substitution (step 4 in Table 1)
Finally, we update the time derivatives of the state variables from time step i to i +1 by averaging the transition functions evaluated with Xi +1 over the structure function with Lagrange multipliers determined from Eqs. 14-18 fixed at time step i , but the transition functions f ,h , q appearing in the exponents of Eq. 19 evaluated at theXi +1:
\(S_{i+1}\sum_{n,\varepsilon}{f\left(n,\varepsilon,{\{X}_{i+1}\right\},\ \{c_{i+1}\})R_{i}(n,\varepsilon,\ \left\{X_{i+1}\right\},\ \{c_{i+1}\})}=\frac{dN_{i+1}}{\text{dt}}\)(21)
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }S_{i+1}\sum_{n,\varepsilon}{h\left(n,\varepsilon,{\{X}_{i+1}\right\},\{c_{i+1}\})R_{i}(n,\varepsilon,\left\{X_{i+1}\right\},\ \{c_{i+1}\})=}\frac{\text{\ d}E_{i+1}}{\text{dt}}\)(22)
\(\text{\ \ }\sum_{n,e\ }{q\left(n,\varepsilon,{\{X}_{i+1}\right\},\{c_{i+1}\})R_{i}(n,\varepsilon,\left\{X_{i+1}\right\},\ \{c_{i+1}\})=\frac{dS_{i+1}}{\text{dt}}}\)(23)
The subscript i on \(R_{i}\) in Eqs. 21-23 signifies that the Lagrange multipliers are those at step i , and thus depend on {Xi } and {dXi /dt}. Eqs. 21-23 implement step D in Fig. 1.
Equations 14-23 comprise DynaMETE. They set forth an iterative procedure for calculating the time evolution of both the state variables and the structure function. From the latter, the time-dependent SAD and MRDI can be calculated.
Supporting Information-A (SI-A) explains the rationale for the term,S , multiplying the summations in Eqs. 16, 17, 21, and 22; SI-B elaborates on the full set of equations 14-23, demonstrating their internal consistency.
We turn next to the model-dependent, mechanistic parent of the hybrid theory: expressions for the transition functions f , h, andq .