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\begin{document}

\title{DynaMETE: A Hybrid MaxEnt-plus-Mechanism Theory of Dynamic Macroecology}



\author[1,2]{John Harte}%
\author[3]{Kaito Umemura}%
\author[4,5]{Micah Brush}%
\affil[1]{Ecology Letters }%
\affil[2]{University of California}%
\affil[3]{University of California Berkeley}%
\affil[4]{University of California, Berkeley}%
\affil[5]{UC Berkeley}%


\vspace{-1em}



  
  \date{May 22, 2024}


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\selectlanguage{english}
\begin{abstract}
The Maximum Entropy Theory of Ecology (METE) predicts the shapes of
macroecological metrics in relatively static ecosystems using
constraints imposed by static state variables. In disturbed ecosystems,
however, with time-varying state variables, its predictions often fail.
We extend macroecological theory from static to dynamic by combining the
MaxEnt inference procedure with explicit mechanisms governing
disturbance. In the static limit, the resulting theory, DynaMETE,
reduces to METE but also predicts new scaling relationships among static
state variables. Under disturbances, expressed as shifts in demographic,
ontogenic growth, or migration rates, DynaMETE predicts the time
trajectories of the state variables as well as the time-varying shapes
of macroecological metrics such as the species abundance distribution
and the distribution of metabolic rates over individuals. An iterative
procedure for completely solving the dynamic theory is presented. In a
lowest-order iteration, characteristic signatures of the deviation from
static predictions of macroecolgoical patterns are shown to result from
different kinds of disturbance. Because DynaMETE combines MaxEnt
inference with explicit dynamical mechanisms, but does not assume any
specific trait distributions over species or individuals, it is widely
applicable across diverse ecosystems. This makes it a promising theory
of macroecology for ecosystems responding to anthropogenic or natural
disturbances.%
\end{abstract}%



\sloppy


\textbf{INTRODUCTION}

One focus of ecology is obtaining insight into the shape and origin of
patterns in abundance, energetics, and spatial distributions of taxa,
across spatial scales, and within different habitats. Macroecology, the
study of such patterns, builds capacity for estimating species diversity
from sparse data, predicting extinction rates under habitat loss, and
deciphering the processes that determine ecosystem structure and
function. (Brown 1995, Rosenzweig 1995, Gaston and Blackburn 2000,
Kitzes and Shirley 2016).

Although the study of dynamic ecosystems is a rising area in ecology
(Hill \& Hamer1998; Dornelas 2010; Turner 2010; Newman 2019),
macroecological theory has largely focused on understanding patterns in
quasi-steady-state ecosystems, ignoring trending patterns in systems
undergoing rapid succession, diversification or collapse (Fisher et al.
2010). Empirical evidence, however, is accumulating that dynamic and
static macroecological patterns differ (e.g., Kempton and Taylor 1974;
Carey et al. 2006; Harte 2011; Supp et al. 2012; Harte and Newman 2014;
Rominger et al. 2015; Newman et al. 2020). Our objective here is the
formulation and initial exploration of a theory, DynaMETE, to predict
macroecological patterns in dynamic systems.

Our starting point is a static theory based on the maximum entropy
(MaxEnt) framework (Harte 2011; Harte and Newman 2014). MaxEnt selects
the flattest, and therefore least informative, probability distributions
compatible with constraints imposed by prior knowledge. Bias, in the
form of assumptions about the distribution that are not compelled by
prior knowledge, is thereby eliminated (Jaynes 1957; 1982). The maximum
entropy form of a probability distribution, \emph{p} (\emph{n} ), is
obtained by maximizing its Shannon information entropy (Shannon
1948),\(-\sum_{n}{p\left(n\right)\log\left(p\left(n\right)\right)},\)subject to imposed constraints.

Ever since Jaynes, the MaxEnt inference procedure has been applied in
many fields, including image reconstruction in medicine and forensics
(Frieden, 1972; Skilling, 1984; Gull and Newton, 1986; Roussev, 2010),
neural net firing patterns (Meshulam, 2017), protein folding (Steinbach
et. al, 2002; Mora et al, 2010), and reconstruction of incomplete
input-output data and other applications in economics (Golan, Judge and
Miller, 1996; Golan, 2018).

The MaxEnt Theory of Ecology (METE) assumes prior knowledge in the form
of static state variables describing a taxonomic group of interest
(e.g., plants or arthropods) in a prescribed location. In the original
version of the theory there are four state variables: area, \emph{A} ,
of the ecosystem, total number of \emph{s} pecies, \emph{S} , within the
broad taxonomic group in that ecosystem, summed number of
individuals,\emph{N} , in those species, and summed metabolic rate,
\emph{E} , of those individuals. From the constraints imposed by the
ratios of the state variables, the forms of many of the metrics of
macroecology can be inferred, with no adjustable parameters, using
MaxEnt.

METE predicts many pervasive patterns in static macroecology, including
the species abundance distribution (SAD) (Harte et al. 2008; Harte and
Kitzes, 2014; White et al. 2012; but see Ulrich et al. 2010), the
species-area relationship (SAR) (Harte et al. 2009), the metabolic rate
distribution over individuals (MRDI) (Harte et al. 2008; 2017; Xiao et
al. 2015), a relationship between the average metabolic rate of the
individuals in a species and the abundance of that species (Harte et al.
2008), and, in an extension of the original theory, the distribution of
species over higher taxonomic categories and the dependence of the
abundance-metabolism relationship on the structure of the taxonomic tree
(Harte et al. 2015).

Just as macroecological patterns shift under disturbance, METE's
predictions also generally fail in ecosystems undergoing relatively
rapid change. In particular, when state variables are changing as a
consequence of succession or anthropogenic disturbance, the values of
the state variables at any moment in time do not accurately predict the
shapes of the macroecological metrics at that same moment in time.

Examples of altered macroecological patterns in disturbed ecosystems
abound. Moth censuses at Rothamsted reveal a log-series SAD (as METE
predicts) at less disturbed locations and a lognormal SAD at more
disturbed locations (Kempton and Taylor 1974). Supp et al. (2012) report
that when the state variables (species richness and total abundance) are
experimentally altered in small-mammal communities, the functional form
of the SAD is altered. Kunin et al. (2018) show that in the highly
fragmented and manipulated UK, METE under-predicts species richness
derived from upscaling data from small plots. Franzman et al. (2020)
show that in an alpine plant community, both the SAR and the SAD
increasingly deviate over time from METE predictions during a period of
drought stress.

In systems recovering from disturbance, macroecological patterns also
change. Carey et al. (2006) show that the shape of the SAR in recovering
subalpine vegetation plots in the aftermath of both an eruption at Mount
St. Helens and a hillslope-erosion event at Gothic CO deviated
systematically from that observed in nearby undisturbed comparison
plots. In the aftermath of a recent fire, Newman et al. (2020) observe
the failure of METE in a fire-adapted Bishop Pine forest site in coastal
California. There, the SAR in a successional post-fire ecosystem
deviates markedly from the METE prediction, in contrast to a control
site that has not burned in many decades. On much longer time scales,
such shifts are also occurring; for example, at younger sites in the
Hawaiian Islands where diversification is occurring more rapidly, both
the SAD and the MRDI show deviations from static theory predictions, in
contrast to sites on the older islands (Rominger et al. 2015).

Across the Smithsonian tropical forest plots, systematic deviations from
MaxEnt predictions appear prominently at the Barro Colorado Island site
in Panama, where the state variables, \emph{S} and \emph{N} , have
declined over the past 30 years, speculatively as a consequence of a
combination of local disturbance and the formation of Gatun Lake
resulting in the semi-isolation of the created island from its
metacommunity (E. Leigh, pers. comm.). The shape of the SAD at BCI is
currently intermediate between a log-series and a lognormal, while other
Smithsonian tropical forest plots that are less disturbed, such as
Cocoli and Bukit Timah, show closer agreement with the log-series SAD
predicted by the static theory (Harte 2011).

The pattern of deviation of macroecological metrics from METE
predictions differs across these investigations of disturbed ecosystems.
Whereas in some of the disturbance sites, the SAD appears to trend
toward a lognormal distribution (Rothamsted moths, BCI trees), in others
the trend is toward a weak inverse power-function,
n\textsuperscript{-a}with a \textless{} 1 (alpine plant community). In
some disturbance sites, the SAR deviates from the METE prediction toward
a power-law (post-burn Bishop pine forest), while in others it deviates
further from power-law behavior (alpine plant community). This
complexity of responses of macroecological patterns to disturbance
provides us with an opportunity to identify drivers of change. Indeed,
DynaMETE predicts departures of macroecological distributions from
steady state that depend in characteristic ways on the specific
mechanisms of disturbance.

METE is static insofar as the state variables are assumed to vary so
slowly that their instantaneous values suffice to derive macroecological
distributions. Plausibly, in a dynamic ecosystem with rapidly changing
state variables, static METE might be inadequate. Analogously in
thermodynamics, pressure, volume, and temperature are the macroscopic
state variables, from which micro-level distributions such as the
Boltzmann distribution of molecular energies can be derived using MaxEnt
(Jaynes 1957; 1982). In an out-of-steady-state ``disturbed'' gas, such
as one with inhomogeneously changing temperature, averaged pressure,
volume, and temperature no longer suffice.

In DynaMETE, we combine the MaxEnt procedure for determining
least-biased probability distributions with explicit mechanisms that
drive the system from steady state. Because the dynamics depends on the
state variables, we require an iterative procedure for up-dating both
constraints and macroecological distributions.

In Methods we review METE and present the theoretical framework for
DynaMETE, including how explicit mechanisms are incorporated and
upscaled from individuals to the community level, and an iteration
procedure for deriving predictions. Under Results we examine predicted
scaling relationships among the state variables in the static limit of
DynaMETE. Then we examine dynamic predictions of the theory near steady
state. Coupled time-differential equations for the state variables are
derived and solved to reveal predicted trajectories of these variables
under various perturbations. We also show predicted deviations of
abundance and metabolic rate distributions in a lowest-order iteration
of the full theory, again for a variety of perturbations. In Discussion,
we assess the current status of DynaMETE and suggest future directions
and applications.

\textbf{METHODS}

\textbf{Review of METE.}

At the core of METE is the time-independent structure function\emph{R}
(\emph{n} , \selectlanguage{greek}\emph{ε} \selectlanguage{english}\textbar{}\emph{S} , \emph{N} , \emph{E} ).\emph{R}
is a joint conditional distribution over abundance, \emph{n} , of a
species, and metabolic rate, \selectlanguage{greek}\emph{ε} \selectlanguage{english}of an individual;\emph{R}
\selectlanguage{ngerman}·d\selectlanguage{greek}\emph{ε} \selectlanguage{english}is the probability that a species picked at random from the
species pool has abundance \emph{n} , and an individual picked at random
from the species with abundance \emph{n} has a metabolic rate in the
interval (\selectlanguage{greek}\emph{ε} , \selectlanguage{english}\selectlanguage{greek}\emph{ε} \selectlanguage{english}+ d\selectlanguage{greek}\emph{ε} \selectlanguage{english}).

We use a discrete notation, with summation, not integral, signs for all
variables \emph{n} , \selectlanguage{greek}\emph{ε} , \selectlanguage{english}\emph{S} , \emph{N} , \emph{E} , and
adopt the units convention that the smallest value for the metabolic
rate is\selectlanguage{greek}ε\selectlanguage{ngerman} = 1, the metabolic rate of the smallest
observed organism in the community (e.g., a germinant tree in a forest,
or a tree with 10 mm dbh if that is the smallest tree censused). The
constraints on the static structure function are:

\(\frac{N}{S}\ =\ \sum_{n,\varepsilon}nR\left(n,\varepsilon\middle|S,N,E\right)\)(1)

and

\(\frac{E}{S}=\ \sum_{n,\varepsilon}\selectlanguage{greek}\text{nε}\selectlanguage{english}R\left(n,\varepsilon\middle|S,N,E\right).\)(2)

The MaxEnt solution (Harte et al., 2008) for \emph{R} (\emph{n}
,\selectlanguage{greek}\emph{ε} \selectlanguage{english}\textbar{}\emph{S} ,\emph{N} ,\emph{E} ) subject these
constraints is:

\(R\left(n,\varepsilon\middle|S,N,E\right)=\selectlanguage{greek}\selectlanguage{english}\frac{e^{-\lambda_{1}n}e^{{-\lambda}_{2}\text{nε}}}{Z}\)(3)

where \emph{Z} \textsuperscript{-1} is a normalization constant (see
below). The \selectlanguage{greek}\emph{λ} \selectlanguage{english}'s are Lagrange multipliers that are determined
from the values of \emph{S} , \emph{N} , \emph{E} . The quantity
\selectlanguage{greek}\emph{β} \selectlanguage{english}=\selectlanguage{greek}\emph{λ} \selectlanguage{english}\textsubscript{1} + \selectlanguage{greek}\emph{λ} \selectlanguage{english}\textsubscript{2} is
determined from

\(\frac{S}{N}\sum_{n=1}^{N}{x^{n}=}\)\(\sum_{n=1}^{N}\frac{x^{n}}{n}\) (4)

where \(x=e^{-\beta}\text{.\ }\) The Lagrange multiplier,\(\lambda_{2},\) is
given by

\(\lambda_{2}=\frac{S}{E-N}.\) (5)

In many ecosystems, the state variables obey the inequalities S
\textless{}\textless{} N\textless{}\textless{} \emph{E} , in which
case\(\beta\ll 1\) and the solution to Eq. 4 is, to a good
approximation,

\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{S}{N}\)\(\approx\beta\ln\left(\frac{1}{1-e^{-\beta}}\right)\approx\)\(\beta\ln\left(\frac{1}{\beta}\right)\) (6)

Moreover, in that approximation,

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Z\approx\frac{ln(\frac{1}{\beta)}}{\lambda_{2}}\), (7)

In all that follows, we assume the validity of these approximations, and
use =, not \(\approx,\) signs in the equations.

We note for later comparison with DynaMETE that the species abundance
distribution (SAD), \(\phi\left(n\right),\) obtained by summing\emph{R} over
metabolic rate, \(\varepsilon,\) is the log-series distribution:

\(\phi\left(n\right)=\frac{e^{-\beta n}}{n\bullet ln(\frac{1}{\beta})}\)(8)

and the distribution of metabolic rates over all individuals (the MRDI),
obtained by summing \emph{nR} S/\emph{N} over abundance, \emph{n} , is:

\(\Psi\left(\varepsilon\right)=\frac{\beta\lambda_{2}e^{-\gamma(\varepsilon)}}{{(1-e^{-\gamma(\varepsilon)})}^{2}}\)(9)

where

\(\gamma\left(\varepsilon\right)=\ \lambda_{1}+\lambda_{2}\varepsilon\). (10)

METE also contains a spatially-explicit component but to focus on the
essential concepts underlying DynaMETE, we ignore the spatial dimension
here.

For further description of MaxEnt and METE, and software for deriving
predictions, see Harte 2011; Bremmer and Newman 2019; Kitzes and Wilbur
2016; Rominger and Merow 2016.

\textbf{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, \emph{n} and
\(\varepsilon,\)to the macro-level state variables, \emph{S} ,
\emph{N} , and \emph{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, \emph{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, \emph{N} /\emph{S} and \emph{E}
/\emph{S} , just as in METE. Three additional Lagrange
multipliers,\(\lambda_{3},\ \lambda_{4}\ \text{and}\ \lambda_{5},\ \)correspond to the constraints of
(1/\emph{S} )d\emph{N} /d\emph{t} , (1/\emph{S} )d\emph{E} /d\emph{t} ,
and d\emph{S} /d\emph{t} .

Consider an ecosystem that has been in steady state up until
time\emph{t} = 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 \emph{S}
, those processes might include extinction, speciation, and immigration;
for\emph{N} , they might include birth, death, and immigration; and
for\emph{E} , they might include ontogenic growth, death of individuals,
and immigration. At \emph{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:

d\emph{n}/d\emph{t} =
\emph{f}(\emph{n},\(\varepsilon\),\{\emph{X}\},\{\emph{c}\}) (11)

Here \{\emph{X} \} refers to the set of state variables and \{\emph{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 \emph{N} . We assume that \emph{f} and the other
transition functions do not depend on the \{d\emph{X} /d\emph{t} \}.

Similarly, the rate of change of the metabolic rate of an individual is:

d\(\varepsilon\)/d\emph{t} = \emph{g} (\emph{n}
,\(\selectlanguage{greek}\text{ε\ }\selectlanguage{english},\ \{X\}\),\{\emph{c} \}). (12)

From Eqs. 11 and 12:

\(\selectlanguage{greek}\frac{d\left(\text{nε}\right)}\selectlanguage{english}{\text{dt}}=ng(n,\varepsilon,\left\{X\right\},\ \{c\})+\varepsilon f(n,\varepsilon,\left\{X\right\},\{c\})\equiv h(n,\varepsilon,\left\{X\right\},\{c\})\)(13)

The function \emph{h} describes processes that contribute to changes in
the total metabolic rate of the species and thus in \emph{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 over\emph{n} and \(\varepsilon\),
yields the time-evolving time derivatives of \emph{N} and \emph{E} and
\emph{S} .

To describe the iterative process we introduce a discrete time-step
index, \emph{i} , and designate the time-dependent state variables,
their time derivatives, the Lagrange multipliers, the transition rate
parameters, and the structure function as \{\emph{X\textsubscript{i}}
\}, \{d\emph{X\textsubscript{i}} /d\emph{t} \},
\(\lambda_{j,i},\)\{\emph{c\textsubscript{i}} \} and
\emph{R\textsubscript{i}}respectively. The index, \emph{j,} designating
the Lagrange multipliers, runs from 1 to 5.

Again consider a system that in the past (\emph{i} \textless{} 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 \emph{i} \textless{} 0,\(\lambda_{3,i},\ \lambda_{4,i},\ \lambda_{5,i}\ \)and the time
derivatives of the state variables vanish. At \emph{i} = 0 a disturbance
is imposed that is expressible as a change in one or more of the
parameters, \emph{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 \emph{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)

\(E_{i}=S_{i}\selectlanguage{greek}\sum_{n,\varepsilon}{\text{nε}R_{i}(n,\varepsilon,\left\{X_{i}\right\},\ \{c_{i}\})}\selectlanguage{english}\)(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:

\(R_{i}(n,\varepsilon,\left\{X_{i}\right\},\{c_{i}\selectlanguage{greek}\})={Z_{i}^{-1}e}^{-\lambda_{1,i}n}e^{{-\lambda}_{2,i}\text{nε}}\selectlanguage{english}e^{{-\lambda}_{3,i}f(n,\varepsilon,\{X_{i}\})}e^{{-\lambda}_{4,i}h(n,\varepsilon,\{X_{i}\})}e^{{-\lambda}_{5,i}q(n,\varepsilon,\{X_{i}\})}\)(19)

where \emph{Z\textsubscript{i}} 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).\selectlanguage{ngerman} 

Finally, we update the time derivatives of the state variables from time
step \emph{i} to \emph{i} +1 by averaging the transition functions
evaluated with \emph{X\textsubscript{i}} \textsubscript{+1} over the
structure function with Lagrange multipliers determined from Eqs. 14-18
fixed at time step \emph{i} , but the transition functions \emph{f}
,\emph{h} , \emph{q} appearing in the exponents of Eq. 19 evaluated at
the\emph{X\textsubscript{i}} \textsubscript{+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 \emph{i} on \(R_{i}\) in Eqs. 21-23 signifies
that the Lagrange multipliers are those at step \emph{i} , and thus
depend on \{\emph{X\textsubscript{i}} \} and
\{d\emph{X\textsubscript{i}} /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,\emph{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 \emph{f} , \emph{h,}
and\emph{q} .

\textbf{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 \emph{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.

\textbf{Contribution of birth and death to\emph{f}(\emph{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
between\emph{n} 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
function\emph{R} (\emph{n} , \(\varepsilon\)) (Harte et al., 2008).

The contribution of death to d\emph{n} /d\emph{t} 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 of\emph{E} , not
\emph{N} or \emph{n} . Thus, we multiply\(\frac{n}{\varepsilon^{1/3}}\ \)by \emph{E}
/\emph{E} \textsubscript{c}, where \emph{E} \textsubscript{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 \emph{E} \textsubscript{c} will also serve as a fundamental
scale parameter, with \emph{E} \textsubscript{c} proportional to plot
area within a given habitat.

\textbf{Contribution of death to\emph{h}(\emph{n},}
\(\mathbf{\ \varepsilon).\ }\ \)Multiplying the death rate in Eq. 25 by \selectlanguage{greek}\emph{ε} , \selectlanguage{english}we
get the following contribution to\emph{h} (\emph{n} ,\(\ \varepsilon):\)

\(h_{\text{death}}={-d}_{0}n\varepsilon^{\frac{2}{3}}\frac{E}{E_{c}}\)(26)

We ignore the contribution of birth to \emph{dE/dt} because birth
primarily partitions, rather than adds to, metabolism.

\textbf{Contribution of immigration to\emph{f}(\emph{n},}
\(\mathbf{\ \varepsilon)}\) \textbf{and\emph{h}(\emph{n},} \(\mathbf{\ \varepsilon).}\)
For a vegetation community, we denote by \emph{m} \textsubscript{0} the
immigration rate of seeds that result in germinants with
\(\varepsilon\)= 1. The probability that the immigrant is in an
existing species with abundance\emph{n} is assumed to be \emph{n}
/\emph{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.

\textbf{Contribution of ontogenic growth to\emph{h}(\emph{n},}
\(\mathbf{\ \varepsilon)}\)\textbf{.} 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 \emph{g} (\emph{n} ,\(\ \varepsilon)\) (see Eq. 12)
is multiplied by \emph{n} to give the contribution of ontogenic growth
to\emph{h} (\emph{n} ,\(\ \varepsilon):\)

\(h_{\text{ontogenic\ growth}}=w_{0}n\varepsilon^{2/3}-w_{1}\selectlanguage{greek}\text{nε}\selectlanguage{english}\)(30)

For reasons of scale consistency, the parameter\emph{w}
\textsubscript{1} equals a scale-independent parameter,\emph{w}
\textsubscript{10}, divided by
ln\textsuperscript{2/3}(1/\(\beta)\). If, instead, we assumed
that\emph{w} \textsubscript{1} and not \emph{w} \textsubscript{10} is
scale independent, then in the resulting equation for d\emph{E} /dt, one
of the terms, -\emph{w} \textsubscript{1}\emph{E} , scales isometrically
with area while the other scales as area/\(\operatorname{}\left(\frac{1}{\beta}\right)\) or roughly
area/ln\textsuperscript{2/3}(area).

\textbf{Contribution of local extinction to\emph{q}(\emph{n},}
\(\mathbf{\varepsilon}\)\textbf{).} 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
\emph{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.

\textbf{Contribution of immigration to\emph{q}(\emph{n},}
\(\mathbf{\varepsilon}\)\textbf{).} 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,\emph{S}
\textsubscript{meta},\textsubscript{,} and total abundance,\emph{N}
\textsubscript{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 to\emph{S}
\textsubscript{meta} -- \emph{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\selectlanguage{greek}\({\mu=ln(\frac{1}{\beta_{\text{meta}}\frac{)}{S_{\text{meta}},}}\text{\ β}}\selectlanguage{english}_{\text{meta}}\)is calculated from Eq. 6 using the values
of\emph{S} \textsubscript{meta} and \emph{N} \textsubscript{meta} which
in turn are estimated SI-E, and \(\gamma\) is Euler's constant,
\textasciitilde{}0.577.

\textbf{Contribution of speciation to\emph{q}(\emph{n},}
\(\mathbf{\varepsilon}\)\textbf{).} 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 \emph{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 \emph{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)

\textbf{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)

\(h\left(n,\varepsilon\right)=w_{0}\selectlanguage{greek}\text{nε}\selectlanguage{english}^{\frac{2}{3}}-\frac{w_{10}}{\ln^{\frac{2}{3}}(\frac{1}{\beta)}}\selectlanguage{greek}\text{nε}\selectlanguage{english}\ {\ d}_{0}n\varepsilon^{\frac{2}{3}}\frac{E}{E_{c}}+\frac{m_{0}n}{N},\)(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.

\textbf{RESULTS}

We first derive predicted state variable trajectories in time, at and
near steady state, from a set of coupled time-differential equations for
the state variables that result from a truncation, at step 2 in Table 1,
of the full iteration procedure. In this approximation, the form of the
structure function in Eq. 3 is retained (\(\lambda_{3},\ \lambda_{4},\ \lambda_{5}=0)\) and we
derive a set of non-linear time-differential equations of motion (Eqs.
38-40) for the state variables that will depend upon both the state
variables themselves and the values of the parameters in the transition
functions. We explore here both the steady state properties of these
equations and the dynamics they predict near steady state. These results
should only be good approximations to the actual DynaMETE predictions
for small deviations from steady state because they derive from the
static structure function.

Then, in a first iteration of the full theory, we use Eqs. 14-23 to
calculate lowest-order effects of various perturbations in the
transition function rate constants on the structure function. From the
perturbed structure function we then derive altered shapes of the
abundance and metabolic rate distributions. We emphasize that these
results may differ considerably from a fully-iterated solution to
DynaMETE.

\textbf{State Variables at and Near Steady State.}

Using Eqs. 21-23, with the transition functions specified in Eqs. 35-37
and \emph{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)

\textbf{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 \textless{}\textless{} 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
\emph{S}\textgreater{}\textgreater{} \emph{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 \emph{S} \textless{}\textless{} \emph{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)

\textbf{The Steady-State Species-Area Relationship (SAR).} We can derive
nested SARs from Eqs 43-46 because in a nested
design,\emph{E\textsubscript{c}} , \emph{E} , and \emph{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, \emph{E\textsubscript{c}} , \emph{E} ,
and\emph{N} scale linearly with area, and in general,
\emph{E}\textgreater{}\textgreater{} \emph{m} \textsubscript{0}. Hence
\emph{S}scales 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.
If\emph{m} \textsubscript{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 \emph{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 \emph{S}
\textasciitilde{}\(m_{0}^{\frac{3}{4}}E_{0}^{\frac{1}{4}}\ \sim\ \text{area}\ \)if\emph{m} \textsubscript{0} scales
linearly with area, and S \textasciitilde{} area\textsuperscript{1/4} if
\emph{m} \textsubscript{0}is scale-independent. This limit applies when
S \textless{} 1/\(\mu\), which for a typical 50 ha tropical
forest plot is approximately \emph{S}\textless{} 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, \emph{f}
,\emph{h} , and \emph{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: \emph{S} \textasciitilde{}
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
\emph{S}\textgreater{}\textgreater{} \emph{K} we get quarter-power
scaling of species richness with \emph{E} (up to a
\(\ln\left(1/\beta\right)\)correction) and therefore with area. In that model,
with S \textless{}\textless{} K, and for any \emph{S} in the second
speciation model, \emph{S} scales linearly with area, up to
a\(\ln\left(1/\beta\right)\) \textasciitilde{} ln(\emph{N} )-ln(\emph{S} )
correction. The first model, with a saturation term \emph{K} that is
small compared to the steady state species richness, thus yields a more
realistic species-area relationship.

\textbf{A Productivity-biomass-diversity-abundance relationship at
steady state.} Metabolic scaling informs us that individual mass
(\emph{m} ) is related to individual metabolism by\(m(\varepsilon)=\ m(1)\varepsilon^{4/3}\).
Summing over the steady-state structure function, total biomass,
\emph{B} , is then (following the methods in SI-C):

\emph{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 \emph{E} as total net
productivity\emph{P} 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 \emph{B} and \emph{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 \emph{P} , and that ln(1/\(\beta)\ \)
varies approximately as ln(\emph{N} ) -- ln(\emph{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.

\textbf{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 \emph{S} at
small\emph{t} , as well as a weak, damped oscillatory response of
\emph{N} , and nearly undetectable change in \emph{E} (Fig. 3a). An
increase in the death rate (Fig. 3b) generates a slight decrease in
\emph{S} , an initial large decline in \emph{N} and a weaker decline in
\emph{E} , followed by damped oscillatory behavior. A decrease in the
ontogenic growth rate (Fig. 3c) generates a nearly indiscernible
increase in \emph{S} , a large damped oscillatory initial rise in
\emph{N} and weak damped oscillatory initial decrease in \emph{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
in\emph{S} , decline and then partial recovery in \emph{N} , and weak
variability in \emph{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
of\emph{S} , the large overshoot and then decline to steady state
of\emph{N} , and a much weaker overshoot and decline of \emph{E} .

If speciation is the driver of diversification, the pattern of recovery
of species richness is markedly different. In the first speciation
model, with \emph{K} \textgreater{}\textgreater{} \emph{S} , recovery
of\emph{S} is sigmoidal and extremely slow, with 90\% recovery taking
approximately 10\textsuperscript{4} years. In the second speciation
model, and in the first with \emph{K} \textless{}\textless{} \emph{S}
,\emph{S} recovers to 90\% of steady state in approximately 4000 years,
and at an ever-slowing, rather than sigmoidal, rate. The recovery
trajectories of \emph{N} and \emph{E} are nearly the same in both
speciation models and very similar to that in the immigration-only case
(Fig. 4).

\textbf{Perturbed Abundance and Metabolic Rate Distributions in
DynaMETE.}

Here we examine, in a first order approximation to a fully iterated
solution, how different types of disturbance give rise to characteristic
departures of the SAD, \(\phi\left(n\right),\) and the
MRDI,\(\psi\left(\varepsilon\right),\) from their steady state form. In particular, we
truncate the iteration procedure, stopping with step 6 in Table 1. A
single iteration at a one-year time step, however, results in changes in
the structure function that are too small to show interesting deviations
from steady state, so to generate a discernible effect for a single
iteration we use a time step of 25 years. Specifically, we assume the
static structure function, with Lagrange multipliers ``frozen'' at their
static numerical values as prescribed in Eqs. 21-23, perturb the
transition functions by changing one or more rates constants, and then
derive from Eqs. 21-23 a set of time-differential equations for the
state variables. These equations differ from Eqs. 38-40 because the
latter were derived by updating the Lagrange multipliers at each time
step. We then ran these equations out to t = 25 and take the values of
the \{\emph{X} (25)\} and the \{d\emph{X} (25)/dt\} as constraints in
Eqs. 14-18 to calculate using Max Ent a perturbed structure function.
That function will be of the form of Eq. 19 and from it we derive
perturbed forms for the species abundance distribution (SAD) and the
metabolic rate distribution over individuals (MRDI) using the same
summations as performed to derive Eqs. 8 and 9.

The results of that calculation are shown in Fig. 5. The five derived
Lagrange multipliers are given in Table 3. Setting the immigration rate
constant, \emph{m} \textsubscript{0}, to zero only slightly alters the
SAD and the MRDI in this first iteration of the full structure function
(Fig. 5a,b), even \emph{S} decreases significantly. A 25\% increase in
the death rate constant, \emph{d} \textsubscript{0}, shifts the SAD
toward a lognormal shape as indicated by the curved rank-log(abundance)
graph at intermediate abundances (Fig. 5c). The rank-log(metabolism)
graph shifts in a more complex manner, weaving around the METE
prediction, and predicting more of the very smallest trees
(\(\varepsilon=1)\), fewer individuals with low
(\(\varepsilon=2-100)\ \)metabolism, and more trees with relatively high
(\(\varepsilon=100-100\)000) metabolism, and a reduction in the sizes of the
very largest individuals (Fig. 5d). A 5\% decrease in the growth rate of
individuals, \emph{w} \textsubscript{0}, generates a roughly
mirror-image shift in the SAD relative to that from an increase in the
death rate; the resulting SAD is approximately described by either an
exponential distribution or an inverse power function with exponent
\textgreater{} 1 (Fig. 5e). Similarly, the shift in the MRDI generated
by a decrease in growth rate is roughly the mirror image of the shift
induced by an increase in the death rate (Fig. 5f).

Figs. 5g,h show the effect of the same combination of changes in the
rate constants used to generate Fig. 3d. We do not attempt here a
detailed comparison to real data because of the first order
approximation used to obtain these theoretical curves, but we do note
their rough similarity to the empirical SAD and MRDI at BCI (see insets
in Figs. 5g,h).

We emphasize that a full iterative solution of Eqs. 14-23 in, say, 25
one-year time steps, over a period of 25 years could result in output
that differs considerably from the truncated solutions in Fig. 5.
Subsequent work will explore this.

\textbf{DISCUSSION}

DynaMETE combines both mechanism and MaxEnt, links the micro-level of
analysis to the macro-level with a novel iterative procedure (Fig. 1,
Table 1), and predicts both static and dynamic behavior. DynaMETE is
thus a hybrid MaxEnt-plus-mechanism extension to dynamic ecosystems of a
static theory, METE, which was based solely on the MaxEnt inference
procedure.

\textbf{Static limit.} In its static limit, DynaMETE recovers the static
METE predictions for the distributions of abundances over species and
metabolic rates over individuals, but also makes new predictions. First,
it predicts a scaling relationship among productivity, biomass, species
richness and abundance in steady state (Eq. 49); this relationship
remains to be tested. Second, in the version of DynaMETE in which only
immigration, not speciation, contributes to diversification, DynaMETE
predicts a static-limit SAR increasing approximately as ln(\emph{A} ),
with a ln(ln(\emph{A} )) correction, in very close agreement with the
static SAR predicted by METE (Fig. 2).

If speciation but not immigration is assumed to drive diversification
(Eqs. 33, 34), then there are problems with the current formulations of
the speciation term in the transition functions. In particular, if the
speciation rate is proportional to the birth rate (Eq. 34) or
proportional to species richness (Eq. 33 with
\emph{K}\textgreater{}\textgreater{} \emph{S} ), then the SAR is
approximately linear at all scales, contrary to observation. On the
other hand, if\emph{S} \textgreater{}\textgreater{} \emph{K} then
DynaMETE predicts a realistic SAR, but this contradicts the original
motivation for\emph{S} -dependent speciation from the fossil record,
which is an increasing rate of diversification over time in the
aftermath of extinction events. DynaMETE suggests a possible solution: a
logarithmic dependence of speciation on species richness. Using the
methods in SI-C, this results in:

\(\frac{S}{\ln^{\frac{3}{4}}(S)}\sim\ {(A)}^{\frac{1}{4}}\ln^{\frac{3}{4}}(\frac{1}{\beta)}\)(50)

and an increasing rate of diversification in the aftermath of
extinction.

\textbf{Dynamics.} Assuming a static structure function (Eq. 3) in a
lowest-order solution to the theory, Eqs. 38-40 predict signature time
trajectories of the state variables for various types of perturbations
in the transition function rate constants (Figs. 3). A combination of
reduced immigration and growth rates and increased death rate generates
trajectories that exhibit some of the features of those observed in the
BCI tropical forest plot (see inset in Fig. 3d). Because these model
simulations are based on extrapolating out in time only the first-order
iteration of the full theory, they are only suggestive; empirical
testing awaits further iteration of Eqs. 14-23.

DynaMETE also predicts trajectories of state variables during the
recovery of an ecosystem from a depauperate state. The predicted
overshoot and then decline to steady state of \emph{N} (Fig. 4) suggests
the ``dog hair'' stage of forest succession in which, post fire or other
disturbance, the abundance of small trees increases rapidly,\emph{E}
/\emph{N} decreases, and then self-thinning brings the system to a quasi
steady state, until the next disturbance.

We also examined in a lowest order approximation the effect on the SAD
and the MRDI that result from a variety of perturbations in the rate
constants (Figs. 5). Importantly, different perturbations result in
different patterns of deviation from static METE predictions for both
the distribution of abundances over species and metabolic rates over
individuals. These signature patterns can provide a way to identify the
processes that are driving ecosystem change under natural or
anthropogenic disturbance. Further analysis will determine whether the
results from a single iteration to t = 25 differ significantly from the
results we would obtain by carrying out 25 one-year iterations.

The SAD in the 50-ha BCI tropical forest plot, which deviates from the
log-series distribution predicted by METE, resembles the DynaMETE
prediction assuming a combined perturbation in death, growth and
migration rates. That same perturbation also results in an MRDI that
improves on the METE prediction for the high-metabolizing (large-size)
individuals, but over-predicts the number of the lowest-metabolism
individuals. Further study of different combinations of parameter
perturbations coupled with analysis of higher order iterations of the
dynamics are needed.

\textbf{Future Work}

\textbf{Flexibility of transition functions.} Our assumptions about the
functional forms of the transition functions are easily replaced with
alternatives. For the extinction process, Eq. 31 can be easily modified
to describe a minimum viable population size greater than 2. Alternative
forms of the ontogenic growth equation (e.g., Makareiva et al., 2004)
can be substituted for Eq. 29. Density dependence can be described by a
--d\textsubscript{1}\emph{n} \textsuperscript{2} term. Moreover, a wide
range of options are possible for modeling the dependence of speciation
rates on \emph{n} , \(\varepsilon,\) and the state variables. A
modification of the birth rate function can improve the realism of the
transition functions when applied to forest census data that are limited
to trees with some minimum threshold dbh, as for example with the
Smithsonian Tropical Forest census data. Entry into the data set arises
not from birth but from ontogenic growth into the smallest censused size
cohort.

\textbf{When the SAD and MRDI are not measured.} If empirical values of
the state variables and their time derivatives are available, they can
be plugged directly into Eqs. 14-18. Initial investigation suggests that
if the unperturbed values of the transition function rate constants are
known, then the range of possible perturbed values of those constants
that permit solutions for the Lagrange multipliers is fairly tightly
constrained by the constraints imposed by the measured time derivatives
of the state variables. Thus the causes of disturbance could be
identified without requiring knowledge of the abundance and metabolic
rate distributions.

\textbf{Space.} To include space explicitly in DynaMETE, one could
modify the assembly (or colonization) rule (Harte, 2011) that generates
the species-level spatial distributions in METE (Conlisk et al. 2007,
Brush \& Harte 2020) by combining the assembly rule with explicit
demographic perturbations incorporated in the transition
function\emph{f} (\emph{n} , \(\varepsilon)\).

\textbf{Traits.} DynaMETE currently incorporates individual metabolic
rate (or body size) and abundance. Other traits are implicitly assumed
to be distributed across species and individuals in such a way as to
result in equal fitness opportunities for each taxon. Under disturbance,
however, some trait values may confer more or less fitness and result in
preferential changes in growth and abundance. Letting the transition
rates, \emph{f} , \emph{h} , and \emph{q} , depend upon additional trait
values, as well as upon \emph{n} and \(\varepsilon\), could allow
investigation of disturbances that affect particular groups of species
or populations more than others, enhancing understanding of how traits
mediate the responses of macroecological metrics to disturbance.

\textbf{Higher taxonomic categories.} The scope and realism of static
METE was enhanced by including an additional state variable
corresponding to the number of families in the community (Harte et al.,
2015). DynaMETE can also accommodate additional state variables
corresponding to the richness of higher taxonomic categories. Applying
such a modified theory at sufficiently long time scales could perhaps
provide insight into the mechanisms of diversification at higher
taxonomic levels, and improve DynaMETE's realism at short time scales..

\textbf{Some broader issues.} Both anthropogenic stresses and natural
disturbances can cause state variables to rapidly change, and thus
systems experiencing either type of disturbance can fall within
DynaMETE's criterion for a dynamic ecosystem. Although evidence reviewed
in the Introduction suggests that systems undergoing either type of
disturbance exhibit macroecological patterns that deviate from METE, it
is unclear whether DynaMETE will be applicable both to ecosystems that
are disturbed anthropogenically, and to ecosystems that are adapted to
natural disturbance regimes. Given the importance of finding early
warning signals that distinguish human impact on ecosystems from the
effects of natural disturbance regimes, this is a high priority.

In a provocative article (Goldenfeld and Woese, 2011) suggest that while
physics makes a clean separation between the state of a physical system
and the equations that govern the time-evolution of the system,
successful biological theory will inevitably be self-referential or
recursive in the sense that the state of the system will strongly enter
into the equations that govern dynamics. We observe that this is true of
DynaMETE (Eqs. 14-23; 35-37); the state variables appear explicitly in
the transition functions, which in turn govern state dynamics.

Many academic fields seek to unify complex micro- and macro-level
dynamics, in a speculative vein we suggest that the proposed iterative
procedure at the core of DynaMETE could be of possible application in,
for example, economics (Golan, 2018) and in statistical physics (Jaynes,
1957; 1982), where in both of these fields, static equilibrial patterns
can be captured by MaxEnt but non-equilibrial dynamics has remained
elusive.

\textbf{CONCLUSION.}

Although ecosystems are dynamic and macroecological patterns do not
remain constant in the face of disturbance, dynamic macroecology has not
been adequately explored. DynaMETE, a theory of dynamic macroecology,
hybridizes explicit mechanisms driving change with a powerful inference
procedure, MaxEnt, from information theory. By predicting how patterns
in macroecology shift under anthropogenic perturbations, or under
natural successional and evolutionary forces, DynaMETE can contribute to
better understanding the fate of disturbed ecosystems, to improving
conservation and management strategies in ecology, to developing early
warning indicators of ecosystems in transition or at the edge of
collapse, to identifying specific processes driving ecological change,
and to clarifying the roles of ecology and evolution in diversifying
ecosystems. DynaMETE is a candidate dynamic theory of macroecology in
the Anthropocene.



\textbf{Acknowledgements}

Funding for this project was provided by grant DEB 1751380 from the US
National Science Foundation. JH thanks the Santa Fe Institute and the
Rocky Mountain Biological Laboratory for intellectual stimulus and
logistic support. MB acknowledges the support of the Natural Sciences
and Engineering Research Council of Canada (NSERC),
{[}PGSD2-517114-452018{]}. We are grateful to Richard Barnes, Juliette
Franzman, Amos Golan, Justin Kitzes, Erica Newman, Andrew Rominger, Roya
Safaeinili, and Jade Zhang for illuminating discussions.

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\begin{longtable}[]{@{}lll@{}}
\toprule
\begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
Step\strut
\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
Known\strut
\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
Derived from known\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Before the perturbation}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Before the perturbation}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Before the perturbation}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
1. In the steady state past, the static structure function is derived
from static state variables; this step generates Eq. 3.\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
\(\left\{X_{\text{past}}\right\};\) \(\left\{\frac{dX_{\text{past}}}{dt}\right\}=0;\text{\ \ }\) \(\lambda_{j,\text{past}}=0\)
\((j=3,4,5)\)\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
\(\lambda_{1,\text{past}};\ \ \lambda_{2,\text{past}};\ R_{\text{past}}\)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Initializing the system right after the perturbation is imposed
at t = 0}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Initializing the system right after the perturbation is imposed
at t = 0}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Initializing the system right after the perturbation is imposed
at t = 0}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
2. A perturbation is now imposed, expressed by a change in one or more
of the parameters, \{\emph{c}\}, in the transition functions,
\emph{k}(\{\emph{X}\},\{\emph{c}\}). The initial time derivatives of the
state variables are calculated from Eqs. 21-23.\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\(\left\{X_{0}\right\}=\left\{X_{\text{past}}\right\};\ {\lambda_{j,0}=\lambda}_{j,\text{past}};\) \emph{R}\textsubscript{0} =
\emph{R}\textsubscript{past}; perturbed transition
functions\(\ \)\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\(\{\frac{dX_{0}}{dt}\}\) \(\neq\) 0\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{After the system is initialized to the Perturbation}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{After the system is initialized to the Perturbation}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{After the system is initialized to the Perturbation}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
3. The state variables are updated using their time derivatives using
Eq. 20.\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\(\left\{X_{0}\right\};\ \left\{\frac{dX_{0}}{dt}\right\}\)\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\(\left\{X_{1}\right\}\)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
4. The transition functions are updated by substituting updated state
variables.\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\{\emph{k}(\{\emph{X}\textsubscript{0}\},\{c\textsubscript{0}\})\}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\{\emph{k}(\{\emph{X}\textsubscript{1}\},\{c\textsubscript{1}\})\}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
5. The time derivatives of the state variables are up-dated using Eqs.
21-23.\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\(\left\{k\left(\left\{X_{1}\right\},\left\{c_{1}\right\}\right)\right\};\ R_{0}\)\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\(\left\{\frac{dX_{1}}{dt}\right\}\ \)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
6. The structure function is updated from the constraints derived above
using Eqs. 14-18.\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\(\left\{X_{1}\right\};\{\frac{dX_{1}}{\text{dt}}\}\);
\{\emph{k}(\{\emph{X}\textsubscript{1}\},\{c\textsubscript{1}\})\}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\({\lambda_{1,1},\ ..,\lambda_{5,1},\ R}_{1}\)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Subsequent steps repeat steps 3-6. With each update of the
structure function, the updated effects of the perturbation on abundance
and metabolic rate distributions can be derived.}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Subsequent steps repeat steps 3-6. With each update of the
structure function, the updated effects of the perturbation on abundance
and metabolic rate distributions can be derived.}\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
\textbf{Subsequent steps repeat steps 3-6. With each update of the
structure function, the updated effects of the perturbation on abundance
and metabolic rate distributions can be derived.}\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

Table 1. Inference dynamics. An iterative process for solving the
dynamic state variable equations. Time is labeled with the
subscript\emph{i} . The notation \{\emph{X\textsubscript{i}} \}, is
shorthand for the set of state variables, \emph{S\textsubscript{i}}
,\emph{N\textsubscript{i}} , and\emph{E\textsubscript{i}}
\textsubscript{;}\{d\emph{X\textsubscript{i}} /d\emph{t} \} is shorthand
for the set of their time derivatives; and \{\emph{k}
(\{X\emph{\textsubscript{i}} \}, \{\emph{c} \textsubscript{i}\})\} is
shorthand for the set of dynamic transition functions, \emph{f} ,
\emph{h} , and \emph{q} .

E\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
Parameter & Value\tabularnewline
\midrule
\endhead
\emph{b}\textsubscript{0} & 0.2\tabularnewline
\emph{d}\textsubscript{0} & 0.2\tabularnewline
\emph{m}\textsubscript{0} & 500\tabularnewline
\emph{w}\textsubscript{0} & 1.0\tabularnewline
\emph{w}\textsubscript{10} & 0.4096\tabularnewline
\emph{E}\textsubscript{c} & 2x10\textsuperscript{7}\tabularnewline
\(\mu=S_{\text{meta}}^{-1}ln(\frac{1}{\beta_{\text{meta}})}\) & 0.0219\tabularnewline
State Variables &\tabularnewline
S & 320\tabularnewline
N & 230,000\tabularnewline
E & 2.04 x 10\textsuperscript{7}\tabularnewline
\bottomrule
\end{longtable}

Table 2. Parameter values for the transition functions and
state-variables in a BCI-like forest ecosystem in which diversification
is driven solely by immigration. The rationale for the parameter values
is given in SI-E.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
\emph{m}\textsubscript{0} = 5000\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
\emph{d}\textsubscript{0} = 0.20.25\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
\emph{w}\textsubscript{0} = 1.00.95\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
\(\mu\) = 0.02190.024 \emph{m}\textsubscript{0} = 500200
\emph{d}\textsubscript{0} = 0.20.235 \emph{w}\textsubscript{0} =
1.00.99\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Figure\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
5a, 5b\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
5c, 5d\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
5e, 5f\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
5g, 5h\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
State variables and their rates of change used for constraints\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
State variables and their rates of change used for constraints\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
State variables and their rates of change used for constraints\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
State variables and their rates of change used for constraints\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
State variables and their rates of change used for constraints\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\emph{S}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
314.2\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
319.6\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
320.8\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
314.9\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\emph{N}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
217962\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
144430\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
262537\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
166984\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\emph{E}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
2.0380 x 10\textsuperscript{7}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
1.7331 x 10\textsuperscript{7}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
1.8280 x 10\textsuperscript{7}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
1.7822 x 10\textsuperscript{7}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
d\emph{S/}d\emph{t}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.242\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.009\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0246\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.196\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
d\emph{N/}d\emph{t}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-418\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-1052\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
2350\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-758\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
d\emph{E/}d\emph{t}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-4349\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-112262\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-48162\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-88709\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Lagrange multipliers\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Lagrange multipliers\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Lagrange multipliers\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Lagrange multipliers\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Lagrange multipliers\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\(\lambda_{1}\)\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.00047481\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0026949\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.0059150\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0015918\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\(\lambda_{2}\)\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.000014104\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
7.8932 x 10\textsuperscript{-6}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.000047504\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.000011608\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\(\lambda_{3}\)\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.10275\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.19247\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.46754\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.18211\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\(\lambda_{4}\)\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.000028926\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.00023261\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.00040369\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.00013171\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
\(\lambda_{5}\)\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0025553\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.0091482\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.22996\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
-0.0032439\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

Table 3. Perturbations, constraints, and resulting Lagrange Multipliers
used to generate Figures 5a-5h. The text describes how the constraints
are derived.

Figure Captions.

Figure 1. The architecture of DynaMETE. A. Selected mechanisms are
incorporated into transition functions. B. The time derivatives of state
variables update the state variables. C. The transition functions, which
depend upon the state variables are updated. D. Updated state variables
and transition functions are averaged over the prior (time \emph{t} )
structure function to update the time derivatives of the state
variables. E. Under updated constraints and transition functions (dashed
box), MaxEnt updates the structure function. F. The macroecological
metrics are updated from the updated structure function. G. Steps B-F
are iterated.

Figure 2. Comparison of up- and down-scaled species richness using METE
and DynaMETE, starting with identical species richness and abundance at
the middle scale shown. The values of \emph{N} , a proxy for area, span
a scale range of 2\textsuperscript{7}. Transition function parameters
and\emph{S} and \emph{N} values for the middle scale are from Table 2;
at larger or smaller scales \emph{E} \textsubscript{c} and\emph{m}
\textsubscript{0} are assumed to scale linearly with area and the other
parameters are held constant.

Figure 3. Responses of state variables to perturbations simulated from
Eqs. 38-40: a. reduction of immigration rate, \emph{m}
\textsubscript{0}; b. increase in death rate, \emph{d}
\textsubscript{0}; c. reduction in growth rate, \(\omega_{0};\ \)d.
increase in the death rate and reduction in the immigration and
ontogenic growth rates. The inset in 3d shows the state variable
trajectories from 1985-2015 in the BCI 50 ha tropical forest plot. The
censuses include trees with dbh\emph{\textgreater{}} 1 cm . Data from
Condit (2019); Hubbell et al., (2005). The inset assumes that the
metabolic rate of individuals scales linearly with basal area.

Figure 4. Predicted recovery of state variables to their steady state
values in Table 1, as predicted from Eqs. 38-40, with steady state
parameters and each initial state variable equal to 80\% of its steady
state value. The monotonic rise in S to steady state, along with the
sizeable overshoot and then damped oscillation in \emph{N} , and the
smaller overshoot and then damped oscillation in \emph{E} occur for a
wide range of initially depleted state variables, steady state state
variables, and parameter choices.

Figure 5. The effect of perturbations on the species abundance
distribution (SAD) and the distribution of metabolic rates over
individuals (MRDI). The relevant perturbation is in each plot's title.
Insets in Figs. 5g and 5h show BCI data (see caption to Fig. 3).



\textbf{Figure 1.}

\textbf{}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/1/1}

\end{center}
\end{figure}

\textbf{Figure 2.}

\textbf{}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/2/2}

\end{center}
\end{figure}

\textbf{Figure 3a.}

\textbf{}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/3a/3a}

\end{center}
\end{figure}

\textbf{Figure 3b.}

\textbf{}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/3b/3b}

\end{center}
\end{figure}

\textbf{Figure 3c.}

\textbf{}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/3c/3c}

\end{center}
\end{figure}

\textbf{Figure 3d.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}



\textbf{Figure 4.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/4/4}

\end{center}
\end{figure}

\textbf{Figure 5a.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}



\textbf{Figure 5b}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}



\textbf{Figure 5c.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}



\textbf{Figure 5d.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}

\end{center}
\end{figure}



\textbf{Figure 5e.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}

\end{center}
\end{figure}



\textbf{Figure 5f.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image13/image13}

\end{center}
\end{figure}

\textbf{Figure 5g.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image14/image14}

\end{center}
\end{figure}

\textbf{Figure 5h.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image15/image15}

\end{center}
\end{figure}

\selectlanguage{english}
\FloatBarrier
\end{document}

