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, p (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, A , of
the ecosystem, total number of s pecies, S , within the
broad taxonomic group in that ecosystem, summed number of individuals,N , in those species, and summed metabolic rate, 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, S and 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-awith a < 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.