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.