John Harte, UC Berkeley INTECOL London August 20, 2013 Maximum Entropy and Mechanism: Prospects for a Happy Marriage

MaxEnt Approach to Macroecology To predict patterns in: abundance distribution energetics network structure across taxonomic groups across spatial scales across habitat categories without adjustable parameters, without arbitrary choice of governing mechanisms and thereby to reach insight into mechanism.

Here “entropy” refers to information entropy, not thermodynamic entropy. Information entropy is a measure of the lack of structure or detail in the probability distribution describing your knowledge of a system. P(x) xx Lower Entropy Higher Entropy Maximum Entropy? Just what is being maximized?

Ingredients of a Fundamental Theory of Macroecology PREDICTIONS (Metrics of Ecology)  Species-Area Relationships  Endemics-Area Relationships  Abundance & Body Size Distributions  Spatial Aggregation Patterns  Web Structure & Dynamics  Species Distribution across Genera, Families, etc. INPUT DATA State Variables: S N E THEORY MaxEnt: An inference procedure based on information theory APPLICATIONS ♣Species Loss under Habitat Loss ♣Reserve design ♣Web Collapse under Deletions ♣Scaling up Biodiversity A Candidate Macroecological Theory: The Maximum Entropy Theory of Ecology (METE)

Examples of Validated Predictions z = 1/4 Harte et al., Ecology Letters, 2010; Harte, Oxford U. Press, 2011 MaxEnt predicts: all species-area curves collapse onto a universal curve log(N(A)/S(A)) MaxEnt predicts: the fraction of species that are rare

S, N, E Resource constraints : Evolutionary constraints : taxonomy/ phylogeny W ater, P hosphorus,.. O rder, F amily, G enus At the Frontier of METE Core theory L inkages Trophic interaction constraints :

Original Theory Alters size-abundance distribution Alters predicted rarity Extending and Generalizing METE

If (S,N,E) (F,S,N,E), then the energy-abundance relationship is modified: Including higher taxonomic levels as constraints m labels the species richness of the family (or order, …) that the species with abundance n is in. Log(metabolic rate) Log(abundance) Families of differing species richness (F = family or other higher order category) The Damuth rule splits apart!

r - 1 = # additional resources Including additional resource constraints (in addition to energy, E) The log-series SAD becomes: The inclusion of additional resource constraints predicts increased rarity

Kempton and Taylor (1974) Abundance Distribution of Rothampsted Moths Relatively undisturbed fields: Fisher log series distribution (predicted by METE) Fields recently left to fallow and in transition: Lognormal distribution 150 y Test of abundance distribution Arthropod abundance distributions from Hawaiian sites of different ages and stages of speciation 4 My Data from Dan Gruner The theory fails to predict patterns in ecosystems undergoing relatively rapid change Similar pattern of success and failure for body size distributions! X Species-area slopes for plants in successional sites Species-area slopes for plants in successional sites (aftermath of an erosion event) lie well above the scatter around the universal curve

SUMMARY: METE is a relatively successful theory of macroecology. Success does not imply mechanism does not matter! Mechanisms are incorporated into the values of the state variables, and we still need to understand what they are. Failure of the core theory tells us that more mechanistic information than is captured by the state variables is needed to predict patterns in ecology. Testing various extensions of the theory allow us to identify the role of particular mechanisms.

Thanks: To my Collaborators: Erin Conlisk Adam Smith Xiao Xiao Mark Wilber Justin Kitzes Andrew Rominger Ethan White Chloe Lewis Erica Newman David Storch Tommaso Zillio Xiao Xiao To Other Sources of Data: J. Green R. Krishnamani J. Godinez W. Kunin R. Condit P. Harnik K. Cherukumilla E. White D. Gruner J. Goddard STRI D. Bartholomew To the Funders: NSF, Miller Foundation,Gordon and Betty Moore Foundation To my Hosts during the development of METE: Santa Fe Institute, Rocky Mountain Biological Laboratory, NCEAS, The Chilean Ecological Society, Charles University, University of Padua

Deviations from the MaxEnt theory x Measure of rapidity of change Hypothesis : But the pattern of deviation of abundance distributions from the predicted Fisher log series depends on whether the system is collapsing or diversifying. This is just the first step in relating the mechanisms that disrupt an ecosystem to patterns predicted by macroecological theory.