Species-Abundance Distribution: Neutral regularity or idiosyncratic stochasticity? Fangliang He Department of Renewable Resources University of Alberta

Law Order

Species-Abundance Relationships Number of species Abundance Speciesabundance Sp1 1 Sp2 1 Sp3 1 Sp4 2 Sp5 2 Sp6 5 Sp7 6 Sp8 6 Sp9 10 Sp10 50 Sp …… Why species cannot be equally abundant?

Logseries distribution where (the biodiversity parameter, Fisher’s  ) Lognormal distribution n x

Neutral Niche w d d x Idiosyncrasy xx Species Ecological equivalence. Individuals are identical in vital rates. Coexistence is determined by drift Each species is unique in its ability to utilize and compete for limiting resources and follows a defined pattern. Niche differentiation is prerequisite for coexist. Any factor can contribute to population dynamics. Each species is unique and follows no defined patterns. Coexist. is determined by multiple factors

Logseries Distribution Derived From Neutral Theory (the biodiversity parameter) where Volkov, Banavar, Hubbell & Maritan Neutral theory and relative species abundance in ecology. Nature 424:

Maximum Entropy Predict species abundance from life-history traits Derive logseries distribution

Entropy: Linking microscopic world to macroscopic worlds n1n1 n3n3 n2n2 N Number of species Abundance H: macroscopic quantity W: microscopic degrees of freedom (multiplicity)

Entropy: the Probability Perspective Entropy measures the degree of uncertainty. p1p1 p3p3 p2p2 N

Tossing a Coin A fair coin has the maximum degrees of freedom (largest W), thus max entropy.

The 2 nd Law of Thermodynamics: Systems tend toward disorder n1n1 n3n3 n2n2 N n1n1 n3n3 n2n2 N n1n1 n3n3 n2n2 N f(x)f(x) x f(x)f(x) x Maximum

n1n1 n3n3 n2n2 The 2 nd Law Constraints Without any prior knowledge, the flattest distribution is most plausible. This is the 2 nd law of thermodynamics. f(x)f(x) x Two Opposite Forces

i =1, 2, …, 6 Predicting Dice Outcome Using MaxEnt

The Boltzmann Distribution Law Probabilities: Scores: The total # of ways that N can be partitioned into a particular set of {n 1, n 2, …, n 6 }, e.g., {2, 3, 1, 4, 0, 2}: Stirling’s approximation:

The Boltzmann Distribution Law Entropy Constraints Math constraints Objective function using Lagrange multipliers:

The Boltzmann Distribution Law Entropy Constraints Math constraints

The Boltzmann Distribution Law

Shipley et al’s work Shipley, Vile & Garnier From plant traits to plant community: A statistical mechanistic approach to biodiversity. Science 314: Use 8 life-history traits to predict abundance for 30 herbaceous species in 12 sites along a 42-yr chronosequence in a vineyard in France. Community-aggregated traits: Probability constraint: trait j sp i site k time x Entropy (degrees of freedom):

Community-aggregated traits: Probability constraint: Entropy (degrees of freedom): Objective function using Lagrange multipliers:

The predicted abundance:

Criticisms Circular argument Entropy is not important Random allocation of traits to species would also predict abundance Species abundance does not follow exponential distribution Roxhurgh & Mokany Science 316:1425b. Marks & Muller-Landau Science 316:1425c.

The Boltzmann Law = Logistic Regression

The Idiosyncratic Theory N individuals belong to S species x

The total # of ways that N can be partitioned into a particular set of S species: Relative Entropy Prior

The two most basic constraints

Maximize H subject to constraints:

Pueyo, He & Zillio. The maximum entropy formalism and the idiosyncratic theory of biodiversity. Ecol. Lett. (in press). Prior Geometric distribution as prior: 1.Species-abundance is invariant at different scales. 2.log(n) is uniform distribution. Logseries Distribution

Lognormal distribution Maximize H subject to constraints:

Conclusions 1.Ecological systems are structured by two opposite forces. One is the Second Law of thermodynamics which drives the systems toward disorder (maximum degrees of freedom). The other is constraints that maintain order by reducing the degrees of freedom. 2.The Boltzmann Law provides a tool to model abundance in terms of traits. The Law is equivalent to logistic regression. 3.Logseries and lognormal distributions are the emerging patterns generated by the balance. 4.Logseries arises from complete noise in idiosyncratic theory, but from strict regularity (identical demographics) in neutral theory. It therefore does not contain information about community assembly. The MaxEnt shows that the neutral theory is just one of a large number of plausible models that lead to the same patterns of diversity. 5.Many biodiversity patterns (Pareto, lognormal) can be readily explained by the idiosyncratic theory.