Physics in Ecology:

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Physics in Ecology

• The Utility of the Useless

• Community and Conservation Ecology Group
• Rampal S. Etienne

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The Utility of the Useless

• Some wisdom from Taoism (Zhuang-Zi) The useless
  tree

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• Hear me out!

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Darwins old Indian ruins
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Darwins old Indian ruins

• Throw up a handful of feathers, and all must
  fall to the ground according to definite laws
  but how simple is the problem where each shall
  fall compared to that of the action and reaction
  of the innumerable plants and animals which have
  determined, in the course of centuries, the
  proportional numbers and kinds of trees now
  growing on the old Indian ruins!

Darwin 1859. On the origin of species by means of
natural selection, or the preservation of
favoured races in the struggle for life. John
Murray
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Physics and ecology

• Can physics explain the diversity on the old
  Indian ruins as well as it can explain the
  falling of a handful of feathers?
• A new philosophy
• from detailed species/system specific
  (simulation) models to
• first-order (analytical) approximations of more
  general systems

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Mind the cultural differences

• Vegetation was quantified using ocular cover
  estimates.
• I just eyeballed the plant cover and might be off
  by an order of magnitude.
• Sample points were subjectively placed using the
  relevé method.
• Id already made my conclusions, and needed some
  data to support them.
• The site was surveyed using random meander
  transects and prioritized based on habitat.
• There was no way I was going to crawl through
  that poison oak looking for rare plants.

Mahony 2009. Ecology meets physics envy.
http//www.lablit.com/article/513
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Phenomena to be explained

• Species abundance distribution (SAD)
• Species-area relationship
• Species-time relationship
• Diversity-productivity relationship
• Diversity-hetereogeneity relationship
• Diversity-habitat (loss) relationship
• Phylogenetic diversity

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Physics in ecology

• Three fields mainly driven by physicists
• I. Neutral theory of biodiversity
• Alonso, Banavar, Chave, Cornell, Etienne,
  Haegeman, Maritan, McKane, ODwyer, Vanpeteghem,
  Volkov, Zillio
• Bell, He, Hubbell, Jabot, Pueyo, Rosindell,
  Walker
• II. Entropy maximization
• Banavar, Dewar, Etienne, Haegeman, Harte,
  Maritan, Porte, Zillio
• Loreau, Pueyo, Shipley
• III. Metabolic ecology

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I. The neutral theory of biodiversity
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Ecology and chance

• When we look at the plants and bushes clothing
  an entangled bank, we are tempted to attribute
  their proportional numbers and kinds to what we
  call chance. But how false a view is this!

Darwin 1859. On the origin of species by means of
natural selection, or the preservation of
favoured races in the struggle for life. John
Murray
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Perspectives in community ecology

• Classical view niche theory
• Functional differences between species cause
  differences in resource and energy requirement
  which define its niche.
• With sufficient variability in resources,
  multiple species can coexist.
• New view neutral theory
• Species are functionally equivalent.
• Species coexist in a stochastic balance between
  speciation / immigration and extinction.
• Motivation Paradox of the plankton

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Neutral theory ecological nihilism1
Stephen P. Hubbell
1Schilthuizen 2006 Bionieuws
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Ingredients of the neutral theory

• Neutrality all individuals in an ecological
  community are functionally equivalent, regardless
  of species
• Stochasticity
• Two sources of diversity
• Speciation
• Immigration

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Neutral theory is not really new

• Grinnells (1922) accidentals (singletons)
• Explained by dispersal
• Gleasons (1926) individualistic concept
• Independent species, chance, dispersal
• MacArthurs (1957) broken stick
• Often seen as niche model, but symmetric and
  stochastic
• Can be derived within neutral framework!
• MacArthur Wilson (1967) island biogeography
• Stochasticity, dispersal, speciation, extinction
• Caswell (1976) first dynamical neutral model
• Borrowing concepts from population genetics
• Lottery and voter models (Fagerström 1988,
  Bramson et al. 1996)

Etienne Alonso 2007. J. Stat. Phys. 128 485-510
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Why then so popular now?

• Because physicists got involved?
• Quantitative / analytical predictions
• Because it was popularized by an ecologist?
• Hubbells motivation was the amazing biodiversity
  in tropical forests unexplained by classical
  niche theory

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Theory vs. model

• Theory the fundamental idea
• Model an implementation of the idea

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Ingredients of the main neutral model

• Neutrality birth, death and immigration rates
  are equal for all individuals, regardless of
  species
• Stochasticity purely demographic
• Two scales mainland-island
• Local (community) immigration
• Regional (metacommunity) speciation by point
  mutation
• Zero-sum individuals saturate all limiting
  resources (e.g. space) ? constant size

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The world according to Hubbell
n
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Neutral theorys predictions

• Species-abundance distributions
• Essential a species abundance caused by
  adaptation or chance?
• Diversity patterns
• With area, time, productivity, heterogeneity,
  habitat loss, .
• Evolutionary characteristics
• speciation rates, species longevity, phylogeny

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Master equation approach (1)

• Master equation for abundance of one species

Vallade Houchmandzadeh 2003. Phys. Rev. E 58
061902
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Master equation approach (2)

• Master equation for abundance vector S

Haegeman Etienne 2009. Bull. Math. Biol. In
press. Etienne Haegeman 2009. Subm.
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Master equation approach (3)

• Master equation for abundance vector N

Allouche Kadmon 2009. Ecol. Lett. In press
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Genealogical approach

• A local community
• 7 individuals
• 2 species (R and G)
• 3 ancestors (1, 2 and 3)
• Events
• Death
• Immigration
• Coalescence

Etienne Olff 2004, Ecol. Lett. 7
170-175Etienne 2005. Ecol. Lett. 8
253-260Etienne 2007. Ecol. Lett. 10
608-618 Rosindell et al. 2008. Ecol. Inf. 3
259-271 Etienne 2009. J. Theor. Biol. 257
510-514 Etienne 2009. Ecology 90 847-852
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Solution for SAD of a sample
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Applications
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Species abundance distribution
Volkov et al. 2003. Nature 438 658-661. Many
other papers
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Species-area relationship
Rosindell Cornell 2007. Ecol. Lett.10 586-595
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Effect of habitat heterogeneity
Kadmon Allouche 2007. Am Nat. 170 443-454
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Effect of disturbance and productivity
Kadmon Benjamini 2006. Am Nat. 167 939-946
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Effect of speciation on the SAD

• Point mutation (blue)
• Constant per individual (solid)
• Constant per species (dashed)
• Random fission (red)
• Constant per individual (solid)
• Constant per species (dashed)
• Effect is weak when dispersal is limited

Etienne et al. 2007. Oikos 116 241-258 Haegeman
Etienne 2009. Bull. Math. Biol. In press.
Etienne Haegeman 2009. Subm.
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Point mutation vs random fission (1)

• Better fit of SADs

Etienne et al. 2007. Oikos 116 241-258 Etienne
Haegeman 2009. Subm.
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Point mutation vs random fission (2)

• But poorer prediction of
• Speciation rate
• Species longevity
• Average species lifetime
• Lifetime of highly abundant species
• Can be resolved by protracted speciation
• Speciation as a process rather than an
  instantaneous event

Ricklefs 2003. Oikos 100 185-192 Nee 2005.
Funct. Ecol. 19 173-176 Etienne Haegeman 2009.
Subm. Rosindell et al. 2009. Subm.
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Robustness of the predicted SAD

• Zero-sum assumption can be relaxed to
• Independent species
• Community-level density dependence
• Neutrality can be relaxed to
• Demographic trade-offs
• Heterogeneous habitats

Etienne et al. 2007. J. Theor. Biol. 248 522-536
Etienne Haegeman 2008. J. Theor. Biol. 252
288-294. Allouche Kadmon 2009. J. Theor. Biol.
258 274-280. Allouche Kadmon 2009. Ecol. Lett.
In press.
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The merits of neutral theory (1)

• New philosophy of science in ecology
• Parsimony Ockhams razor
• Hypothesis testing against null model
• Classical or Bayesian
• To what extent do species differences matter?
• First order approximation
• Allows studying the effect of a particular
  mechanism without confounding / complicating
  differences between species
• Difference between theory and model

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The merits of neutral theory (2)

• Information content in / confrontation to data
• Sampling theory predictions are for samples
• Connection between evolutionary and ecological
  factors
• Effect of speciation mode on community structure
• Matter of scale of perspective
• Population ecologists treat individuals of same
  species as functionally equivalent in simplest
  model
• Metapopulation ecologists treat populations of
  species as functionally equivalent in simplest
  model

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The merits of neutral theory (3)

• Emergent neutrality (Holt 2006)
• Speciation (e.g. allopatric) consistent with
  functionally equivalent species
• At large spatial scales, dispersal limitation may
  be overwhelmingly dominating
• Convergent evolution because of similar
  constraints
• Emerging guilds of functionally equivalent
  species (Scheffer Van Nes 2006, Bonsall et al.
  2004)

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Dont throw out the baby with the bathwater!
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II. Entropy maximization
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Ecology and thermodynamics (1)

• Ecological complexity poses challenges to
  con-ventional scientific ways of knowing. Ecology
  is not like thermodynamics, in which complexity
  can be simplified through statistical averaging
  of large numbers of identically behaving
  components

Taylor 2005. Unruly Complexity Ecology,
Interpretation, Engagement. University of Chicago
Press
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Ecology and thermodynamics (2)

• Aims to understand and predict the macroscopic
  behaviour of complex systems consisting of large
  numbers of interacting microscopic components.
• Due to the large number of components
  (individual organisms), the same macroscopic
  behaviour can be realised in many different ways
  microscopically.
• Ecological patterns are expressions of the
  community-level behaviour that can be realised in
  the greatest number of ways at the individual
  level.

Dewar Porte 2008. J. Theor. Biol. 251 389403
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Ecology and thermodynamics (3)

• Ecosystems have been characterized as
  medium-number systems for which both the
  approaches of mechanistic and statistical
  modelling are problematic there are too many
  components to describe each of the components
  explicitly, and there are not enough components
  to work with averaged properties.

Haegeman Loreau 2008. Oikos 117 1700-1710
citing ONeill et al. 1986. A hierarchical
concept of ecosystems. Princeton Univ. Press

• Empirical studies should settle the question
  whether this method is useful.

Haegeman Loreau 2008. Oikos 117 1700-1710
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Entropy maximization (1)

• Aim predicting macroscopic behavior of complex
  systems consisting of large numbers of
  microscopic degrees of freedom, subject to given
  constraints
• Constraints (C) are generally insufficient to
  determine the microstate i
• Focus on the probability pi of microstate i
• Macroscopic quantity Q is expectation value

• Problem to be solved construct p that satisfies
  constraint but contains no other information

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Entropy maximization (2)

• Maximizing entropy H does exactly that

• Only p that maximizes H encodes just information
  on C relative to prior information q

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Example 1 vineyards

• In 12 vineyards during 42 years, measurements of
• relative abundance of 30 species
• average trait value for 8 traits
• Constraints

Shipley et al. 2006. Science 314 812814
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Example 1 performance
Shipley et al. 2006. Science 314 812814
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Example 1 problems (1)

• Circularity
• Averaged trait values were computed using species
  traits and relative abundances
• Low-dimensionality
• One-individual formulation shows only part of
  power of EM
• Triviality
• Constraints almost completely determine p

Haegeman Loreau 2008. Oikos 117
1700-1710 Haegeman Loreau 2009. Oikos. In press.
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Example 1 problems (2)
Haegeman Loreau 2008. Oikos 117 1700-1710
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Example 2 nitrogen-limited grasslands

• S 26 species constrained by space (62
  individuals per m2) and resources (5.9 g nitrogen
  m-2 yr-1)

• Solution

Dewar Porte 2008. J. Theor. Biol. 251 389-403
using data of Harpole Tilman 2006. Ecol. Lett.
9 1523.
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Example 2 performance
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Example 2 analogies

• ß and µ are analogous to inverse temperature and
  chemical potential (of a physical system with
  constraints on average number of particles and
  energy)

• Consequence Two interacting communities will
  eventually have equal ß and µ.

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Example 3 general macro-ecology (1)

• S0 species in area A0 with total number of
  individuals N0 and energy E0, probability density
  R(n,e), spatial abundance distribution PA(n)

Harte et al. 2008. Ecology 89 27002711
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Example 3 performance (1)

• Species-abundance distribution

• Species-level spatial abundance distribution

Harte et al. 2008. Ecology 89 27002711
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Example 3 performance (2)

• Species-area curve

• Endemics-area curve

Harte et al. 2008. Ecology 89 27002711
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Example 3 discussion

• No prior q needed 1/n behavior predicted, not
  assumed
• Energy distribution is predicted, not assumed
• With more constraints, EM entails non-realistic
  species-abundance distribution

Harte et al. 2008. Ecology 89 27002711
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Example 4 spatial abundance distribution (1)

• Different formulations
• State of system is single-cell abundance
• State of system is abundance vector for all cells
• With abundance vectors several EM solutions are
  possible, differing in
• Constraints
• Hard total number of individuals is N
• Soft average total number of individuals is N
• Configurations
• Labeled
• Unlabeled

Haegeman Etienne. 2009. Subm.
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Example 4 spatial abundance distribution (2)
multinomial, RP model
binomial

• Joint distribution Marginal distribution

geometric
uniform
geometric
negative hypergeo-metric
SD model
negative binomial
Haegeman Etienne. 2009. Subm.
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Example 4 spatial abundance distribution (3)

• Joint distribution approach reveals that there is
  not one unique EM solution
• Harte et als different models can all be seen as
  EM solutions

• Unclear how Hartes model fits in
• Mostly like unlabeled-hard/soft EM problem
• But these are not scale-consistent
• Choice of prior and system description important
  (and exchangeable)

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A future for physics in ecology

• Macro-ecological patterns may have simple,
  unifying physics-like explanations
• Many ecological details inessential
• Future work
• Neutral theory
• Space
• Time
• Phylogenetics
• Entropy maximization
• Prior
• Constraints

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Thanks to