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Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Reading the Secrets of Biological Fluctuations
Carl Boettiger
UC Davis
June 27, 2008
Carl Boettiger, UC Davis Biological Fluctuations 1/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
1 Noisy Biology
2 Fluctuation Regimes
3 Model Choice
4 Macroscopic Phenomena
5 Fluctuation Dominance
Carl Boettiger, UC Davis Biological Fluctuations 2/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Why study fluctuations?
Biology is noisy and we want to understand it.
Stochasticity can drive phenomena we would miss indeterministic models.
Fluctuations hold the key to deeper biological understanding?
Grenfell et al. (1998) Nature
Carl Boettiger, UC Davis Biological Fluctuations 3/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Variables at the Macroscopic and Individual Levels
Deterministic models describe macroscopic behavior
Individual based model are described by transition ratesbetween states – a Markov process
Macroscopic variable φ is independent of details of system(intensive), i.e. population density
Individual-based variable n depends on system size(extensive), i.e. population number
In a given area Ω with a macroscopic density φ, we’d expectto find 〈n〉 = φΩ on average, which is more accurate withlarger Ω.
Carl Boettiger, UC Davis Biological Fluctuations 4/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Theory of Fluctuations
Markov process Linear Noise Approximation
=⇒ n = Ωφ+Ω1/2ξ =⇒
Fundamental Equations
dφdt
= α1,0(φ)+α′′1,0(φ)σ2 (1)
dσ2
dt= 2α′
1,0(φ)σ2 + α2,0(φ) (2)
α1,0(φ) = b(φ)− d(φ), α2,0 = b(φ) + d(φ)
Carl Boettiger, UC Davis Biological Fluctuations 5/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
1 Noisy Biology
2 Fluctuation Regimes
3 Model Choice
4 Macroscopic Phenomena
5 Fluctuation Dominance
Carl Boettiger, UC Davis Biological Fluctuations 6/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Distinct Fluctuation Regimes
dn
dt= c
n
N
“1− n
N
”| z
bn
− en
N|zdn
dσ2
dt= 2α′1,0(φ)σ2 + α2,0(φ)
Carl Boettiger, UC Davis Biological Fluctuations 7/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Near Equilibrium: Fluctuation Dissipation Regime
In the dissipation regime, fluctuations exponentially relax to theequilibrium level
σ2 = b(n)+d(n)2[d′(n)−b′(n)]
N = 1000, e = 0.2,c = 1
n = Nˆ1− e
c
˜= 800
σ2 = N ec
= 200
Dots are simulationaverages, lines aretheoretical prediction
Carl Boettiger, UC Davis Biological Fluctuations 8/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Fluctuation Enhancement
With an initial condition starting deep in the enhancement regime,fluctuations grow exponentially. At N = 400, dissipation takes overand fluctuations return to the same equilibrium as before.
Carl Boettiger, UC Davis Biological Fluctuations 9/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
1 Noisy Biology
2 Fluctuation Regimes
3 Model Choice
4 Macroscopic Phenomena
5 Fluctuation Dominance
Carl Boettiger, UC Davis Biological Fluctuations 10/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Which model best describes this data?
dn
dt= c
n
N
“1− n
N
”| z
bn
− en
N|zdn
dn
dt= rn|z
bn
− rn2
K|zdn
. . . and why does it matter?
Carl Boettiger, UC Davis Biological Fluctuations 11/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Using the Information Hidden in the Fluctuations
1 Independently parameterizebirth & death rates, see which isdensity dependent
2 Works with single realization atequilibrium
3 With replicates: The dynamicequations can determinefunctions b(n) and d(n)
4 Uses more information to informmodel choice
5 Can discount weights of pointsfrom high-variance regions whenmodel-fitting
Predicted fluctuations
Carl Boettiger, UC Davis Biological Fluctuations 12/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
1 Noisy Biology
2 Fluctuation Regimes
3 Model Choice
4 Macroscopic Phenomena
5 Fluctuation Dominance
Carl Boettiger, UC Davis Biological Fluctuations 13/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Stochastic Corrections: Deflation and Inflation
α′′1,0(φ) < 0 =⇒ Fluctuations
suppress the average relative tothe deterministic approximation.
Our theory accurately predictsthe extent of this effect.
Recall α2,0 = bn + dn controlsthe magnitude of this effect.
Ecological and evolutionaryconsequences for whenvariability is favorable?
dφdt
= α1,0(φ)+α′′1,0(φ)σ2
Carl Boettiger, UC Davis Biological Fluctuations 14/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Fluctuation Phenomena: Deflation
dn
dt= c
n
N
“1− n
N
”| z
bn
− en
N|zdn
dφ
dt= α1,0(φ)+α′′1,0(φ)σ2
Carl Boettiger, UC Davis Biological Fluctuations 15/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
1 Noisy Biology
2 Fluctuation Regimes
3 Model Choice
4 Macroscopic Phenomena
5 Fluctuation Dominance
Carl Boettiger, UC Davis Biological Fluctuations 16/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Fluctuation Dominance
Far from equilibrium, enhancement can expand the fluctuationsuntil they reach the macroscopic scale.
Variance equation fails dramatically
Mean trajectory need not follow the deterministic trajectory
Bimodal distribution of trajectories can emerge
Conjecture: occurs when neighborhood exists for whichα1,0 ≈ 0 and α′
1,0 ≈ 0
Carl Boettiger, UC Davis Biological Fluctuations 17/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Breakdown of the approximation
Carl Boettiger, UC Davis Biological Fluctuations 18/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Breakdown of the Canonical Equation of AdaptiveDynamics
Carl Boettiger, UC Davis Biological Fluctuations 19/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Further Topics
This approach can be applied to a variety of stochastic processes inbiology. . .
The multivariate theory: multiple species or age structuredpopulations. Predicts covariances as well.
Macroevolutionary theory: inferring speciation andextinction rates from phylogenetic trees
Adaptive dynamics: quantifying uncertainty in the canonicalequation, correcting for fluctuations.
Carl Boettiger, UC Davis Biological Fluctuations 20/21
Noisy Biology Fluctuation Regimes Model Choice Macroscopic Phenomena Fluctuation Dominance
Acknowledgments
Advisors & Advice
Alan HastingsJoshua WeitzMany here for helpfuldiscussions!
Funding
DOE CSGFUC Davis PopulationBiology Graduate Group
Carl Boettiger, UC Davis Biological Fluctuations 21/21