Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models

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Unifying evolutionary dynamics From individual
stochastic processes to macroscopic models
ANR MAEVGrenoble, 19-20 octobre 2006

• Nicolas Champagnat (INRIA Sophia-Antipolis)
• Regis Ferriere (UPMC-ENS)
• Sylvie Meleard (Ecole Polytechnique)
• Theor. Pop. Biol. 69297-321 (2006)

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Ecological and evolutionary dynamics of
populations

• Accepted wisdom
• Ecological change occurs on generational
  timescale
• Evolutionary change is longer by orders of
  magnitude
• But rapid evolution in adaptive traits is common
• Industrial melanism
• Evolution of resistance in pathogens
• Genetic adaptation to harvesting
• Experimental evolution in microorganisms

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Ecological and evolutionary dynamics of
populations

• Short-term directional evolution is usually rapid
• Evolution can be so fast as to become part of
  ecological processes
• Documented effects on population dynamics
  (Yoshida et al 2003), trophic interactions
  (Hairston et al. 1999), ecosystem processes
  (Elser et al. 2000)
• Why is long-term evolution apparently so slow?
  Interspersed stasis and reversals
• Best documented examples in the wild
• Darwins finches responding to fluctuating
  rainfall (body size, beak size)
• Freshwater invertebrates (daphnia) and
  vertebrates (guppies) responding to fluctuating
  fish predation

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Major implications for biodiversity management

• Majority of documented cases of rapid evolution
  involve anthropogenic pressures
• Adaptations morphological traits, physiology,
  life history, phenology, behavior
• Will adaptive evolution necessarily rescue
  populations from anthropogenic pressures?
• Identifying ecological and evolutionary factors
  of population viability
• Identifying adverse ecological effects of
  evolutionary responses
• How can management and restoration practices
  account for/use evolutionary processes?
• Need for mathematical models

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Biological framework for eco-evolutionary
dynamics The Environment Feedback Loop
Ferriere, Le Galliard in Dispersal (OUP,
2001) Metz et al. TREE 1992 Day Taylor JTB
1998 van Baalen Rand JTB 1998
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Modelling eco-evolutionary dynamics

• Stochastic individual processes
• Individuals reproduce die
• Individuals interact w/ each other
• Interaction, reproduction, death influenced by
• Individual phenotype adaptive trait x
• Other individuals differential ecological
  success
• Genetic variation of trait x
• Here, asexual reproduction

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Example Eco-evolutionary dynamics of cell
populations

• Cell size evolution under asymmetrical
  competition
• Cells characterized by size x at division
• Size x under genetic control
• Larger x ? lower division rate
• Larger x ? dominates smaller x in resource
  competition
• More similar xs ? competition more intense
• Predicting the evolution of x distribution
  through time

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Importance of scales

• Scales of mutational change
• Mutations can be more or less likely (?), their
  effect can be very small or not so small (?2)
• Demographic scales
• Growth, birth (b), death (d) timescales, scaling
  with system size (K)
• How timescale of population size fluctuation
  compares with timescale of individual growth,
  reproduction and death
• Mathematical modelling strategy
• Large system size rescaling stochastic
  individual processes

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In silico experiments usingindividual-based
models

• Evolution of cell size w/asymmetrical competition

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In silico experiments usingindividual-based
models

• Evolution of cell size w/asymmetrical competition

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In silico experiments usingindividual-based
models

• Evolution of cell size w/asymmetrical competition

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Rescaling-1 Large system size

• Making system size very large
• Rescaling interaction effect
• Larger population of smaller individuals, keep
  biomass of competitors of same order
• Macroscopic model
• This is Kimuras equation for continuum-of-allele
  s model, extended to density-dependent selection

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Rescaling-2 Large system size accelerated
birth-death

• Making system size very large
• Rescaled interaction kernels
• Birth and death rates K?
• Many mutations, due to accelerated birth
• Mutation effect ? ?2 (1/K?)
• Many mutations, but very small effects
• Reference timescale demography
• b d r independent of K
• Two cases ? lt 1 or ? 1

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• Accelerated birth and death K?
• Many mutations of small effect
• Macrosocopic model, ? lt 1
• This is Fishers equation Kimuras
  approximation for small mutation steps
• Laplacian term Brownian approximation of
  mutation process
• Typical amount of phenotypic change generated by
  mutation process/unit time ?2(K) b(K)
  ? ? ?2 r ? ?, independent of ?

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• Accelerated birth and death K?
• Many mutations of small effect
• Macroscopic model, ? 1
• New class of eco-evolutionary dynamics
• Birth death stochasticity impact population
  dynamics on demographic timescale, hence white
  noise term
• IBM simulations often go extinct evolutionary
  suicide

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Rescaling-3 Large system size extra rare
mutations

• Mutation timescale much longer than demographic
  timescale
• Characteristic evolutionary timescale 1/(K ?)
  with
• Log K ltlt 1/ (K ?) ltlt exp (C K)
• Log K time of growth and stabilization of
  invading mutant population
• Mutation should be rare enough to leave enough
  time to a mutant for invading or going extinct
• exp(C K) time over which resident population
  likely to drift away from equilibrium
• Mutation should be frequent enough for resident
  population to be still at equilibrium when a
  mutant appears

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• Mutation-selection process jump process
• Monomorphic population
• Converges toward deterministic process when ? ? 0
• Macroscopic model Canonical Equation of
  Adaptive Dynamics
• Equilibria evolutionary singularities
• Potential attractors of evolutionary trajectories
  in trait space
• Branching points attracting singularities
  where population ceases to be monomorphic

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• General insights from Brownian motion theory
• If ancestral trait surrounded by ESS and
  branching point, evolution almost surely reaches
  ( stop at) the ESS first
• General insights from Large Deviation theory
• Evolutionary dynamics when dim(trait space) gt 1
  multiple attracting singularities
• Model of punctuated equilibria
• Characteristic stasis time exp(1/?)
• Predicting the sequence of visits of evolutionary
  singularities

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Summary

• Unfolding macroscopic models by rescaling single
  set of individual stochastic processes
• Different rescalings yield
• Kimuras equation, diffusion approximation
• Trait Substitution Sequence, Adaptive Dynamics
• Allow for detailed account of ecological
  mechanisms
• New applications of math (martingales, large
  deviations), new classes of models

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New class of eco-evolutionary models

• Being small and living fast, but slow demography
• Relevant for microorganisms in colonies
• Individual stochasticity scales up to
  eco-evolutionary dynamics
• Evolution cant be ignored to explain ecological
  fluctuations
• Diversification-extinction dynamics fractal
  geometry?
• Universal laws?

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Wide range of biological applications

• Adaptive evolution of mutagenesis
• mutation rate, step, asymmetry
• Adaptive evolution of allometries
• scaling laws, e.g. birth rate/body mass
• Macroevolutionary patterns
• ecological conditions conducive to radiation vs
  punctuated equilibria predicting
  speciation/extinction rates in relation with
  individual and ecological features
• Multi-species coevolution in changing
  environments

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Mathematical perspectives MAEV at work

• Short-term and long-term dynamics
• Finite populations
• N Champagnat A Lambert, Annals of Applied
  Probability
• Eco-evolutionary dynamics in time and space
• C Prevost, L Desvillettes, R Ferriere
• N Champagnat S Meleard
• Structured populations
• S Meleard, C Viet Tran
• Sexual reproduction
• L Desvillettes, S Meleard
• Multilevel dynamics
• A Lambert, V Bansaye
• S Meleard, S Roelly, R Ferriere
• Random environments

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Thanks

• Sylvie Meleard (Ecole Polytechnique)
• Nicolas Champagnat (INRIA Nice)
• Laurent Desvillettes (ENS Cachan)
• Amaury Lambert (UPMC)
• Our sponsors ANR, EU, NSF Biomath
• Our friends in Grenoble for hosting us
• Champagnat, Ferriere, Meleard (2006) Theoretical
  Population Biology 69297-321