Download Models of Evolutionary Dynamics

Models of Evolutionary Dynamics:
An Integrative Perspective
Ulf Dieckmann
Evolution and Ecology Program
International Institute for Applied Systems Analysis
Laxenburg, Austria
Mechanisms of Adaptation
 Imitation
copy successful behavior
 Learning
iteratively refine behavior
 Deduction
derive optimal behavior
Increasing cognitive demand
 Natural selection survival of the fittest
Theories of Adaptation
Population
genetics
Quantitative
genetics
Adaptive
dynamics
Evolutionary
game
theory
1930
1950
1990
1970
Overview
Models of Adaptive Dynamics
Connections with…
Optimization Models
Pairwise Invasibility Plots
Quantitative Genetics
Matrix Games
Models of
Adaptive
Dynamics
Four Models of Adaptive Dynamics
PS
MS
MD
PD
These models describe…
 either polymorphic or monomorphic populations
 either stochastic or deterministic dynamics
Individual-based Evolution
Polymorphic and Stochastic
Dieckmann (1994)
Species N
Species 1
...
Death
Coevolutionary
community
Environment:
density and
frequency
dependence
Birth
without
mutation
Birth
with
mutation
Illustration of Individual-based Evolution
Trait 2
Viability region
Evolutionary trajectories
Global evolutionary attractor
Trait 1
Effect of Mutation Probability
Small: 0.1%
Mutation-selection equilibrium
Mutation-limited evolution
“Moving cloud”
“Steps on a staircase”
Evolutionary time
Evolutionary time
Trait
Large: 10%
Evolutionary Random Walks
Monomorphic and Stochastic
 Mutation
Dieckmann & Law (1996)
Survival probability of rare mutant f+ / (f + d)
Population
dynamics

Invasion
Branching
process theory
 Fixation
Invasion
implies fixation
Death rate d
Fitness advantage f
Invasion probabilities based on the Moran process, on diffusion approximations,
or on graph topologies are readily incorporated.
Illustration of Evolutionary Random Walks
Initial condition
Trait 2
Bundles of
evolutionary trajectories
Trait 1
Illustration of Averaged Random Walks
Trait 2
Mean
evolutionary trajectories
Trait 1
Gradient-Ascent on Fitness Landscapes
Monomorphic and Deterministic
Dieckmann & Law (1996)
 Canonical equation of adaptive dynamics
d
1

*
2
xi  i ( xi )ni ( x) i ( xi )
fi ( xi, x) x  x
i
i
dt
2
xi
evolutionary
rate in species i
mutation
probability
equilibrium
population
size
mutational
variance-covariance
local
invasion
selection fitness
gradient
Trait 2
Illustration of Deterministic Trajectories
Evolutionary isoclines
Evolutionary fixed point
Trait 1
Reaction-Diffusion Dynamics
Polymorphic and Deterministic
Kimura (1965)
Dieckmann (unpublished)
 Kimura limit
2
d
1

2
pi ( xi )  fi ( xi , p) pi ( xi )  i ( xi ) i ( xi )  2 bi ( xi , p) pi ( xi )
dt
2
xi
 Finite-size corrections
pi
Additional per
capita death rate
results in
compact support
xi
Summary of Derivations
large population size
small mutation probability small mutation variance
PS
MS
MD
large population size
large mutation probability
PD
Optimization
Models
Evolutionary Optimization
Fitness
Phenotype
Envisaging evolution as a hill-climbing process on a static fitness landscape
is attractively simple, but essentially wrong for most systems.
Frequency-Dependent Selection
Fitness
Phenotype
Generically, fitness landscapes change in dependence on a population’s
current composition.
Evolutionary Branching
Metz et al. (1992)
Fitness
Phenotype
Convergence to a fitness minimum
Phenotype
Evolutionary Branching
Branching point
Directional selection
Disruptive selection
Time
Pairwise
Invasibility
Plots
Invasion Fitness
Metz et al. (1992)
Population size
 Definition
Initial per capita growth rate of a small
mutant population within a resident population at
ecological equilibrium.
+
–
Time
Pairwise Invasibility Plots
Mutant trait
–
+
–
Resident trait
+
+
–
Geritz et al. (1997)
Invasion of the mutant
into the resident population
possible
Invasion impossible
One trait substitution
Singular phenotype
Reading PIPs:
Comparison with Recursions
 Recursion relations
 Trait substitutions
Next state
Mutant trait
–
Current state
Size of vertical steps deterministic
+
+
–
Resident trait
Size of vertical steps stochastic
Reading PIPs:
Four Independent Properties
 Evolutionary Stability
 Convergence Stability
 Invasion Potential
 Mutual Invasibility
Geritz et al. (1997)
Reading PIPs:
Eightfold Classification
Pairwise Invasibility Plot
Geritz et al. (1997)
Classification Scheme
Resident trait







Evolutionary bifurcations
Mutant trait
(1) (2) (3) (4)
(1) Evolutionary instability, (2) Convergence stability, (3) Invasion potential, (4) Mutual invasibility.
Two Especially Interesting Types of PIP
–
+
 Branching Point
+
–
Resident trait
Evolutionarily stable,
but not convergence stable
Mutant trait
Mutant trait
 Garden of Eden
+
–
–
+
Resident trait
Convergence stable,
but not evolutionarily stable
Quantitative
Genetics
An Alternative Limit
large population size
small mutation probability small mutation variance
PS
MS
MD
large population size
large mutation probability
PD
given moments
Infinite Moment Hierarchy
 0th moments: Total population densities
d
dt
ni 
n
x
σ
2
 1st moments: Mean traits
d
dt
xi 
n
x
σ
2
 2nd moments: Trait variances and covariances
d
dt
σ 
2
i
n
x
σ
2
skewness
Quantitative Genetics: Lande’s Equation
 Lande (1976, 1979) & Iwasa et al. (1991)
d
2 
2
xi  σi
fi ( xi, x, n, σ ) 
xi  xi
dt
xi
rate of mean trait
in species i
local
selection
gradient
fitness
current population
variance-covariance
Population densities, variances, and covariances are all assumed to be fixed.
Note that evolutionary rates here are not proportional to population densities.
Game Theory: Strategy Dynamics
 Brown and Vincent (1987 et seq.)
d
dt
ni  ni fi ( xi , x, n, σ )
2
d
dt
xi  σ
d
dt
σ 
2 
i xi
fi ( xi, x, n,σ )
2
xi  xi
2
i
Variance-covariance matrices may be assumed to vanish, be fixed,
or undergo their own dynamics.
Matrix
Games
Replicator Equation: Definition
 Assumption: The abundances ni of strategies i = A, B, …
increase according to their average payoffs:
d
ni  (Wn)i
dt
 Their relative frequencies pi then follow the
replicator equation:
d
pi  (Wp )i  p Wp
dt
Average payoff
in entire population
Replicator Equation: Limitations
 Since the replicator equation cannot include innovative
mutations, it describes short-term, rather than long-term,
evolution.
 The replicator equation for frequencies naturally arises as
a transformation of arbitrary density dynamics.
 Owing to the focus on frequencies, the replicator equation
cannot capture density-dependent selection.
 Interpreted as an equation for densities, the replicator
equation assumes a very specific kind of density
regulation. Other regulations will have altogether different
evolutionary implications.
 Bilinear payoff functions based on matrix games imply
additional limitations…
Bilinear Payoff Functions
Meszéna et al. (2001)
Dieckmann & Metz (2006)
 Mixed strategies in matrix games have bilinear payoff
functions,
W ( p, p)  p Wp ,
and an invasion fitness that is linear in the variant’s trait,
f ( p, p)  p Wp  p Wp .
 For example, for the hawk-dove game, we have
f ( p, p)  12 ( p  p)(V  pC ) .
Meszéna et al. (2001)
Dieckmann & Metz (2006)
Degenerate PIPs
 The PIPs implied by a matrix game are thus highly
degenerate:
–
+
+
–
–
+
–
+
–
+
+
–
 This degeneracy is the basis for the Bishop-Cannings
theorem: All pure strategies, and all their mixtures,
participating in an ESS mixed strategy have equal fitness.
Example:
Fluctuating Rewards
Dieckmann & Metz (2006)
 If we assume rewards in the hawk-dove
game to fluctuate between rounds (taking
one of two similar values with equal
probability), the PIP’s degeneracy
immediately vanishes:
–
+
+
–
 Accordingly, the structurally unstable neutrality of
invasions at the ESS is overcome.
 This resolves the ambiguity between population-level
and individual-level mixed strategies.
Two-Dimensional Unfolding
Dieckmann & Metz (2006)
of Degeneracy
Game-theoretical
case straddles
two bifurcation
curves and thus
acts as the
organizing centre
of a rich
bifurcation
structure.
Mixtures of
Mixed Strategies
Dieckmann & Metz (2006)
 Evolutionary outcomes can now be more subtle:
+
+
One mixed
strategy
+
+
Two pure
strategies
+
+
A pure and a mixed
strategy
+
+
Two mixed
strategies
 The interplay between population-level polymorphisms
and individual-level probabilistic strategy mixing thus
becomes amenable to evolutionary analysis.
Summary
 Models of adaptive dynamics offer a flexible toolbox
for studying phenotypic evolution: Simplified models
are systematically deduced from a common individualbased underpinning, providing an integrative
perspective.
 The resultant models are particularly helpful for
investigating the evolutionary implications of complex
ecological settings: Frequency-dependent selection is
essential for understanding the evolutionary formation
and loss of biological diversity.