Models of Evolutionary Dynamics:An Integrative Perspective

Ulf DieckmannEvolution and Ecology Program

International Institute for Applied Systems Analysis

Laxenburg, Austria

Mechanisms of Adaptation

Natural selection survival of the fittest

Increasing cognitive demand

Imitation copy successful behavior

Learning iteratively refine behavior

Deduction derive optimal behavior

Theories of Adaptation

Populationgenetics

1930

Quantitativegenetics

1950

Adaptivedynamics

1990

Evolutionarygametheory

1970

Overview

Models of Adaptive Dynamics

Connections with…

Optimization Models

Pairwise Invasibility Plots

Quantitative Genetics

Matrix Games

Models ofAdaptive Dynamics

Four Models of Adaptive Dynamics

PSPS MSMS MDMD PDPD

These models describe… either polymorphic or monomorphic populations either stochastic or deterministic dynamics

Death Birthwithoutmutation

Birthwithmutation

Environment:density andfrequencydependence

. . .

Species 1 Species N Coevolutionarycommunity

Dieckmann (1994)Individual-based EvolutionPolymorphic and Stochastic

Illustration of Individual-based Evolution

Trait 1

Trai

t 2

Viability region

Evolutionary trajectories

Global evolutionary attractor

Effect of Mutation Probability

Small: 0.1%Mutation-limited evolution

“Steps on a staircase”

Evolutionary time

Large: 10%Mutation-selection equilibrium

“Moving cloud”

Evolutionary time

Trai

t

Mutation

Invasion

Fixation

Populationdynamics

Branchingprocess theory

Invasionimplies fixation

{ Survival probability of rare mutant

Fitness advantage Death rate

f+ / (f + d)

d f

Invasion probabilities based on the Moran process, on diffusion approximations,or on graph topologies are readily incorporated.

Evolutionary Random WalksMonomorphic and Stochastic Dieckmann & Law (1996)

Trait 1

Trai

t 2

Illustration of Evolutionary Random Walks

Bundles ofevolutionary trajectories

Initial condition

Illustration of Averaged Random Walks

Trait 1

Trai

t 2 Meanevolutionary trajectories

Gradient-Ascent on Fitness LandscapesMonomorphic and Deterministic

Canonical equation of adaptive dynamics

* 21( ) ( ) ( ) ( , )

2 i ii i i i i i i i x x

i

dx x n x x f x x

dt x

evolutionaryrate in species i

equilibriumpopulation

size mutationalvariance-covariance

invasionfitness

mutationprobability

localselectiongradient

Dieckmann & Law (1996)

Trait 1

Trai

t 2

Illustration of Deterministic Trajectories

Evolutionary isoclines

Evolutionary fixed point

Reaction-Diffusion DynamicsPolymorphic and Deterministic

Kimura limit2

22

1( ) ( , ) ( ) ( ) ( ) ( , ) ( )

2i i i i i i i i i i i i i ii

dp x f x p p x x x b x p p x

dt x

Finite-size correctionsAdditional percapita death rateresults incompact support

xi

pi

Kimura (1965)Dieckmann (unpublished)

Summary of Derivations

PSPS MSMS

large population sizesmall mutation probability

MDMD

small mutation variance

PDPD

large population sizelarge mutation probability

OptimizationModels

Evolutionary Optimization

Fitness

Phenotype

Envisaging evolution as a hill-climbing process on a static fitness landscapeis 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

Convergence to a fitness minimum

Metz et al. (1992)

Fitness

Phenotype

Branching point

Time

Phen

otyp

e

Evolutionary Branching

Directional selection Disruptive selection

Pairwise Invasibility Plots

Invasion Fitness

DefinitionInitial per capita growth rate of a smallmutant population within a resident population at ecological equilibrium.

Popu

latio

n si

ze

Time

+–

Metz et al. (1992)

Pairwise Invasibility Plots

+ +–

–Resident trait

Mut

ant t

rait

+–

Invasion of the mutantinto the resident populationpossible

Invasion impossible

One trait substitution

Singular phenotype

Geritz et al. (1997)

Recursion relations

Current state

Nex

t sta

te

Size of vertical steps deterministic

Trait substitutions

Resident trait

Mut

ant t

rait

Size of vertical steps stochastic

+ +–

–

Reading PIPs:Comparison with Recursions

Reading PIPs:Four Independent Properties

Evolutionary Stability

Convergence Stability

Invasion Potential

Mutual Invasibility

Geritz et al. (1997)

Mut

ant t

rait

Resident trait

Pairwise Invasibility Plot

Classification Scheme(1) (2) (3) (4)

(1) Evolutionary instability, (2) Convergence stability, (3) Invasion potential, (4) Mutual invasibility.

Evol

utio

nary

bifu

rcat

ions

Geritz et al. (1997)

Reading PIPs:Eightfold Classification

Two Especially Interesting Types of PIP

Garden of Eden Branching Point

+

+

–

–Resident trait

Mut

ant t

rait +

+

–

–Resident trait

Mut

ant t

rait

Evolutionarily stable,but not convergence stable

Convergence stable,but not evolutionarily stable

Quantitative Genetics

An Alternative Limit

PSPS MSMS MDMD PDPD

large population sizesmall mutation probability small mutation variance

large population sizelarge mutation probability

given moments

Infinite Moment Hierarchy

0th moments: Total population densities

2σdidtx n x

2σdidtn n x

2 2σ σdidt

n x

1st moments: Mean traits

2nd moments: Trait variances and covariances

skewness

Quantitative Genetics: Lande’s Equation

Lande (1976, 1979) & Iwasa et al. (1991)

rate of mean traitin species i

current populationvariance-covariance

fitnesslocalselectiongradient

2 2σ ( , , ,σ )i i

i i i i x xi

dx f x x n

dt x

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.)

2 2σ ( , , ,σ )i i i

di i i idt x x xx f x x n

2( , , ,σ )di i i idtn n f x x n

2σd idt

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:

( )i i

dn Wn

dt

Their relative frequencies pi then follow thereplicator equation:

( )i i

dp Wp p Wp

dt

Average payoffin 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

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 have12( , ) ( )( )f p p 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.

Meszéna et al. (2001)Dieckmann & Metz (2006)

Example:Fluctuating Rewards

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.

Dieckmann & Metz (2006)

Two-Dimensional Unfoldingof Degeneracy

Game-theoretical case straddles two bifurcation

curves and thus acts as the

organizing centre of a rich

bifurcation structure.

Dieckmann & Metz (2006)

Mixtures ofMixed Strategies

Evolutionary outcomes can now be more subtle:

++

One mixedstrategy

++

Two purestrategies

++

A pure and a mixedstrategy

++

Two mixedstrategies

The interplay between population-level polymorphisms and individual-level probabilistic strategy mixing thus becomes amenable to evolutionary analysis.

Dieckmann & Metz (2006)

Summary

Models of adaptive dynamics offer a flexible toolbox for studying phenotypic evolution: Simplified models are systematically deduced from a common individual-based 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.