Models of Evolutionary Dynamics An Integrative Perspective Ulf

Models of Evolutionary Dynamics: An Integrative Perspective Ulf Dieckmann Evolution and Ecology Program International Institute for Applied Systems Analysis Laxenburg, Austria

Mechanisms of Adaptation survival of the fittest n Imitation copy successful behavior n Learning iteratively refine behavior n Deduction derive optimal behavior Increasing cognitive demand n Natural selection

Theories of Adaptation Population Quantitative Adaptive genetics dynamics 1930 1950 1990 Evolutionary game theory 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… n either polymorphic or monomorphic populations n 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 attra Trait 1

Effect of Mutation Probability Large: 10% Small: 0. 1% Trait Mutation-selection equilibrium. Mutation-limited evolution “Moving cloud” Evolutionary time “Steps on a staircase” Evolutionary time

Evolutionary Random Walks Monomorphic and Stochastic Dieckmann & Law (1996) { Mutation Population f+ / (f + d) Survival probability of rare mutant dynamics Invasion Branching process theory Fixation Invasion implies fixation Death rate d Fitness advantage f Invasion probabilities based on the Moran process, on diffusion approx or on graph topologies are readily incorporated.

Illustration of Evolutionary Random Walks Trait 2 Initial condition Bundles of evolutionary trajectories Trait 1

Trait 2 Illustration of Averaged Random Walks Mean evolutionary trajectories Trait 1

Gradient-Ascent on Fitness Landscapes Monomorphic and Deterministic Dieckmann & Law (1996) n Canonical equation of adaptive dynamics evolutionary rate in species i mutation probability local invasion selection fitness gradient equilibrium population size mutational variance-covariance

Trait 2 Illustration of Deterministic Trajectories Evolutionary isoclines Evolutionary fixed point Trait 1

Reaction-Diffusion Dynamics Kimura (1965) Polymorphic and Deterministic Dieckmann (unpublished) n Kimura limit n Finite-size corrections pi Additional per capita death rat results in compact suppo 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 lands 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 point Directional selection Disruptive selection Time

Pairwise Invasibility Plots

Invasion Fitness Metz et al. (1992) Population size n Definition Initial per capita growth rate of a small mutant population within a resident population at ecological equilibrium. + – Time

Mutant trait Geritz et al. (1997) Pairwise Invasibility Plots – + – Resident trait + + – Invasion of the mutant into the resident populatio possible Invasion impossible One trait substitution Singular phenotype

Reading PIPs: Comparison with Recursions n Recursion n Trait substitutions Next state Mutant trait relations Current state Size of vertical steps deterministic – + + – Resident trait Size of vertical steps stochastic

Reading PIPs: Geritz et al. (1997) Four Independent Properties n Evolutionary Stability n Convergence Stability n Invasion Potential n Mutual Invasibility

Evolutionary bifurcations Reading PIPs: Eightfold Classification Geritz et al. (1997) (1) Evolutionary instability, (2) Convergence stability, (3) Invasion potential, (4) Mutua

Two Especially Interesting Types of PIP + – Resident trait Evolutionarily stable, but not convergence stable n Branching Point Mutant trait n 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 PD large population size given moments large mutation probability

Infinite Moment Hierarchy n 0 th moments: Total population densities n 1 st moments: Mean traits n 2 nd moments: Trait variances and covariances skewness

Quantitative Genetics: Lande’s Equation n Lande (1976, 1979) & Iwasa et al. (1991) rate of mean trait in species i local fitness selection gradient current population variance-covariance Population densities, variances, and covariances are all assumed to Note that evolutionary rates here are not proportional to population

Game Theory: Strategy Dynamics n Brown and Vincent (1987 et seq. ) Variance-covariance matrices may be assumed to vanish, be fixed, or undergo their own dynamics.

Matrix Games

Replicator Equation: Definition n Assumption: The abundances ni of strategies i = A, B, … increase according to their average payoffs: n Their relative frequencies pi then follow the replicator equation: Average payoff in entire population

Replicator Equation: Limitations n Since the replicator equation cannot include innovative mutations, it describes short-term, rather than long-term, evolution. n The replicator equation for frequencies naturally arises as a transformation of arbitrary density dynamics. n Owing to the focus on frequencies, the replicator equation cannot capture densitydependent selection. n Interpreted as an equation for densities, the replicator equation assumes a very specific kind of density regulation. Other regulations will have altogether different evolutionary

Meszéna et al. (2001) Dieckmann & Metz (2006) Bilinear Payoff Functions n Mixed strategies in matrix games have bilinear payoff functions, , and an invasion fitness that is linear in the varian. n For example, for the hawk-dove game, we have .

Degenerate PIPs Meszéna et al. (2001) Dieckmann & Metz (2006) n The PIPs implied by a matrix game are thus highly degenerate: – + + – – + – + + – n 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: Dieckmann & Metz (2006) Fluctuating Rewards n 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: n Accordingly, the structurally unstable neutrality of invasions at the ESS is overcome. n This resolves the ambiguity between population-level and individual-level mixed strategies.

Two-Dimensional Unfolding Dieckmann & Metz (2006) of Degeneracy Gametheoretical case straddles two bifurcation curves and thus acts as the organizing centre of a rich bifurcation structure.

Mixtures of Mixed Strategies Dieckmann & Metz (2006) n Evolutionary outcomes can now be more subtle: + + One mixed strategy + + + Two pure A pure and a mixed Two mixed strategies strategy strategies n The interplay between population-level polymorphisms and individual-level probabilistic strategy mixing thus becomes amenable to evolutionary analysis.

Summary n 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. n 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.