Models of Evolutionary Dynamics: An Integrative Perspective Ulf Dieckmann Evolution 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 Population genetics 1930 Quantitative genetics 1950 Adaptive dynamics 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 PSMSMDPD These models describe… either polymorphic or monomorphic populations either stochastic or deterministic dynamics
DeathBirth without mutation Birth with mutation Environment: density and frequency dependence... Species 1 Species N Coevolutionary community Dieckmann (1994) Individual-based Evolution Polymorphic and Stochastic
Illustration of Individual-based Evolution Trait 1 Trait 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 Trait
Mutation Invasion Fixation Population dynamics Branching process theory Invasion implies fixation { Survival probability of rare mutant Fitness advantageDeath 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 Walks Monomorphic and Stochastic Dieckmann & Law (1996)
Trait 1 Trait 2 Illustration of Evolutionary Random Walks Bundles of evolutionary trajectories Initial condition
Illustration of Averaged Random Walks Trait 1 Trait 2 Mean evolutionary trajectories
Gradient-Ascent on Fitness Landscapes Monomorphic and Deterministic Canonical equation of adaptive dynamics evolutionary rate in species i equilibrium population size mutational variance-covariance invasion fitness mutation probability local selection gradient Dieckmann & Law (1996)
Trait 1 Trait 2 Illustration of Deterministic Trajectories Evolutionary isoclines Evolutionary fixed point
Reaction-Diffusion Dynamics Polymorphic and Deterministic Kimura limit Finite-size corrections Additional per capita death rate results in compact support xixi pipi Kimura (1965) Dieckmann (unpublished)
Summary of Derivations PS MS large population size small mutation probabilityMD small mutation variance PD large population size large mutation probability
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 Convergence to a fitness minimum Metz et al. (1992) Fitness Phenotype
Branching point Time Phenotype Evolutionary Branching Directional selectionDisruptive selection
Pairwise Invasibility Plots
Invasion Fitness Definition Initial per capita growth rate of a small mutant population within a resident population at ecological equilibrium. Population size Time + – Metz et al. (1992)
Pairwise Invasibility Plots + + – – Resident trait Mutant trait + – Invasion of the mutant into the resident population possible Invasion impossible One trait substitution Singular phenotype Geritz et al. (1997)
Recursion relations Current state Next state Size of vertical steps deterministic Trait substitutions Resident trait Mutant trait 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)
(1) Evolutionary instability, (2) Convergence stability, (3) Invasion potential, (4) Mutual invasibility. Evolutionary bifurcations Geritz et al. (1997) Reading PIPs: Eightfold Classification
Two Especially Interesting Types of PIP Garden of Eden Branching Point + + – – Resident trait Mutant trait + + – – Resident trait Mutant trait Evolutionarily stable, but not convergence stable Convergence stable, but not evolutionarily stable
Quantitative Genetics
An Alternative Limit PSMSMDPD large population size small mutation probabilitysmall mutation variance large population size large mutation probability given moments
Infinite Moment Hierarchy 0 th moments: Total population densities 1 st moments: Mean traits 2 nd moments: Trait variances and covariances skewness
Quantitative Genetics: Lande’s Equation Lande (1976, 1979) & Iwasa et al. (1991) rate of mean trait in species i current population variance-covariance fitnesslocal selection gradient 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.) Variance-covariance matrices may be assumed to vanish, be fixed, or undergo their own dynamics.
Matrix Games
Replicator Equation: Definition Assumption : The abundances n i of strategies i = A, B, … increase according to their average payoffs: Their relative frequencies p i then follow the replicator equation: 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 Mixed strategies in matrix games have bilinear payoff functions, and an invasion fitness that is linear in the variant’s trait,,. For example, for the hawk-dove game, we have. 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 Unfolding of Degeneracy Game-theoretical case straddles two bifurcation curves and thus acts as the organizing centre of a rich bifurcation structure. Dieckmann & Metz (2006)
Mixtures of Mixed Strategies 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. 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.