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  • Author or Editor: É. Kisdi x
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We review mechanisms that lead to cyclic evolution with alternating levels of diversity. Such cycles involve directional evolution towards a so-called evolutionary branching point, where selection becomes disruptive and splits the population into two strategies. Coevolution of these strategies eventually leads to the extinction of one of them. The remaining strategy evolves back to the evolutionary branching point, and a new cycle begins. There are a number of different evolutionary mechanisms that can produce this kind of cycles including chance extinction, switching between population dynamical attractors, and coevolution with an ecologically distinct species. We also present an example for branching-extinction cycles where the direction of evolution changes between monomorphic and dimorphic populations solely due to the different levels of diversity. The latter cycles exhibit a novel feature: Even though extinction is deterministic in the sense that it is unavoidable and always occurs at the same trait values, it is random which of the two coexisting strategies goes extinct. As a result, long and short cycles alternate in a random sequence.

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Matrix game theory and optimisation models offer two radically different perspectives on the outcome of evolution. Optimisation models consider frequency-independent selection and envisage evolution as a hill-climbing process on a constant fitness landscape, with the optimal strategy corresponding to the fitness maximum. By contrast, in evolutionary matrix games selection is frequency-dependent and leads to fitness equality among alternative strategies once an evolutionarily stable strategy has been established. In this review we demonstrate that both optimisation models and matrix games represent limiting cases of the general framework of nonlinear frequency-dependent selection. Adaptive dynamics theory considers arbitrary nonlinear frequency and density dependence and envisages evolution as proceeding on an adaptive landscape that changes its shape according to which strategies are present in the population. In adaptive dynamics, evolutionarily stable strategies correspond to conditional fitness maxima: the ESS is characterised by the fact that it has the highest fitness if it is the established strategy. In this framework it can also be shown that dynamical attainability, evolutionary stability, and invading potential of strategies are pairwise independent properties. In optimisation models, on the other hand, these properties become linked such that the optimal strategy is always attracting, evolutionarily stable and can invade any other strategy. In matrix games fitness is a linear function of the potentially invading strategy and can thus never exhibit an interior maximum: Instead, the fitness landscape is a plane that becomes horizontal once the ESS is established. Due to this degeneracy, invading potential is part of the ESS definition for matrix games and dynamical attainability is a dependent property. We conclude that nonlinear frequency-dependent theory provides a unifying framework for overcoming the traditional divide between evolutionary optimisation models and matrix games.

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