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  • Author or Editor: G. Meszéna x
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The growth of many artificial replicators is approximately parabolic (sub-exponential) in solution, due to the self-inhibition through duplex formation by the association of single-stranded molecules. This type of growth implies “survival of everybody” under a selection constraint. Parabolic growth requires high enough concentration so that the single strands can find one another. The selective outcome is more complicated when spontaneous decay of molecules is also taken into account. When double strands decompose at a slower rate than single strands, coexistence or survival of the fittest becomes a quantitative issue. Here we investigate the evolution of parabolic replicators by the methods of adaptive dynamics. Directional selection for higher replication rate in general results in a “parabolic quasi-species”, due to the fact that the fittest template is followed by a moving shadow of inferior templates that owe their presence to parabolic growth. Under the assumption of cross-hybridisation between non-identical templates molecular coexistence disappears when such pairing is sufficiently non-selective, because replicators do not inhibit themselves more than they limit the others. At intermediate specificity of pairing adaptive branching of the population becomes feasible, due to the fact that distant enough sequences are able to escape from cross-limitation by other sub-populations.

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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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