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Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective. (English) Zbl 1473.35590

Summary: We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii), and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimisation of free energy functionals.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
60J25 Continuous-time Markov processes on general state spaces
92D15 Problems related to evolution
92D25 Population dynamics (general)
58E30 Variational principles in infinite-dimensional spaces
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