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Gradient Flow Formulations of Discrete and Continuous Evolutionary Models: A Unifying Perspective
- Source :
- Acta Applicandae Mathematicae. 171
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- 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.
- Subjects :
- Partial differential equation
Markov chain
Applied Mathematics
010102 general mathematics
Evolutionary game theory
Computer Science::Computational Geometry
Heavy traffic approximation
01 natural sciences
010101 applied mathematics
Replicator equation
Moran process
Quantitative Biology::Populations and Evolution
Applied mathematics
Limit (mathematics)
0101 mathematics
Balanced flow
Mathematics
Subjects
Details
- ISSN :
- 15729036 and 01678019
- Volume :
- 171
- Database :
- OpenAIRE
- Journal :
- Acta Applicandae Mathematicae
- Accession number :
- edsair.doi...........9ce74969d18843d2cf5c7706e3084ccd