On the stochastic evolution of finite populations
| Autor: | Max O. Souza, Fabio A. C. C. Chalub |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
0106 biological sciences
0301 basic medicine Population Dynamics Population Evolutionary game theory 010603 evolutionary biology 01 natural sciences Evolution Molecular 03 medical and health sciences Game Theory Animals Statistical physics Selection Genetic Quantitative Biology - Populations and Evolution education Probability Mathematics Stochastic Processes education.field_of_study Models Genetic Markov chain Stochastic process Applied Mathematics Populations and Evolution (q-bio.PE) Mathematical Concepts State (functional analysis) Agricultural and Biological Sciences (miscellaneous) Markov Chains Term (time) Fixation (population genetics) Genetics Population 030104 developmental biology Modeling and Simulation FOS: Biological sciences Mutation Mutation (genetic algorithm) Genetic Fitness 92D15 92D25 15B51 60J10 |
| DOI: | 10.48550/arxiv.1602.00478 |
| Popis: | This work is a systematic study of discrete Markov chains that are used to describe the evolution of a two-types population. Motivated by results valid for the well-known Moran (M) and Wright-Fisher (WF) processes, we define a general class of Markov chains models which we term the Kimura class. It comprises the majority of the models used in population genetics, and we show that many well-known results valid for M and WF processes are still valid in this class. In all Kimura processes, a mutant gene will either fixate or become extinct, and we present a necessary and sufficient condition for such processes to have the probability of fixation strictly increasing in the initial frequency of mutants. This condition implies that there are WF processes with decreasing fixation probability --- in contradistinction to M processes which always have strictly increasing fixation probability. As a by-product, we show that an increasing fixation probability defines uniquely an M or WF process which realises it, and that any fixation probability with no state having trivial fixation can be realised by at least some WF process. These results are extended to a subclass of processes that are suitable for describing time-inhomogeneous dynamics. We also discuss the traditional identification of frequency dependent fitnesses and pay-offs, extensively used in evolutionary game theory, the role of weak selection when the population is finite, and the relations between jumps in evolutionary processes and frequency dependent fitnesses. |
| Databáze: | OpenAIRE |
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