Abstract
We revisit some problems arising in the context of multiallelic discrete‐time evolutionary dynamics driven by fitness. We consider both the deterministic and the stochastic setups and for the latter both the Wright‐Fisher and the Moran approaches. In the deterministic formulation, we construct a Markov process whose Master equation identifies with the nonlinear deterministic evolutionary equation. Then, we draw the attention on a class of fitness matrices that plays some role in the important matter of polymorphism: the class of strictly ultrametric fitness matrices. In the random cases, we focus on fixation probabilities, on various conditionings on nonfixation, and on (quasi)stationary distributions.
Highlights
Population genetics aims at elucidating the fate of genotype frequencies undergoing the basic evolutionary processes when various driving “forces” such as fitness, mutation, or recombination are at stake in the gene pool
This paper studies a classical population genetic model describing a one-locus multiallelic population subject to natural selection, random mating and random genetic drifts as from the Wright-Fisher and Moran models
Considering first the deterministic updating mechanisms driven by selection, we underline that it has the form of a nonlinear Master equation suggesting that it is possible to construct an underlying Markov process governed by this Master equation
Summary
Population genetics aims at elucidating the fate of genotype frequencies undergoing the basic evolutionary processes when various driving “forces” such as fitness, mutation, or recombination are at stake in the gene pool This requires to clarify the updating mechanisms of the gene frequency-distributions over time. The updates of the allele frequency distributions are driven by the relative fitnesses of the alleles, ending up in a state where only the fittest monomorphic state will survive This state is an extremal point of the simplex over which the dynamics takes place. When dealing with the class of strictly potential fitness matrices, the mean fitness quadratic form is definite-positive; we derive a related class of fitness matrices leading to a definite-negative mean fitness quadratic form For this class of matrices, there will be a unique polymorphic equilibrium state for the diploid dynamics and it will be stable. We exploit the reversible character of this process to derive a new explicit product formula for its invariant probability measure
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