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DDC: 510 × Subject: fixation probability ×

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    Discrete logistic branching populations and the canonical diffusion and the canonical diffusion of adaptive dynamics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Champagnat, Nicolas; Lambert, Amaury
    The biological theory of adaptive dynamics proposes a description of the long-time evolution of an asexual population, based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE, called `canonical equation of adaptive dynamics'. However, in order to include the effect of genetic drift in this description, we have to apply a limit of weak selection to a finite stochastically fluctuating discrete population subject to competition in the logistic branching fashion. We start with the study of the particular case of two competing subpopulations (resident and mutant) and seek explicit first-order formulae for the probability of fixation of the mutant, also interpreted as the mutant's fitness, in the vicinity of neutrality. In particular, the first-order term is a linear combination of products of functions of the initial mutant frequency times functions of the initial total population size, called invasibility coefficients (fertility, defence, aggressiveness, isolation, survival). Then we apply a limit of rare mutations to a population subject to mutation, birth and competition where the number of coexisting types may fluctuate, while keeping the population size finite. This leads to a jump process ...
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    Duality and fixation in Xi-Wright-Fisher processes with frequency-dependent selection
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Casanova, Adrian González; Spanò, Dario
    A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of potential parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the populations ancestral process. The scaling limits are, respectively, a two-types Xi-Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branchingcoalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process ergodic properties.