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Adaptive dynamics in logistic branching populations

Nicolas Champagnat;  Amaury Lambert — 2008

Banach Center Publications

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. Under these assumptions, the evolution of a quantitative dominant trait in an isolated population is described by a deterministic differential equation called 'canonical equation of adaptive dynamics'. In this work, in order to include the effect of genetic drift in this model, we consider instead...

Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics

Mireille Bossy;  Nicolas Champagnat;  Sylvain Maire;  Denis Talay — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of ℝ d . This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator...

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