A microscopic interpretation for adaptive dynamics trait substitution sequence models

By Nicolas Champagnat. 2005

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We consider an interacting particle Markov process for Darwinian evolution in an asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate, and a probability [mu] of mutation at each birth event. We introduce a renormalization parameter K scaling the size of the population, which leads, when K-->+[infinity], to a deterministic dynamics for the density of individuals holding a given trait. By combining in a non-standard way the limits of large population (K-->+[infinity]) and of small mutations ([mu]-->0), we prove that a timescale separation between the birth and death events and the mutation events occurs and that the interacting particle microscopic process converges for finite dimensional distributions to the biological model of evolution known as the "monomorphic trait substitution sequence" model of adaptive dynamics, which describes the Darwinian evolution in an asexual population as a Markov jump process in the trait space.

Subfields: Genetics, Public Health, Environmental and Occupational Health, Sociology and Political Science

View Paper: https://www.semanticscholar.org/paper/b0388ec7078e606459ccc780051271178b19bcfd

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Citations in the corpus

Polymorphic evolution sequence and evolutionary branching (2008, methodology): “Its skeleton is similar to the one in [4] for monomorphic populations.”
Dirac concentrations in Lotka-Volterra parabolic PDEs (2007, unknown): “Such models, together with related asymptotic, can be derived form individual based stochastic processes in the limit of large populations; we refer to [13, 14].”
From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution (2008, result): “In the case of linear logistic density-dependence introduced in Section 2.2 (b(x , ) = b(x) and d(x , ) = d(x) + (x) ), the equilibrium monomorphic density n̄(x) writes (b(x) − d(x))/ (x)U (0) and the condition (H6)-(ii) is actually equivalent to f (y, x) > 0 and f (x , y) 0 (see Ref.”
Population Dynamics and Random Genealogies (2008, methodology): “Former models are used in mathematical population genetics[27,42], whereas latter models, usually called “branching models,” are more popular in mathematical ecology[76,96] and adaptive dynamics[21–23].”