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Unifying evolutionary dynamics: From individual stochastic processes to macroscopic evolution
 Theor. Pop. Biol
"... Running head: From individual processes to evolutionary dynamics ..."
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Cited by 90 (13 self)
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Running head: From individual processes to evolutionary dynamics
From individual stochastic processes to macroscopic models in adaptive evolution
 Stoch. Models
, 2008
"... We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the microscopic, stochastic description of a population of discrete individuals characterized by one or several ad ..."
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Cited by 44 (4 self)
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We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the microscopic, stochastic description of a population of discrete individuals characterized by one or several adaptive traits. The population is modelled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual’s trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. We look for tractable large population approximations. By combining various scalings on population size, birth and death rates, mutation rate, mutation step, or time, a single microscopic model is shown to lead to contrasting macroscopic limits, of different nature: deterministic, in the form of ordinary, integro, or partial differential equations, or probabilistic, like stochastic partial differential equations or superprocesses. In the limit of rare mutations, we show that a possible approximation is a jump process, justifying rigorously the socalled trait substitution sequence. We thus unify different points of view concerning mutationselection evolutionary models. Keywords: Darwinian evolution, birthdeathmutationcompetition point process, mutationselection dynamics, nonlinear integrodifferential equations, nonlinear partial differential
Dirac concentrations in LotkaVolterra parabolic PDEs
 Indiana Univ. Math. J
, 2008
"... We consider parabolic partial differential equations of LotkaVolterra type, with a nonlocal nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to sele ..."
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Cited by 39 (6 self)
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We consider parabolic partial differential equations of LotkaVolterra type, with a nonlocal nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained HamiltonJacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a BV estimate in time on the nonlocal nonlinearity, a characterization of the concentration point (in a monomorphic situation) and, surprisingly, some counterexamples showing that jumps on the Dirac locations are indeed possible. KeyWords: Integral parabolic equations, adaptive dynamics, asymptotic behavior, Dirac concentrations, population dynamics.
Evolution of discrete populations and the canonical diffusion of adaptive dynamics
, 2006
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The evolutionary limit for models of populations interacting competitively with many resources
, 2010
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Stochastic and deterministic models for agestructured populations with genetically variable traits
 ESAIM: Proceedings
"... with genetically variable traits ..."
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Adaptation in a stochastic multiresources chemostat model
, 2013
"... We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. Adaptive dynamics so far has been put on a rigorous footing only ..."
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Cited by 8 (0 self)
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We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. Adaptive dynamics so far has been put on a rigorous footing only for direct competition models (LotkaVolterra models) involving a competition kernel which describes the competition pressure from one individual to another one. We extend this to a multiresources chemostat model, where the competition between individuals results from the sharing of several resources which have their own dynamics. Starting from a stochastic birth and death process model, we prove that, when advantageous mutations are rare, the population behaves on the mutational time scale as a jump process moving between equilibrium states (the polymorphic evolution sequence of the adaptive dynamics literature). An essential technical ingredient is the study of the long time behavior of a chemostat multiresources dynamical system. In the small mutational steps limit this process in turn gives rise to a differential equation in phenotype space called canonical