# From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution

@article{Champagnat2008FromIS, title={From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution}, author={Nicolas Champagnat and R{\'e}gis Ferri{\`e}re and Sylvie M{\'e}l{\'e}ard}, journal={Stochastic Models}, year={2008}, volume={24}, pages={2 - 44} }

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…

## 109 Citations

Lévy flights in evolutionary ecology

- MathematicsJournal of mathematical biology
- 2012

It is shown that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of non-local fractional reaction–diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator.

Random Modeling of Adaptive Dynamics and Evolutionary Branching

- Biology
- 2011

It is proved that under a good combination of these two scales, the population process is approximated in an evolution long time scale by a Markov pure jump process describing successive equilibria of the population.

Evolutionary dynamics from deterministic microscopic ecological processes.

- MathematicsPhysical review. E
- 2020

This work develops a theory for a class of deterministic individual-based models describing individual births and deaths, which captures the essential features of standard stochastic individual- based models and becomes identical to the latter under maximal competition.

From adaptive dynamics to adaptive walks.

- MathematicsJournal of mathematical biology
- 2019

This work considers an asexually reproducing population on a finite type space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event, and modelled as a measure-valued Markov process.

Dynamics of lineages in adaptation to a gradual environmental change

- MathematicsAnnales Henri Lebesgue
- 2022

We investigate a simple quantitative genetics model subjet to a gradual environmental change from the viewpoint of the phylogenies of the living individuals. We aim to understand better how the past…

Polymorphic evolution sequence and evolutionary branching

- Biology
- 2011

It is proved that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population.

Probabilistic and deterministic analysis of the evolution : influence of a spatial structure and a mating preference.

- Environmental Science
- 2016

A probabilistic model to study the effect of the sexual preference on the speciation of species and gives numerical results and a detailed biological behavior analysis around two issues: the co-evolution of phenotypic and spatial niches of mutualistic species and the invasion dynamics of a homogeneous space by these species.

From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one step

- Mathematics
- 2015

A "stochastic Euler scheme" based on coupling arguments that allows to control the time evolution of the stochastic system over time-scales that diverge with $K$, proving convergence to the canonical equation of adaptive dynamics (CEAD).

Crossing a fitness valley as a metastable transition in a stochastic population model

- MathematicsThe Annals of Applied Probability
- 2019

This work focuses on the limit of large population and rare mutations at several speeds, and chooses parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness.

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