# Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence

@article{Leman2015ConvergenceOA,
title={Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence},
author={H{\'e}l{\e}ne Leman},
journal={Stochastics and Partial Differential Equations: Analysis and Computations},
year={2015},
volume={4},
pages={791-826}
}`
• H. Leman
• Published 7 September 2015
• Mathematics
• Stochastics and Partial Differential Equations: Analysis and Computations
We consider an individual-based spatially structured population for Darwinian evolution in an asexual population. The individuals move randomly on a bounded continuous space according to a reflected brownian motion. The dynamics involves also a birth rate, a density-dependent logistic death rate and a probability of mutation at each birth event. We study the convergence of the microscopic process in a long time scale when the population size grows to $$+\infty$$+∞ and the mutation probability…
8 Citations
• Mathematics
Journal 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.
Probabilistic and deterministic analysis of the evolution : influence of a spatial structure and a mating preference.
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.
Selection-mutation dynamics with age structure : long-time behaviour and application to the evolution of life-history traits
This thesis is divided into two parts connected by the same thread. It concerns the theoretical study and the application of mathematical models describing population dynamics. The individuals
Crossing a fitness valley as a metastable transition in a stochastic population model
• Mathematics
The 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.
A spatial measure-valued model for chemical reaction networks in heterogeneous systems
• Mathematics
• 2020
We propose a novel measure valued process which models the behaviour of chemical reaction networks in spatially heterogeneous systems. It models reaction dynamics between different molecular species
A stochastic model for speciation by mating preferences
• Computer Science
Journal of mathematical biology
• 2018
A stochastic model of population where individuals, with type a or A, are equivalent from ecological, demographical and spatial points of view, and differ only by their mating preference, showing that mating preferences by themselves, even if they are very small, are enough to entail reproductive isolation between patches.
Reaction Networks and Population Dynamics Organised
Reaction systems and population dynamics constitute two highly developed areas of research that build on well-defined model classes, both in terms of dynamical systems and stochastic processes.

## References

SHOWING 1-10 OF 47 REFERENCES
LARGE POPULATION LIMIT AND TIME BEHAVIOUR OF A STOCHASTIC PARTICLE MODEL DESCRIBING AN AGE-STRUCTURED POPULATION
We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of
Invasion and adaptive evolution for individual-based spatially structured populations
• Mathematics
Journal of mathematical biology
• 2007
This work considers a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals, and shows the important role of parameter scalings on clustering and invasion.
Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system
• Mathematics
• 2014
To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still
Stochastic eco-evolutionary model of a prey-predator community
• Mathematics
Journal of mathematical biology
• 2016
An individual-based model of the community that takes into account both prey and predator phenotypes is introduced and it is proved the existence of a unique globally asymptotically stable equilibrium under specific conditions on the interaction among prey individuals.
Adaptive Dynamics: A Geometrical Study of the Consequences of Nearly Faithful Reproduction
• Mathematics
• 1995
We set out to explore a class of stochastic processes called, 'adaptive dynamics', which supposedly capture some of the essentials of the long-term biological evolution. These processes have a strong
Evolution of Phenotypic Clusters Through Competition and Local Adaptation Along an Environmental Gradient
• Computer Science
Evolution; international journal of organic evolution
• 2008
Analysis of the evolution of a quantitative trait in populations that are spatially extended along an environmental gradient, with gene flow between nearby locations shows that the shape of competition kernels is important for clustering: the sign structure of the Fourier transform of a competition kernel determines whether the kernel promotes clustering.
Spatial aspects of interspecific competition.
• Economics, Mathematics
Theoretical population biology
• 1998
It is proved rigorously that if one species has a competitive advantage over each of the others, then eventually it takes over all the sites in the system and produces a stationary distribution with an interesting spatial structure.