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… 
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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
TLDR
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
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
TLDR
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
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
TLDR
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.
TLDR
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.
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