• Corpus ID: 119138361

The deterministic limit of the Moran model: a uniform central limit theorem

@article{Cordero2015TheDL,
  title={The deterministic limit of the Moran model: a uniform central limit theorem},
  author={Fernando Cordero},
  journal={arXiv: Probability},
  year={2015}
}
  • F. Cordero
  • Published 21 August 2015
  • Mathematics
  • arXiv: Probability
We consider a Moran model with two allelic types, mutation and selection. In this work, we study the behaviour of the proportion of fit individuals when the size of the population tends to infinity, without any rescaling of parameters or time. We first prove that the latter converges, uniformly in compacts in probability, to the solution of an ordinary differential equation, which is explicitly solved. Next, we study the stability properties of its equilibrium points. Moreover, we show that the… 
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References

SHOWING 1-10 OF 15 REFERENCES
Limit theorems and diffusion approximations for density dependent Markov chains
One parameter families of Markov chains X A (t) with infinitesimal parameters given by q k,k+l A =Af(A −1 k,l) k, l ∈Z′ l≠0 are considered. Under appropriate conditions X A (t)/A converges in
Mathematical properties of mutation-selection models
TLDR
The fundamental results about existence and stability of equilibria for classical mutation-selection models with a finite number of alleles, for models like the stepwise-mutation model, and for the continuum-of-alleles model are reviewed.
Mutation-selection models solved exactly with methods of statistical mechanics.
TLDR
The mean number of mutations, the mutation load, and the variance in fitness under mutation-selection balance are investigated, and some insight is yielded into the 'error threshold' phenomenon, which occurs in some, but not all, examples.
Mutation, selection, and ancestry in branching models: a variational approach
TLDR
The quasispecies model of sequence evolution with mutation coupled to reproduction but independent across sites, and a fitness function that is invariant under permutation of sites is used, and the fitness of letter compositions is worked out explicitly.
Solutions of ordinary differential equations as limits of pure jump markov processes
In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are
Some Mathematical Models from Population Genetics
TLDR
This work provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics and falls into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time model that trace out the genealogical relationships between individuals in a sample from the population.
Continuous martingales and Brownian motion
0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.-
Convergence of a Moran model to Eigen's quasispecies model.
Solutions of Riccati-Abel equation in terms of third order trigonometric functions
Solutions of the generalized Riccati equations with third order nonlinearity, named as Riccati-Abel equation, are expressed via third order trigonometric functions. It is shown, as the ordinary
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