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Over the past century, nonlinear difference and differential equations have been used to understand conditions for species coexistence. However, these models fail to account for random fluctuations due to demographic and environmental…
Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the…
We study a population model of fixed size undergoing strong selection where individuals accumulate beneficial mutations, namely the Moran model with selection. In a specific setting with strong selection, Schweinsberg showed that the…
We consider an interacting particle Markov process for Darwinian evolution in an asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate, and a probability $\mu$ of…
We consider a metapopulation made up of $K$ demes, each containing $N$ individuals bearing a heritable quantitative trait. Demes are connected by migration and undergo independent Moran processes with mutation and selection based on trait…
We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation, with linear birth and death rates, as well as a density-dependent competition. To depict repeating changes of the…
We give a closed form of the discrete-time evolution of a recombination transformation in population genetics. This decomposition allows to define a Markov chain in a natural way. We describe the geometric decay rate to the limit…
The first chapter concerns monotype population models. We first study general birth and death processes and we give non-explosion and extinction criteria, moment computations and a pathwise representation. We then show how different scales…
We study the role of demographic fluctuations in typical endemics as exemplified by the stochastic SIRS model. The birth-death master equation of the model is simulated using exact numerics and analysed within the linear noise…
We consider 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. On the individual-based level this…
The influence of time-dependent fitnesses on the infinite population dynamics of simple genetic algorithms (without crossover) is analyzed. Based on general arguments, a schematic phase diagram is constructed that allows one to characterize…
In this paper we propose a Moran model that describes the population dynamics of two types: While the first type has a selective advantage during reproduction, the second type can avoid replacement during reproduction with some positive…
Several theoretical frameworks have been proposed to explain observed biodiversity patterns, ranging from the classical niche-based theories, mainly employing a continuous formalism, to neutral theories, based on statistical mechanics of…
Two mathematical models of macroevolution are studied. These models have population dynamics at the species level, and mutations and extinction of species are also included. The population dynamics are updated by difference equations with…
We construct a constant size population model allowing for general selective interactions and extreme reproductive events. It generalizes the idea of (Krone and Neuhauser 1997) who represented the selection by allowing individuals to sample…
Moran or Wright-Fisher processes are probably the most well known model to study the evolution of a population under various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one…
Evolution in finite populations is often modelled using the classical Moran process. Over the last ten years this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such…
Evolutionary forces shape patterns of genetic diversity within populations and contribute to phenotypic variation. In particular, recurrent positive selection has attracted significant interest in both theoretical and empirical studies.…
We consider the spatial Lambda-Fleming-Viot process model for frequencies of genetic types in a population living in R^d, with two types of individuals (0 and 1) and natural selection favouring individuals of type 1. We first prove that the…
I study a population model in which the reproduction rate lambda is inherited with mutation, favoring fast reproducers in the short term, but conflicting with a process that eliminates agglomerations of individuals. The model is a variant…