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Recent theoretical studies have shown that demographic stochasticity can greatly increase the tendency of asexually reproducing phenotypically diverse organisms to spontaneously evolve into localised clusters, suggesting a simple mechanism…
We introduce a simple model of population dynamics which considers birth and death rates for every individual that depend on the number of particles in its neighborhood. The model shows an inhomogeneous quasistationary pattern with many…
We discuss stochastic dynamics of populations of individuals playing games. Our models possess two evolutionarily stable strategies: an efficient one, where a population is in a state with the maximal payoff (fitness) and a risk-dominant…
Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and…
The dynamics of adaptation is difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from number fluctuations, called genetic drift, arising in the small number of particularly fit…
Motivated by a general principle governing regulation mechanisms in biological cells, we investigate a general interaction scheme between different populations of particles and specific particles, referred to as agents. Assuming that each…
We review models of biological evolution in which the population frequency changes deterministically with time. If the population is self-replicating, although the equations for simple prototypes can be linearised, nonlinear equations arise…
We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric…
We investigate the large population dynamics of a family of stochastic particle systems with three-state cyclic individual behaviour and parameter-dependent transition rates. On short time scales, the dynamics turns out to be approximated…
Evolution occurs in populations of reproducing individuals. The structure of a biological population affects which traits evolve. Understanding evolutionary game dynamics in structured populations is difficult. Precise results have been…
A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the…
When studying the dynamics of trait distribution of populations in a heterogeneous environment, classical models from quantitative genetics choose to look at its system of moments, specifically the first two ones. Additionally, in order to…
The population is composed of individuals characterised by their genetic strings, phenotypes and ages. We discuss the influence of probabilities of survival of the individuals on the dynamics and phenotypic variability of the population. We…
Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we summarise and adapt a deterministic approach, as valid for infinite populations, which assumes…
Frequency-dependent selection reflects the interaction between different species as they battle for limited resources in their environment. In a stochastic evolutionary game the species relative fitnesses guides the evolutionary dynamics…
We propose a stochastic model for evolution through mutation and natural selection of a population that evolves on a $\bbT_d^+$ tree. We think of this model as a way of describing the evolution fitness landscape of a population. We obtain…
We study the stability of non-conservative deterministic cross diffusion models and prove that they are approximated by stochastic population models when the populations become locally large. In this model, the individuals of two species…
Stochastic models of sequential mutation acquisition are widely used to quantify cancer and bacterial evolution. Across manifold scenarios, recurrent research questions are: how many cells are there with $n$ alterations, and how long will…
We present a general framework to describe the evolutionary dynamics of an arbitrary number of types in finite populations based on stochastic differential equations (SDE). For large, but finite populations this allows to include…
Finding the most powerful node in a dynamic random network, the largest set in a partition-valued stochastic process, or the largest family in an evolving population at a given time, can be a very difficult problem. This is particularly the…