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Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…
The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. We study the higher-order connectivity of such a network by investigating the topological…
We present a statistical analysis of biological evolution processes. Specifically, we study the stochastic replication-mutation-death model where the population of a species may grow or shrink by birth or death, respectively, and…
We describe the simulation method of modelling the population evolution using Monte Carlo based on the Penna model. Individuals in the populations are represented by their diploid genomes. Genes expressed after the minimum reproduction age…
The case-control design is often used to test associations between the case-control status and genetic variants. In addition to this primary phenotype a number of additional traits, known as secondary phenotypes, are routinely recorded and…
We propose a new deterministic growth model which captures certain features of both the Gompertz and Korf laws. We investigate its main properties, with special attention to the correction factor, the relative growth rate, the inflection…
Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright-Fisher diffusion and the Kingman coalescent, where the former describes the stochastic…
We consider a model of asexually reproducing individuals. The birth and death rates of the individuals are affected by a fitness parameter. The rate of mutations that cause the fitnesses to change is proportional to the population size, N.…
The distributions of the times to the first common ancestor t_mrca is numerically studied for an ecological population model, the extended Moran model. This model has a fixed population size N. The number of descendants is drawn from a beta…
We consider a model of stationary population with random size given by a continuous state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure of the genealogical tree of…
Distance-based approaches in phylogenetics such as Neighbor-Joining are a fast and popular approach for building trees. These methods take pairs of sequences from them construct a value that, in expectation, is additive under a stochastic…
We introduce a two-dimensional growth model where every new site is located, at a distance $r$ from the barycenter of the pre-existing graph, according to the probability law $1/r^{2+\alpha_G} (\alpha_G \ge 0)$, and is attached to (only)…
We introduce a new analytical framework for modelling degree sequences in individual communities of real-world networks, e.g., citations to papers in different fields. Our work is inspired by a recent modification of the Price's model,…
The growth of a population divided among spatial sites, with migration between the sites, is sometimes modelled by a product of random matrices, with each diagonal elements representing the growth rate in a given time period, and…
Using Monte Carlo model of biological evolution we have discovered that populations can switch between two different strategies of their genomes' evolution; Darwinian purifying selection and complementing the haplotypes. The first one is…
Evolutionary models used for describing molecular sequence variation suppose that at a non-recombining genomic segment, sequences share ancestry that can be represented as a genealogy--a rooted, binary, timed tree, with tips corresponding…
In large asexual populations, beneficial mutations have to compete with each other for fixation. Here, I derive explicit analytic expressions for the rate of substitution and the mean beneficial effect of fixed mutations, under the…
This paper proposes a non-phenomenological model of population growth that is based on the interactions between the individuals that compose the system. It is assumed that the individuals interact cooperatively and competitively. As a…
Although extensive behavioral changes often exist between closely related animal species, our understanding of the genetic basis underlying the evolution of behavior has remained limited. Here, we propose a new framework to study behavioral…
Reconstructing past population size from present day genetic data is a major goal of population genetics. Recent empirical studies infer population size history using coalescent-based models applied to a small number of individuals. Here we…