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Understanding the evolution of individuals which live in a structured and fluctuating environment is of central importance in mathematical population genetics. Here we outline some of the mathematical challenges arising from modelling structured populations, primarily focussing on the interplay between forwards in time models for the evolution of the population and backwards in time models for the genealogical trees relating individuals in a sample from that population. In addition to classical...
The present paper introduces a tumor
model with two time scales, the time t during which the tumor
grows and the cycle time of individual cells. The model also
includes the effects of gene mutations on the population density
of the tumor cells. The model is formulated as a free boundary
problem for a coupled system of elliptic, parabolic and hyperbolic
equations within the tumor region, with nonlinear and nonlocal
terms. Existence and uniqueness theorems are proved, and
properties of the free boundary...
The subject of the paper is reciprocal influence of pure mathematics and applied sciences. We illustrate the idea by giving a review of mathematical results obtained recently, related to the model of stochastic gene expression due to Lipniacki et al. [38]. In this model, featuring mRNA and protein levels, and gene activity, the stochastic part of processes involved in gene expression is distinguished from the part that seems to be mostly deterministic, and the dynamics is expressed by means of a...
We consider a simple model for the immune system in which virus are able to undergo mutations and are in competition with leukocytes. These mutations are related to several other concepts which have been proposed in the literature like those of shape or of virulence – a continuous notion. For a given species, the system admits a globally attractive critical point. We prove that mutations do not affect this picture for small perturbations and under strong structural assumptions. Based on numerical...
We consider a simple model for the immune system
in which virus are able to undergo mutations and are in competition
with leukocytes. These mutations are related to several other concepts which have
been proposed in the literature like those of shape or of
virulence – a continuous notion. For a given species, the system admits a
globally attractive critical point. We prove that mutations do not affect this
picture for small perturbations and under strong structural assumptions.
Based on numerical...
High-dimensional data models abound in genomics studies, where often inadequately small sample sizes create impasses for incorporation of standard statistical tools. Conventional assumptions of linearity of regression, homoscedasticity and (multi-) normality of errors may not be tenable in many such interdisciplinary setups. In this study, Kendall's tau-type rank statistics are employed for statistical inference, avoiding most of parametric assumptions to a greater extent. The proposed procedures...
We consider the spatial -Fleming–Viot process model (Electron. J. Probab.15(2010) 162–216) for frequencies of genetic types in a population living in , in the special case in which there are just two types of individuals, labelled and . At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type . We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the dynamics...
Population geneticists study the variability of gene sequences within a species, whereas phylogeneticists compare gene sequences between species and usually have only one representative sequence per species. Stochastic models in population genetics are used to determine probability distributions for gene frequencies and to predict the probability that a new mutation will become fixed in a population. Stochastic models in phylogenetics describe the substitution process in the single sequence that...
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