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Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

Pierre Pudlo (2010)

ESAIM: Probability and Statistics

To establish lists of words with unexpected frequencies in long sequences, for instance in a molecular biology context, one needs to quantify the exceptionality of families of word frequencies in random sequences. To this aim, we study large deviation probabilities of multidimensional word counts for Markov and hidden Markov models. More specifically, we compute local Edgeworth expansions of arbitrary degrees for multivariate partial sums of lattice valued functionals of finite Markov...

Large population limit and time behaviour of a stochastic particle model describing an age-structured population

Viet Chi Tran (2008)

ESAIM: Probability and Statistics


We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound...

Large scale behaviour of the spatial 𝛬 -Fleming–Viot process

N. Berestycki, A. M. Etheridge, A. Véber (2013)

Annales de l'I.H.P. Probabilités et statistiques

We consider the spatial 𝛬 -Fleming–Viot process model (Electron. J. Probab.15(2010) 162–216) for frequencies of genetic types in a population living in ℝ d , in the special case in which there are just two types of individuals, labelled 0 and 1 . At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type 0 . 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...

Large time behavior in a density-dependent population dynamics problem with age structure and child care

Vladas Skakauskas (2003)

Banach Center Publications

Two asexual density-dependent population dynamics models with age-dependence and child care are presented. One of them includes the random diffusion while in the other the population is assumed to be non-dispersing. The population consists of the young (under maternal care), juvenile, and adult classes. Death moduli of the juvenile and adult classes in both models are decomposed into the sum of two terms. The first presents death rate by the natural causes while the other describes the environmental...

Lévy Processes, Saltatory Foraging, and Superdiffusion

J. F. Burrow, P. D. Baxter, J. W. Pitchford (2008)

Mathematical Modelling of Natural Phenomena

It is well established that resource variability generated by spatial patchiness and turbulence is an important influence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulations indicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment can be formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly,...

Limitation and Regulation of Ecological Populations: a Meta-analysis of Tipula paludosa Field Data

R. P. Blackshaw, S. V. Petrovskii (2010)

Mathematical Modelling of Natural Phenomena

Whether the size of an animal population is environmentally limited or regulated by density dependent negative feedback mechanisms is of ecological interest. Proponents of limitation theory have issued a set of specific challenges which are addressed in this paper using field data for the insect Tipula paludosa. This species is known to be subject to population crashes caused by adverse environmental conditions and assumed to be limited. We re-examine published data in support of this hypothesis...

Linking population genetics to phylogenetics

Paul G. Higgs (2008)

Banach Center Publications

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...

Local Collapses in the Truscott-Brindley Model

I. Siekmann, H. Malchow (2008)

Mathematical Modelling of Natural Phenomena

Relaxation oscillations are limit cycles with two clearly different time scales. In this article the spatio-temporal dynamics of a standard prey-predator system in the parameter region of relaxation oscillation is investigated. Both prey and predator population are distributed irregularly at a relatively high average level between a maximal and a minimal value. However, the slowly developing complex pattern exhibits a feature of “inverse excitability”: Both populations show collapses which occur...

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