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Genealogies of regular exchangeable coalescents with applications to sampling

Vlada Limic (2012)

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

This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as 𝛯 -coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number of active lineages were obtained by the author in a previous work. Analogous results for the number of active mutation-free lineages and the combined lineage lengths are derived using the same martingale-based technique. They are given in terms of convergence in...

Generation of Interface for an Allen-Cahn Equation with Nonlinear Diffusion

M. Alfaro, D. Hilhorst (2010)

Mathematical Modelling of Natural Phenomena

In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property.

Global attractivity of the equilibrium of a nonlinear difference equation

John R. Graef, C. Qian (2002)

Czechoslovak Mathematical Journal

The authors consider the nonlinear difference equation x n + 1 = α x n + x n - k f ( x n - k ) , n = 0 , 1 , ⋯ . 1 where α ∈ ( 0 , 1 ) , k ∈ { 0 , 1 , ⋯ } and f ∈ C 1 [ [ 0 , ∞ ) , [ 0 , ∞ ) ] ( 0 ) with f ' ( x ) < 0 . They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.

Global Existence and Boundedness of Solutions to a Model of Chemotaxis

J. Dyson, R. Villella-Bressan, G. F. Webb (2008)

Mathematical Modelling of Natural Phenomena

A model of chemotaxis is analyzed that prevents blow-up of solutions. The model consists of a system of nonlinear partial differential equations for the spatial population density of a species and the spatial concentration of a chemoattractant in n-dimensional space. We prove the existence of solutions, which exist globally, and are L∞-bounded on finite time intervals. The hypotheses require nonlocal conditions on the species-induced production of the chemoattractant.

Global exponential stability of almost periodic solutions for a delayed single population model with hereditary effect

Qiyuan Zhou, Jianying Shao (2015)

Annales Polonici Mathematici

This paper is concerned with a delayed single population model with hereditary effect. Under appropriate conditions, we employ a novel argument to establish a criterion of the global exponential stability of positive almost periodic solutions of the model. Moreover, an example and its numerical simulation are given to illustrate the main result.

Global stability of steady solutions for a model in virus dynamics

Hermano Frid, Pierre-Emmanuel Jabin, Benoît Perthame (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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...

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