Methods of small parameter in mathematical biology
type: Item
instance of: scholarly article
title: Methods of small parameter in mathematical biology
zbMATH Open document ID: 1309.92012
DOI: 10.1007/978-3-319-05140-6
author: Jacek Banasiak
author: Mirosław Lachowicz
published in: Modeling and Simulation in Science, Engineering and Technology
publication date: 2014-03-31T00:00:00Z
review text: In the introductory Chapter 1, the basic ideas of asymptotic analysis are presented with information on models of mathematical biology, chemical kinetics and Brownian motion, population dynamics, epidemiology, models of fluid dynamics with initial and boundary layer phenomena, to which asymptotic analysis methods and in particular small parameters methods shall be applied. The basic idea is the subdivision of the variables describing the model into two groups: variables that evolve slowly and variables that evolve quickly. A systematic way of asymptotic analysis application is suggested here: {\parindent=6mm\begin{itemize}\item[1.] Finding an appropriate set of aggregated macro-variables, which describes the dynamics of the model at this level. \item[2.] Proving that these aggregated macro-variables approximate the micro-variables. \item[3.] Determining the relations between the parameters at the macro-level and the original micro-variables. \end{itemize}} The subsequent chapters present several realizations of this program. Chapter 2 gives the presentation of the Chapman-Enskog-type asymptotic expansion, with the basic techniques for the proof of its convergence on linear ODEs with a preliminary survey of the necessary information from linear algebra to the theory of finite-dimensional dynamical systems. As a concrete application, the authors consider the simplified linear model of interacting populations that can reside in one of two patches with migration between them much faster than their demographic processes. The techniques developed here can be applied to nonlinear processes modeled by ODEs, PDEs and also integro-differential equations with a proof for each application. In Chapter 3 presents a theory which is based on some results due to \textit{N. A. Tikhonov} and \textit{A. B. Vasilyeva} (cf.\ [\textit{A. N. Tikhonov} et al., Differential equations. English edition. Berlin etc.: Springer-Verlag (1985; Zbl 0553.34001)]), which provides constrictive estimate…
Mathematics Subject Classification ID: 92-02
Mathematics Subject Classification ID: 92B05
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 92D30
zbMATH DE Number: 6276056
zbMATH Keywords: small parameter methods
zbMATH Keywords: Chapman-Enskog method
zbMATH Keywords: Tikhonov-Vasilyeva theory with applications to Allee model
zbMATH Keywords: SIS model with age-structure
zbMATH Keywords: population problems with fast migrations
zbMATH Keywords: singularly perturbed McKendrick problem
zbMATH Keywords: diffusion limit of the telegraph equation
zbMATH Keywords: kinetic model of alignment
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-05140-6
OpenAlex ID: W179936242
description: scientific article