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Methods of Small Parameter in Mathematical BiologyFrom Microscopic to Macroscopic Descriptions

Methods of Small Parameter in Mathematical Biology: From Microscopic to Macroscopic Descriptions [This chapter provides a general overview of multiscale descriptions of natural phenomena and, in contrast to the previous chapters, spans all three scales, from the micro- to the macro-scale. It begins with the microscopic, the so-called individually based, models in which each individual in the population (agent) is characterized by certain properties. The models at this level are represented by (large) systems of linear integro-differential equations describing appropriate jump Markov processes. The passage to the meso-scale is accomplished by means of an asymptotic limit when a small parameter, which here is related to the (inverse) of the size of the population, tends to 0 (i.e. the size of the population tends to infinity). In the resulting limit the population is described by a distribution function which is a solution of a bilinear, Boltzmann-like, integro-differential equation. Finally, the micro-scale description of the population is provided by a diffusion-type equation obtained in the asymptotic limit of the mesoscopic bilinear equation, when the range of the interactions tends to 0. The chapter also contains an extensive survey of models fitting into the framework of the theory and of their properties.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Methods of Small Parameter in Mathematical BiologyFrom Microscopic to Macroscopic Descriptions

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Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2014
ISBN
978-3-319-05139-0
Pages
223 –270
DOI
10.1007/978-3-319-05140-6_8
Publisher site
See Chapter on Publisher Site

Abstract

[This chapter provides a general overview of multiscale descriptions of natural phenomena and, in contrast to the previous chapters, spans all three scales, from the micro- to the macro-scale. It begins with the microscopic, the so-called individually based, models in which each individual in the population (agent) is characterized by certain properties. The models at this level are represented by (large) systems of linear integro-differential equations describing appropriate jump Markov processes. The passage to the meso-scale is accomplished by means of an asymptotic limit when a small parameter, which here is related to the (inverse) of the size of the population, tends to 0 (i.e. the size of the population tends to infinity). In the resulting limit the population is described by a distribution function which is a solution of a bilinear, Boltzmann-like, integro-differential equation. Finally, the micro-scale description of the population is provided by a diffusion-type equation obtained in the asymptotic limit of the mesoscopic bilinear equation, when the range of the interactions tends to 0. The chapter also contains an extensive survey of models fitting into the framework of the theory and of their properties.]

Published: Mar 12, 2014

Keywords: Macro-models; Meso-models; Micro-models; Markov processes; Reaction diffusion equation

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