Mathematical Aspects of Pattern Formation in Biological Systems

by Juncheng Wei, Matthias Winter

Overview

This monograph is concerned with the mathematical analysis of patterns which are encountered in biological systems. It summarises, expands and relates results obtained in the field during the last fifteen years. It also links the results to biological applications and highlights their relevance to phenomena in nature. Of particular concern are large-amplitude patterns far from equilibrium in biologically relevant models.

The approach adopted in the monograph is based on the following paradigms:

• Examine the existence of spiky steady states in reaction-diffusion systems and select as observable patterns only the stable ones

• Begin by exploring spatially homogeneous two-component activator-inhibitor systems in one or two space dimensions

• Extend the studies by considering extra effects or related systems, each motivated by their specific roles in developmental biology, such as spatial inhomogeneities, large reaction rates, altered boundary conditions, saturation terms, convection, many-component systems.

Mathematical Aspects of Pattern Formation in Biological Systems will be of interest to graduate students and researchers who are active in reaction-diffusion systems, pattern formation and mathematical biology.

Product Details

ISBN-13:	9781447172611
Publisher:	Springer London
Publication date:	10/03/2014
Series:	Applied Mathematical Sciences , #189
Edition description:	Softcover reprint of the original 1st ed. 2014
Pages:	319
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

Introduction.- Existence of spikes for the Gierer-Meinhardt system in one dimension.- The Nonlocal Eigenvalue Problem (NLEP).- Stability of spikes for the Gierer-Meinhardt system in one dimension.- Existence of spikes for the shadow Gierer-Meinhardt system.- Existence and stability of spikes for the Gierer-Meinhardt system in two dimensions.- The Gierer-Meinhardt system with inhomogeneous coefficients.- Other aspects of the Gierer-Meinhardt system.- The Gierer-Meinhardt system with saturation.- Spikes for other two-component reaction-diffusion systems.- Reaction-diffusion systems with many components.- Biological applications.- Appendix.

Editorial Reviews

From the book reviews:

“This book deals with the mathematical analysis of patterns encountered in biological systems, using a variety of functional analysis methods to prove the existence of solutions. … It is indeed written for advanced graduates and experts interested in the mathematics of pattern formation and reaction-diffusion equations. … this is a good reference source for various advanced theories and mathematical applications in this field.” (J. Michel Tchuenche, zbMATH, Vol. 1295, 2014)

From the Publisher