Mathematical aspects of pattern formation in biological systems
 Responsibility
 Juncheng Wei, Matthias Winter.
 Language
 English.
 Publication
 London : Springer, [2014]
 Physical description
 xii, 319 pages : illustrations (black and white) ; 24 cm.
 Series
 Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 189.
Access
Creators/Contributors
 Author/Creator
 Wei, Juncheng, 1968 author.
 Contributor
 Winter, Matthias (Mathematician), author.
Contents/Summary
 Bibliography
 Includes bibliographical references and index.
 Contents

 Introduction
 Reactiondiffusion systems
 The GiererMeinhardt system for Hydra
 Turing's diffusiondriven instability
 Amplitude equations and order parameters
 Analytical methods for spiky patterns
 Existence of spikes for the GiererMeinhardt system in one dimension
 Symmetric multispike solutions : a rigorous proof of existence
 Asymmetric multispike solutions : a formal derivation
 Existence of symmetric and asymmetric multiple spikes : a unified rigorous approach
 Some preliminaries
 Study of the approximate solutions
 The LiapunovSchmidt reduction method
 The reduced problem
 Clustered multiple spikes
 Notes on the literature
 The nonlocal eigenvalue problem (NLEP)
 A basic theorem for tau=0
 The method of continuation
 Hopf bifurcation
 The method of hypergeometric functions
 Notes on the literature
 Stability of spikes for the GiererMeinhardt system in one dimension
 Symmetric multiple spikes : stability
 Large eigenvalues
 Small eigenvalues
 The spectrum of the matrices B and M
 Notes on the literature
 Existence of spikes for the shadow GiererMeinhardt system
 The shadow GiererMeinhardt system
 The existence proof
 Technical analysis
 The LiapunovSchmidt reduction method
 The reduced problem : a finitedimensional maximisation problem
 The completion of the existence proof
 Notes on the literature
 Existence and stability of spikes for the GiererMeinhardt system in two dimensions
 Symmetric multiple spikes : existence
 The amplitudes of the peaks
 Reduction to finite dimensions
 The reduced problem
 Symmetric multiple spikes : stability
 Large eigenvalues
 Small eigenvalues
 Asymmetric multiple spikes : existence
 Analysing the algebraic system for the amplitudes
 The reduced problem
 Asymmetric multiple spikes : stability
 Large eigenvalues
 Small eigenvalues
 Notes on the literature
 The GiererMeinhardt system with inhomogeneous coefficients
 Precursors
 Results on existence and stability
 Numerical computations
 Discontinuous diffusivities
 Existence and stability of interior spike
 A spike near the jump discontinuity of the inhibitor diffusivity
 Numerical simulations
 Notes on the literature
 Other aspects of the GiererMeinhardt system
 The GiererMeinhardt system with finite diffusivity
 Some properties of the function Wl
 Nonlocal eigenvalue problems
 Extensions to higher dimensions
 The GiererMeinhardt system with large reaction rates
 Construction of the steady state
 Stability
 Large eigenvalues
 Small eigenvalues
 The GiererMeinhardt system with Robin boundary conditions
 Study of the NLEP
 Eigenvalue estimates
 Numerical simulations
 The GiererMeinhardt system on manifolds
 Introduction
 The geometric setting
 The main results
 Notes on the literature
 The GiererMeinhardt system with saturation
 The parametrised ground state
 Stability of spikes
 Notes on the literature
 Spikes for other twocomponent reactiondiffusion systems
 The Schnakenberg model
 The GrayScott model
 Flowdistributed spikes
 Notes on the literature
 Reactiondiffusion systems with many components
 The hypercycle of eigen and schuster
 Mutual exclusion of spikes
 Multiple activators and substrates
 Exotic spiky patterns for a consumer chain model
 Notes on the literature
 Biological applications
 Biological, chemical and ecological applications of reactiondiffusion systems
 Hydra : transplantation of head
 Embryology : formation of body axes for newt and Drosophila, segmentation for Drosophila
 Pigmentation patterns on sea shells, fish and mammals
 Patterns on growing domains : stripes on angelfish and tooth formation in alligators
 Appendix
 Sobolev spaces and linear operators
 Uniqueness, nondegeneracy and spectrum of the ground state
 References
 Index.
 Publisher's Summary
 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 largeamplitude 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 reactiondiffusion systems and select as observable patterns only the stable ones * Begin by exploring spatially homogeneous twocomponent activatorinhibitor 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, manycomponent systems. Mathematical Aspects of Pattern Formation in Biological Systems will be of interest to graduate students and researchers who are active in reactiondiffusion systems, pattern formation and mathematical biology.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2014
 Series
 Applied mathematical sciences, 00665452 ; volume 189
 Note
 Also published electronically.
 Terms
 Current copyright fee: GBP19.32 42\0.
 Available in another form
 9781447155263 (online)
 (GyWOH)har130448195
 ISBN
 9781447155256 (cased)
 1447155254 (cased)