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Mathematical aspects of pattern formation in biological systems / Juncheng Wei, Matthias Winter.

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Author/Creator:
Wei, Juncheng, 1968- author.
Language:
English.
Publication date:
2014
Publication:
London : Springer, [2014]
Format:
  • Book
  • xii, 319 pages : illustrations (black and white) ; 24 cm.
Bibliography:
Includes bibliographical references and index.
Contents:
  • Introduction
  • Reaction-diffusion systems
  • The Gierer-Meinhardt system for Hydra
  • Turing's diffusion-driven instability
  • Amplitude equations and order parameters
  • Analytical methods for spiky patterns
  • Existence of spikes for the Gierer-Meinhardt system in one dimension
  • Symmetric multi-spike solutions : a rigorous proof of existence
  • Asymmetric multi-spike solutions : a formal derivation
  • Existence of symmetric and asymmetric multiple spikes : a unified rigorous approach
  • Some preliminaries
  • Study of the approximate solutions
  • The Liapunov-Schmidt 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 Gierer-Meinhardt 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 Gierer-Meinhardt system
  • The shadow Gierer-Meinhardt system
  • The existence proof
  • Technical analysis
  • The Liapunov-Schmidt reduction method
  • The reduced problem : a finite-dimensional maximisation problem
  • The completion of the existence proof
  • Notes on the literature
  • Existence and stability of spikes for the Gierer-Meinhardt 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 Gierer-Meinhardt 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 Gierer-Meinhardt system
  • The Gierer-Meinhardt system with finite diffusivity
  • Some properties of the function Wl
  • Nonlocal eigenvalue problems
  • Extensions to higher dimensions
  • The Gierer-Meinhardt system with large reaction rates
  • Construction of the steady state
  • Stability
  • Large eigenvalues
  • Small eigenvalues
  • The Gierer-Meinhardt system with Robin boundary conditions
  • Study of the NLEP
  • Eigenvalue estimates
  • Numerical simulations
  • The Gierer-Meinhardt system on manifolds
  • Introduction
  • The geometric setting
  • The main results
  • Notes on the literature
  • The Gierer-Meinhardt system with saturation
  • The parametrised ground state
  • Stability of spikes
  • Notes on the literature
  • Spikes for other two-component reaction-diffusion systems
  • The Schnakenberg model
  • The Gray-Scott model
  • Flow-distributed spikes
  • Notes on the literature
  • Reaction-diffusion 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 reaction-diffusion 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.
Note:
Also published electronically.
Terms:
Current copyright fee: GBP19.32 42\0.
Contributor:
Winter, Matthias (Mathematician), author.
Available in another form:
9781447155263 (online)
(GyWOH)har130448195
Series:
Applied mathematical sciences, 0066-5452 ; volume 189
Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 189.
Subjects:
ISBN:
9781447155256
1447155254

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