Mutually Exclusive Spiky Pattern and Segmentation Modeled by the Five-Component Meinhardt--Gierer System.

SIAM Journal on Applied Mathematics (Impact Factor: 1.41). 01/2008; 69(2):419-452. DOI: 10.1137/060673138
Source: DBLP

ABSTRACT We consider the five-component Meinhardt-Gierer model for mutually exclusive patterns and segmentation, which was proposed in [H. Meinhardt and A. Gierer, J. Theoret. Biol., 85 (1980), pp. 429-450]. We prove rigorous results on the existence and stability of mutually exclusive spikes which are located in different positions for the two activators. Sufficient conditions for existence and stability are derived, which depend in particular on the relative size of the various diffusion constants. Our main analytical methods are the Liapunov-Schmidt reduction and nonlocal eigenvalue problems. The analytical results are confirmed by numerical simulations.

  • [Show abstract] [Hide abstract]
    ABSTRACT: Mathematical modeling is quite a burgeoning field of study. Representing a natural phenomenon is challenging, a non-ending process that does evolve as we expand interdisciplinary studies. This book deals with mathematical analysis of patterns encountered in biological systems, using a variety of functional analysis methods to prove the existence of solutions. It is the authors own account and expansion of their results over a fifteen year period, which contributes to initiating a systematic programme of rigorous mathematical investigation into pattern formation for large scale-amplitude patterns far from equilibrium in biologically relevant models. By reviewing the bifurcation of patterned states from unstable homogeneous steady states which differ from the standard instability approach by Turing, one can investigate large-amplitude patterns in the case of multiple spikes which concentrate at certain points of the underlying domain. Previous studies have introduced non-local eigenvalue problems in one-dimensional settings, this books extends that concept to general partial differential equations without any symmetry assumptions, but with only a smoothness requirement at the boundary. The book is not for lay readers as stated therein “... of interest to graduate students and researchers who are ACTIVE ...” It is indeed written for advanced graduates and experts interested in the mathematics of pattern formation and reaction diffusion equations. The linkage of results in the book to biological applications as well as highlighting their relevance to natural phenomena have nothing obvious for a beginner in this field. Also, little background information are provided at the beginning of most chapters. The mathematical proofs are quite robust and readers interested in the study of pattern formation will find it quite interesting to have a compilation of such amount of information in one place. I would recommend other textbooks to someone interested or active in mathematical biology. However, this is a good reference source for various advanced theory and mathematical applications in this field. The book layout is excellent, but its greatest interest is the specific context in which it is written; to provide the reader with advanced theories on - examining existence of spiky steady states in reaction-diffusion systems - exploring spatially homogeneous two-component activator-inhibitor system in 1 or 2-D and finally - extending the theories by adding extra effects with specific roles in developmental biology such as spatial inhomogeneity, large reaction rates, convection, modified or altered boundary conditions and saturation terms (to name a few).Reviewer: J. Michel Tchuenche (Atlanta)
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30-39, 1972] and Schnakenberg (J Theor Biol, 81:389-400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order [Formula: see text]. Further, there are small eigenvalues of order [Formula: see text] which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order [Formula: see text] leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order [Formula: see text] needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant [Formula: see text].
    Journal of Mathematical Biology 11/2012; DOI:10.1007/s00285-012-0616-8 · 2.39 Impact Factor


Available from