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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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
We introduce general two-component reaction-diffusion systems and Turing instability. Then we specialise on the Gierer-Meinhardt system for hydra. We discuss amplitude equations, order parameters and analytical methods for spiky patterns.
Juncheng Wei, Matthias Winter

Chapter 2. Existence of Spikes for the Gierer-Meinhardt System in One Dimension

Abstract
We give a full account of the existence of multiple spikes for the Gierer-Meinhardt system in an interval on the real line. We present a unified rigorous approach based on the Liapunov-Schmidt method which is very flexible and consider the cases of symmetric and asymmetric multi-spike solutions. We also study clustered multiple spikes.
Juncheng Wei, Matthias Winter

Chapter 3. The Nonlocal Eigenvalue Problem (NLEP)

Abstract
We rigorously study the stability of nonlocal eigenvalue problems by using a variety of approaches: Quadratic forms, the method of continuation and hypergeometric functions.
Juncheng Wei, Matthias Winter

Chapter 4. Stability of Spikes for the Gierer-Meinhardt System in One Dimension

Abstract
We study the stability of multi-spike steady states in one space dimension. There are two types of solutions: symmetric and asymmetric multiple spikes. We will show that symmetric spikes in one dimension may be stable or unstable. Asymmetric spikes in one dimension are always unstable, however they may be metastable. Large eigenvalues of unity order are studied by nonlocal eigenvalue problems, small eigenvalues are investigated by a projection similar to Liapunov-Schmidt reduction.
Juncheng Wei, Matthias Winter

Chapter 5. Existence of Spikes for the Shadow Gierer-Meinhardt System

Abstract
We study the existence of boundary spikes for the shadow Gierer-Meinhardt system in higher dimensions. We use Liapunov-Schmidt reduction and the Localised Energy Method in combination with geometric computations depending on the neighbourhood of the spike.
Juncheng Wei, Matthias Winter

Chapter 6. Existence and Stability of Spikes for the Gierer-Meinhardt System in Two Dimensions

Abstract
We prove results on the existence and stability of multiple spikes for the Gierer-Meinhardt system in a bounded, smooth two-dimensional domain. The case of symmetric spikes is studied in detail and for asymmetric spikes the results are stated.
Juncheng Wei, Matthias Winter

Chapter 7. The Gierer-Meinhardt System with Inhomogeneous Coefficients

Abstract
We extend the rigorous approach for the existence and stability of spikes to reaction-diffusion systems with spatially inhomogeneous coefficients. Two particular cases are studied: precursors (inhomogeneous decay rate of activator) and discontinuous diffusivities. Results on the instability due to a precursor and on the existence and stability of a spike near the jump discontinuity of the inhibitor diffusivity are derived.
Juncheng Wei, Matthias Winter

Chapter 8. Other Aspects of the Gierer-Meinhardt System

Abstract
Others aspects of spikes for reaction-diffusion systems are presented: Stability for finite diffusivity, existence and stability for large reaction rates, Robin instead of Neumann boundary conditions, the system on Riemannian manifolds.
Juncheng Wei, Matthias Winter

Chapter 9. The Gierer-Meinhardt System with Saturation

Abstract
We investigate the Gierer-Meinhardt system with saturation. The shape of the spike changes and it is now determined as the solution of a parametrised differential equation. To investigate the stability we study a parametrised nonlocal eigenvalue problem.
Juncheng Wei, Matthias Winter

Chapter 10. Spikes for Other Two-Component Reaction-Diffusion Systems

Abstract
We present results on the existence and stability of multiple spikes for reaction-diffusion systems of the activator-substrate type: The Schnakenberg and Gray-Scott models. We will consider the Schnakenberg model in one space dimension and the Gray-Scott model in two space dimensions. We will conclude by considering flow-distributed spikes, namely the influence of convection on the existence and stability of spikes in the case of the Schnakenberg model. In this chapter we focus on the main results and their biological relevance but skip most of the proofs.
Juncheng Wei, Matthias Winter

Chapter 11. Reaction-Diffusion Systems with Many Components

Abstract
We consider some large reaction-diffusion systems which consist of more than two components. We begin with the hypercycle of Eigen and Schuster which has an arbitrary number of components. For this system we determine the maximum number of components for which a stable cluster is possible. Next we study a five-component system for which we will prove the existence and stability of mutually exclusive spikes, i.e. spikes which for different components are located at different positions. Then we consider systems with multiple activators and substrates and derive conditions for stable spiky patterns. Finally, we investigate a consumer chain model, which is a three-component system with two small diffusion constants and prove the existence and stability of a new type of clustered spiky pattern.
Juncheng Wei, Matthias Winter

Chapter 12. Biological Applications

Abstract
We discuss a number of biological, chemical and ecological applications of pattern formation in reaction-diffusion systems. These include the head development and regeneration in Hydra, embryology for newt and Drosophila, pigmentation patterns on sea shells, fish and mammals and, finally, patterns on growing domains for angelfish body patterns and alligator tooth formation.
Juncheng Wei, Matthias Winter

Chapter 13. Appendix

Abstract
In this Appendix we provide some basic mathematical definitions and results which are used throughout the book. We begin by introducing Sobolev spaces and linear operators and stating some of their properties. Then we give a self-contained proof of the uniqueness, nondegeneracy and spectrum of the ground state of a semi-linear elliptic partial differential equation.
Juncheng Wei, Matthias Winter

Backmatter

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