Méndez V., Fedotov S., Horsthemke W. Reaction Transport Systems

Heidelberg: Springer, 2010. — 456 p.The book is organized in three parts. Part I lays the foundation. Chapter 1 provides an introduction to rate equations and their stability analysis. It also presents several important chemical and biological models. In Chap. 2 we discuss the standard reaction–diffusion equation and introduce two deviations from normal diffusion, namely transport with inertia and anomalous diffusion. We present a phenomenological approach of standard diffusion, transport with inertia, and anomalous diffusion. This chapter also contains a first mesoscopic description of the transport in terms of random walk models. We strongly recommend such a mesoscopic approach to ensure that the reaction–transport equations studied are physically and mathematically sound.We present a comprehensive review of the mesoscopic foundations of reaction–transport equations in Chap. 3, which is at the heart of Part I.
Part II focuses on front propagation. Chapter 4 provides an overview of front propagation in standard reaction–diffusion systems. We discuss pulled vs pushed fronts and provide tools to determine the front velocity for both cases. Chapter 5 deals with the effect of deviations from standard diffusion or Brownian motion on front propagation into unstable states. The effect of spatial heterogeneities on
front propagation is studied in Chap. 6; Chapters 7 and 8 contain applications to ecology, namely human migrations, avian range expansions, and plant invasions, and biomedical systems, namely cancer invasion, virus dispersal, and propagation in spiny dendrites.
Part III focuses on spatial instabilities and patterns. We examine the simplest type of spatial pattern in standard reaction–diffusion systems in Chap. 9, namely patterns in a finite domain where the density vanishes at the boundaries. We discuss methods to determine the smallest domain size that supports a nontrivial steady state, known as the critical patch size in ecology. In Chap. 10, we provide first an overview of the Turing instability in standard reaction–diffusion systems. Then we explore how deviations from standard diffusion, namely transport with inertia and anomalous diffusion, affect the Turing instability. Chapter 11 deals with the effects of temporally or spatially varying diffusivities on the Turing instability in reaction–diffusion systems. We present applications of Turing systems to chemical reactions
and biological systems in Chap. 12; Chapter 13 deals with spatial instabilities and patterns in spatially discrete systems, such as diffusively and photochemically coupled reactors.
This book can serve as a text for a special topics course for advanced undergraduate and beginning graduate students. With this purpose in mind, we have included a set of exercises at the end of each chapter.General Concepts
Reaction Kinetics
Reactions and Transport: Diffusion, Inertia, and Subdiffusion
RandomWalks and Mesoscopic Reaction–Transport Equations
Reaction–Diffusion Fronts
Reaction–Transport Fronts Propagating into Unstable States
Reaction–Diffusion Fronts in Complex Structures
Ecological Applications
Biomedical Applications
Persistence and Extinction of Populations in Finite Domains
Turing Instabilities in Homogeneous Systems
Turing Instabilities in Reaction–Diffusion Systems with Temporally or Spatially Varying Parameters
Chemical and Biological Applications of Turing Systems
Pattern Formation in Spatially Discrete Systems