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We now turn our attention to steady states of reaction–transport systems. We focus first on steady states that arise in RD models on finite domains. Such models are important from an ecological point of view, since they describe population dynamics in island habitats. The main problem consists in determining the critical patch size, i.e., the smallest patch that can minimally sustain a population. As expected intuitively, the critical patch size depends on a number of factors, such as the population dynamics in the patch, on the nature of the boundaries, the patch geometry, and the reproduction kinetics of the population. The first critical patch model was studied by Kierstead Kierstead, H. and Slobodkin Slobodkin, L. B. [228] and Skellam Skellam, J. G. [414] and is now called the KISS problem. A significant amount of work has focused on systems with partially hostile boundaries, where individuals can cross the boundary at some times but not at others, or systems where individuals readily cross the boundary but the region outside the patch is partially hostile, or a combination of the above. In this chapter we deal with completely hostile boundaries and calculate the critical patch size for different geometries, reproduction processes, and dynamics. [ABSTRACT FROM AUTHOR]

The preceding chapters have dealt with the spatiotemporal behavior of spatially continuous systems. We now turn our attention to the dynamical behavior and stability properties of spatially discrete systems. A wide variety of phenomena in chemistry, biology, physics, and other fields involve the coupling between nonlinear, discrete units. Examples include arrays of Josephson junctions, chains of coupled diode resonators, coupled chemical or biochemical reactors, myelinated nerve fibers, neuronal networks, and patchy ecosystems. Such networks of coupled nonlinear units often combine dynamical and structural complexity [422]. Cells in living tissues, for example, are arranged in a variety of geometries. One-dimensional rings of cells were already considered by Turing [440]. Other types of lattices, such as open-ended linear arrays, tubes, rectangular and hexagonal arrays, and irregular arrangements in two or three dimensions are also found, see for example [5]. Cells interact with adjacent cells in various distinct ways. For example, signaling between cells may occur via diffusion through gap junctions [352, 230] or by membrane-bound proteins, juxtacrine signaling [339, 340, 471]. [ABSTRACT FROM AUTHOR]

Reaction–transport equations have found many applications to biological processes of interest in medicine and microbiology. Examples range from cancer invasion to virus dispersal and transport in spiny dendrites. [ABSTRACT FROM AUTHOR]

In Chap. 10 we considered the Turing instability in systems where the kinetic parameters and the transport coefficients are constant in space and time. While the vast majority of theoretical work on Turing patterns deals with such systems, there are good reasons from applications in biology and ecology to account for the effect of spatial or temporal variations on the threshold of the Turing instability. Chemical or biological systems are rarely completely uniform. Pattern formation in the Drosophila egg, for example, occurs in the presence of maternally imposed gradients of gene products [106]. Experimental studies of Turing patterns in the CIMA and CDIMA reactions use continuously fed unstirred reactors continuously fed unstirred reactor (CFURs), see Chap. 12, which unavoidably exhibit gradients in the concentrations of the feed reactants. The problem of determining diffusion-driven instabilities in reacting systems with spatially or temporally varying parameters is in general a rather difficult one. The tools of the linear 10 cannot be extended to such systems, since they do not posses a uniform steady state in most cases. Reaction–diffusion systems with weak heterogeneities can be studied with perturbation techniques [19, 50, 92, 40, 341, 342]. Lengyel Lengyel, I. and coworkers [249] used an approximation of the reaction–diffusion equation to study the effect of the gradients in CFURs CFUR on the position and the possible three-dimensional character of the Turing structures. [ABSTRACT FROM AUTHOR]

Turing΄s paper on diffusion-driven instabilities in nonequilibrium reaction–diffusion systems as a means of biological pattern formation [440] attracted little attention for about two decades, as shown by the citation histogram in Fig. 12.1. One of the first scientists to be intrigued by Turing΄s ideas was Wardlaw, a botanist who thought about ways to test the mechanism experimentally [468, 470, 469]. By the early 1970s theoretical biologists and biomathematicians began to explore in earnest if Turing instabilities could explain spatial pattern formation in a variety of living systems and a considerable body of theoretical work was produced, see for example [157, 279, 231, 239, 182, 183, 264, 261, 308]. Morphogen-based pattern formation, where the long-range influence of signaling molecules induces structure, is a well-established phenomenon in developmental biology [26]. However, definitive evidence for a Turing mechanism of pattern formation within a morphogen system is still lacking. Several promising candidate systems exist and are discussed in Sect. 12.2. [ABSTRACT FROM AUTHOR]

Alan Turing΄s Turing, A. paper entitled ˵The Chemical Basis of Morphogenesis″ [440] ranks without doubt among the most important papers of the last century. In that seminal work Turing laid the foundation for the theory of chemical pattern formation. Turing showed that diffusion can have nontrivial effects in nonequilibrium systems. The interplay of diffusion with nonlinear kinetics can destabilize the uniform steady state of reaction–diffusion systems and generate stable, stationary concentration patterns. [ABSTRACT FROM AUTHOR]

Reaction–transport equations allow us to model the spread of invading populations. Traveling front solutions describe the invasive process, and their velocity is a quantity that can be obtained from observational data. In this chapter we review some recent reaction–transport models and compare the theoretical predictions with the observed values for the rate of population invasions. [ABSTRACT FROM AUTHOR]

Traveling waves are typical nonequilibrium phenomena encountered in numerous instances in physics, chemistry, biology, and other areas [129, 82, 309, 310]. Reacting and diffusing systems described by the RD equation (2.3) represent a particular well-studied class of applications. [ABSTRACT FROM AUTHOR]

Most systems in nature are heterogeneous. Porous media, fractals, random or fluctuating environments, and ecological patchiness are only a few examples. Homogeneous media are, in fact, an idealization. In this chapter we discuss how fronts propagate in these more complex structures. Our main goal is to present various techniques that allow us to calculate the front velocity. When the heterogeneity is weak, we can employ perturbative methods, but in most cases only the Hamilton–Jacobi method works effectively in the large-time limit. [ABSTRACT FROM AUTHOR]

Particles, such as molecules, atoms, or ions, and individuals, such as cells or animals, move in space driven by various forces or cues. In particular, particles or individuals can move randomly, undergo velocity jump processes or spatial jump processes [333]. The steps of the random walk can be independent or correlated, unbiased or biased. The probability density function (PDF) for the jump length can decay rapidly or exhibit a heavy tail. Similarly, the PDF for the waiting time between successive jumps can decay rapidly or exhibit a heavy tail. We will discuss these various possibilities in detail in Chap. 3. Below we provide an introduction to three transport processes: standard diffusion, transport with inertia, and anomalous diffusion. [ABSTRACT FROM AUTHOR]

As discussed in Sect. 2.1, the standard reaction–diffusion equation for the particle density ]> has the form 3.1 ]> This equation is an example of a macroscopic reaction–transport equation that can be obtained in the long-time large-scale limit of mesoscopic equations. Recall that the mesoscopic approach is based on the idea that one can introduce mean-field equations for the particle density involving a detailed description of the movement of particles on the microscopic level. At the same time, random fluctuations around the mean behavior can be neglected due to a large number of individual particles. [ABSTRACT FROM AUTHOR]

In this chapter we consider the problem of propagating fronts traveling into an unstable state of a reaction–transport system. The purpose is to present the general formalism for the asymptotic analysis of traveling fronts. The method relies on the hyperbolic scaling procedure, the theory of large deviations, and the Hamilton–Jacobi technique. A generic model that describes phenomena of this type is the RD equation (2.3) with appropriate kinetics, such as the FKPP equation (4.1). The propagation velocity of fronts of this equation has been studied in Chap.4. The RD equation involves implicitly a long-time large-scale parabolic scaling, while as far as propagating fronts are concerned, the appropriate scaling must be a hyperbolic one. The macroscopic transport process arises from the overall effect of many particles performing complex random movements. Classical diffusion is simply an approximation for this transport in the long-time large-scale parabolic limit. In general, this approximation is not appropriate for problems involving propagating fronts. The basic idea is that the kinetic term in the RD equation with KPP kinetics is very sensitive to the tails of a density profile. These tails are typically ˵non-universal,″ ˵non-diffusional,″ and dependent on the microscopic details of the underlying random walk. The purpose of this chapter is to demonstrate that the macroscopic dynamics of the front for a reaction–transport system are dependent on the choice of the underlying random walk model for the transport process. To illustrate the idea of an alternative description of front propagation into an unstable state of reaction–transport system, we consider several models including discrete-in-time or continuous-in-time Markov models with long-distance dispersal kernels, non-Markovian models with memory effects, etc., instead of the RD equation. Let us give a few examples of such models. [ABSTRACT FROM AUTHOR]

If spatial effects can be neglected, many systems in chemistry, biology, ecology, physics, and other areas can be described by rate equations, a set of ordinary differential equations. Systems may be homogeneous because the system is small enough and transport is efficient enough to eliminate spatial gradients. Or the spatial homogeneity may be imposed from the outside, for example by stirring in chemical reactors. In the following we collect some basic facts about rate equations and discuss various model schemes whose dynamics we investigate in later chapters. [ABSTRACT FROM AUTHOR]