ABSTRACT. In this paper we study the existence and stability of asymptotically large stationary multi-pulse solutions in a family of singularly perturbed reaction-diffusion equations. This family includes the generalized Gierer-Meinhardt equation. The existence of N-pulse homoclinic orbits (N ≥ 1) is established by the methods of geometric singular perturbation theory. A theory, called the NLEP (=NonLocal Eigenvalue Problem) approach, is developed, by which the stability of these patterns can be studied explicitly. This theory is based on the ideas developed in our earlier work on the Gray-Scott model. It is known that the Evans function of the linear eigenvalue problem associated to the stability of the pattern can be decomposed into the product of a slow and a fast transmission function. The NLEP approach determines explicit leading order approximations of these transmission functions. It is shown that the zero/pole cancellation in the decomposition of the Evans function, called the NLEP paradox, is a phenomenon that occurs naturally in singularly perturbed eigenvalue problems. It follows that the zeroes of the Evans function, and thus the spectrum of the stability problem, can be studied by the slow transmission function. The key ingredient of the analysis of this expression is a transformation of the associated nonlocal eigenvalue problem into an inhomogeneous hypergeometric differential equation. By this transformation it is possible to determine both the number and the position of all elements in the discrete spectrum of the linear eigenvalue problem. The method is applied to a special case that corresponds to the classical model proposed by Gierer and Meinhardt. It is shown that the one-pulse pattern can gain (or lose) stability through a Hopf bifurcation at a certain value µ Hopf of the main parameter µ. The NLEP approach not only yields a leading order approximation of µ Hopf , but it also shows that there is another bifurcation value, µ edge , at which a new (stable) eigenvalue bifurcates from the edge of the essential spectrum. Finally, it is shown that the N-pulse patterns are always unstable when N ≥ 2. 443

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In this work, we analyse a pair of one-dimensional coupled reaction-diffusion equations known as the Gray-Scott model, in which self-replicating patterns have been observed. We focus on stationary and travelling patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. Single-pulse and multi-pulse stationary waves are shown to exist in the appropriately scaled equations on the infinite line. A (single) pulse is a narrow interval in which the concentration U of one chemical is small, while that of the second, V , is large, and outside of which the concentration U tends (slowly) to the homogeneous steady state U ≡ 1, while V is everywhere close to V ≡ 0. In addition, we establish the existence of a plethora of periodic steady states consisting of periodic arrays of pulses interspersed by intervals in which the concentration V is exponentially small and U varies slowly. These periodic states are spatially inhomogeneous steady patterns whose length scales are determined exclusively by the reactions of the chemicals and their diffusions, and not by other mechanisms such as boundary conditions. A complete bifurcation study of these solutions is presented. We also establish the non-existence of travelling solitary pulses in this system. This non-existence result reflects the system's degeneracy and indicates that some event, for example pulse splitting, 'must' occur when two pulses move apart from each other (as has been observed in simulations): these pulses evolve towards the non-existent travelling solitary pulses. The main mathematical techniques employed in this analysis of the stationary and travelling patterns are geometric singular perturbation theory and adiabatic Melnikov theory. Finally, the theoretical results are compared to those obtained from direct numerical simulation of the coupled partial differential equations on a 'very large' domain, using a moving grid code. It has been checked that the boundaries do not influence the dynamics. A subset of the family of stationary single pulses appears to be stable. This subset determines the boundary of a region in parameter space in which the self-replicating process takes place. In that region, we observe that the core of a time-dependent self-replicating pattern turns out to be precisely a stationary periodic pulse pattern of the type that we construct. Moreover, the simulations reveal some other essential components of the pulse-splitting process and provide an important guide to further analysis.

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationThe dynamics of modulated wave trains A. Doelman, B. Sandstede, A. Scheel, G. Schneider The dynamics of modulated wave trains ABSTRACT We investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg--Landau equation, we establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to Burgers equation over the natural time scale. In addition to the validity of Burgers equation, we show that the viscous shock profiles in Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, we establish the existence and stability of waves that are time-periodic in appropriately moving coordinate frames which separate regions in physical space that are occupied by wave trains of different, but almost identical, wave number. The speed of these shocks is determined by the Rankine--Hugoniot condition where the flux is given by the nonlinear dispersion relation of the wave trains. The group velocities of the wave trains in a frame moving with the interface are directed toward the interface. Using pulse-interaction theory, we also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results represented here are then applied to the FitzHugh--Nagumo equation and to hydrodynamic stability problems. REPORT MAS-E0504 JANUARY 2005 AbstractWe investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg-Landau equation, we establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to Burgers equation over the natural time scale. In addition to the validity of Burgers equation, we show that the viscous shock profiles in Burgers equation for the wave number can be found as genuine modulated waves in the underlying reactiondiffusion system. In other words, we establish the existence and stability of waves that are time-periodic in appropriately moving coordinate frames which separate regions in physical space that are occupied by wave trains of different, but almost identical, wave number. The speed of these shocks is determined by the Rankine-Hugoniot condition where the flux is given by the nonlinear dispersion relation of the wave trains. The group velocities of the wave trains in a frame moving with the interface are directed toward the interface. Using pulse-interaction theory, we also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results represented here are then applied to the FitzHugh-Nagumo equation and to hydrodynamic stability problems.

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In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781-3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

The origin of regular spatial patterns in ecological systems has long fascinated researchers. Turing's activator-inhibitor principle is considered the central paradigm to explain such patterns. According to this principle, local activation combined with longrange inhibition of growth and survival is an essential prerequisite for pattern formation. Here, we show that the physical principle of phase separation, solely based on density-dependent movement by organisms, represents an alternative class of self-organized pattern formation in ecology. Using experiments with self-organizing mussel beds, we derive an empirical relation between the speed of animal movement and local animal density. By incorporating this relation in a partial differential equation, we demonstrate that this model corresponds mathematically to the wellknown Cahn-Hilliard equation for phase separation in physics. Finally, we show that the predicted patterns match those found both in field observations and in our experiments. Our results reveal a principle for ecological self-organization, where phase separation rather than activation and inhibition processes drives spatial pattern formation.he activator-inhibitor principle, originally conceived by Turing in 1952 (1), provides a potential theoretical mechanism for the occurrence of regular patterns in biology (2-6) and chemistry (7-9), although experimental evidence in particular for biological systems has remained scarce (3, 4, 10). In the past decades, this principle has been applied to a wide range of ecological systems, including arid bush lands (11-15), mussel beds (16, 17), and boreal peat lands (18,19). The principle, in which a local positive activating feedback interacts with large-scale inhibitory feedback to generate spatial differentiation in growth, birth, mortality, respiration, or decay, explains the spontaneous emergence of regular spatial patterns in ecosystems even under near-homogeneous starting conditions. Physical theory offers an alternative mechanism for pattern formation, proposed by Cahn and Hilliard in 1958 (20). They identified that density-dependent rates of dispersal can lead to separation of a mixed fluid into two phases that are separated in distinct spatial regions, subsequently leading to pattern formation. The principle of density-dependent dispersal, switching between dispersion and aggregation as local density increases, has become a central mathematical explanation for phase separation in many fields (21) such as multiphase fluid flow (22), mineral exsolution and growth (23), and biological applications (24-28). Although aggregation due to individual motion is a commonly observed phenomenon within ecology, application of the principles of phase separation to explain pattern formation in ecological systems is absent both in terms of theory and experiments (25,26).Here, we apply the concept of phase separation to the formation of spatial patterns in the distribution of aggregating mussels. On intertidal flats, establishing mussel beds exhibit spatial self-org...