Arjen Doelman

Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach

Abstract In this work, we analyze the linear stability of singular homoclinic stationary solutions and spatially periodic stationary solutions in the one-dimensional Gray-Scott model. This stability analysis has several implications for understanding the recently discovered phenomena of self-replicating pulses. For each solution constructed in A. Doelman et al. [Nonlinearity 10 (1997) 523–563], we analytically find a large open region in the space of the two scaled parameters in which it is stable. Specifically, for each value of the scaled inhibitor feed rate, there exists an interval, whose length and location depend on the solution type, of values of the activator (autocatalyst) decay rate for which the solution is stable. The upper boundary of each interval corresponds to a subcritical Hopf bifurcation point, and the lower boundary is explicitly determined by finding the parameter value where the solution ‘disappears’, i.e., below which it no longer exists as a solution of the steady state system. Explicit asymptotic formulae show that the one-pulse homoclinic solution gains stability first as the second parameter is decreased, and then successively, the spatially periodic solutions (with decreasing period) become stable. Moreover, the stability intervals for different solutions overlap. These stability results are derived via the reduction of a fourth-order slow-fast eigenvalue problem to a second-order nonlocal eigenvalue problem (NLEP). Explicit determination of these stability intervals plays a central role in understanding pulse self-replication. Numerical simulations confirm that the spatially periodic stationary solutions are attractors in the pulse-splitting regime; and, moreover, whenever, for a given solution, the value of the activator decay rate was taken to lie in the regime below that solution s stability interval, initial data close to that solution were observed to evolve toward a different spatially periodic stationary solution, one whose stability interval inclucded the parameter value. The main analytical technique used is that of matched asymptotic expansions.

On the nonlinear dynamics of free bars in straight channels

A simple morphological model is considered which describes the interaction between a unidirectional flow and an erodible bed in a straight channel. For sufficiently large values of the width-depth ratio of the channel the basic state, i.e. a uniform current over a flat bottom, is unstable. At near-critical conditions growing perturbations are confined to a narrow spectrum and the bed profile has an alternate bar structure propagating in the downstream direction. The timescale associated with the amplitude growth is large compared to the characteristic period of the bars. Based on these observations a weakly nonlinear analysis is presented which results in a GinzburgLandau equation. It describes the nonlinear evolution of the envelope amplitude of the group of marginally unstable alternate bars. Asymptotic results of its coefficients are presented as perturbation series in the small drag coefficient of the channel. In contrast to the Landau equation, described by Colombini et al. (1987), this amplitude equation also allows for spatial modulations due to the dispersive properties of the wave packet. It is demonstrated rigorously that the periodic bar pattern can become unstable through this effect, provided the bed is dune covered, and for realistic values of the other physical parameters. Otherwise, it is found that the periodic bar pattern found by Colombini et al. (1987) is stable. Assuming periodic behaviour of the envelope wave in a frame moving with the group velocity, simulations of the dynamics of the Ginzburg-Landau equation using spectral models are carried out, and it is shown that quasi-periodic behaviour of the bar pattern appears.

The dynamics of modulated wave trains.

The authors of this title investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg - Landau equation, they establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to the Burgers equation over the natural time scale. In addition to the validity of the Burgers equation, they show that the viscous shock profiles in the Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, they 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, the authors also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results presented here are applied to the FitzHugh - Nagumo equation and to hydrodynamic stability problems.

Phase separation explains a new class of self- organized spatial patterns in ecological systems

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 long-range 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 well-known 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.

Periodic and quasi-periodic solutions of degenerate modulation equations

Abstract In some circumstances (degenerations) it is essential to add higher-order nonlinear coefficients to a Ginzburg-Landau type modulation equation (which only has one cubic nonlinearity). In this paper we study these degenerate modulation equations. We consider the important situation in which the equation has real coefficients and the case of coefficients with small imaginary parts. First we determine the stability of periodic solutions. The stationary problem is, like in the non-degenerate case, integrable: there exist families of quasi-periodic and homoclinic solutions. This system is perturbed by considering modulation equations with coefficients with small imaginary parts. We establish that there exists an unbounded domain in parameter space in which the modulation equation has quasi-periodic solutions. Moreover, we show that there is a manifold of codimension 1 (in parameter space) on which the homoclinic solutions survive the perturbation.

Regularity of solutions and the convergence of the galerkin method in the ginzburg-landau equation

In this paper an analytical explanation is given for two phenomena observed in numerical simulations of the Ginzburg-Landau equation on the domain [0, 1]D (D = 1, 2, 3) with periodic boundary conditions. First, it is shown that the solutions with initial data become analytic (in the spatial variable). This behavior accounts for the numerically observed exponential decay of the Fourier-modes. Then, based on the regularity result, it is shown that the (linear) Galerkin method has an exponential rate of convergence. This gives an explanation of simulations which show that the Ginzburg-Landau equation can be approximated by very low dimensional Galerkin projections.Furthermore, we discuss the influence of the parameters in the Ginzburg-Landau equation on the decay rate of the Fourier-modes and on the rate of convergence of the Galerkin approximations.

Homoclinic stripe patterns

In this paper, we study homoclinic stripe patterns in the two-dimensional generalized Gierer--Meinhardt equation, where we interpret this equation as a prototypical representative of a class of singularly perturbed monostable reaction-diffusion equations. The structure of a stripe pattern is essentially one-dimensional; therefore, we can use results from the literature to establish the existence of the homoclinic patterns. However, we extend these results to a maximal domain in the parameter space and establish the existence of a bifurcation that forms a new upper bound on this domain. Beyond this bifurcation, the Gierer--Meinhardt equation exhibits self-replicating pulse, respectively, stripe patterns in one, respectively, two dimension(s). The structure of the self-replication process is very similar to that in the Gray--Scott equation. We investigate the stability of the homoclinic stripe patterns by an Evans function analysis of the associated linear eigenvalue problem. We extend the recently develope...

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component.To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances.In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.

Slowly Modulated Two-Pulse Solutions in the Gray--Scott Model II: Geometric Theory, Bifurcations, and Splitting Dynamics

With great sadness, we note the passing away of our mentor and colleague Wiktor Abstract. In this second paper, we develop a geometrical method to systematically study the singular perturbed problem associated to slowly modulated two-pulse solutions. It enables one to see that the characteristics of these solutions are strongly determined by the flow on a slow manifold and, hence, also to identify the saddle-node bifurcations and bifurcations to classical traveling waves in which the solutions constructed in part I are created and annihilated. Moreover, we determine the geometric origin of the critical maximum wave speeds discovered in part I. In this paper, we also study the central role of the slowly varying inhibitor component of the two-pulse solutions in the pulse-splitting bifurcations. Finally, the validity of the quasi-stationary approximation is established here, and we relate the results of both parts of this work to the literature on self-replication.

Nonlinear Asymptotic Stability of the Semistrong Pulse Dynamics in a Regularized Gierer-Meinhardt Model

We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semistrong regime of two‐pulse interactions in a regularized Gierer–Meinhardt system. In the semistrong limit the localized activator pulses interact strongly through the slowly varying inhibitor. The interaction is not tail‐tail as in the weak interaction limit, and the pulse amplitudes and speeds change as the pulse separation evolves on algebraically slow time scales. In addition the point spectrum of the associated linearized operator evolves with the pulse dynamics. The RG approach employed here validates the interaction laws of quasi‐steady two‐pulse patterns obtained formally in the literature, and establishes that the pulse dynamics reduce to a closed system of ordinary differential equations for the activator pulse locations. Moreover, we fully justify the reduction to the nonlocal eigenvalue problem (NLEP) showing that the large difference between the quasi‐steady NLEP operator and the operator arisi...