Adiabatic stability under semi-strong interactions: The weakly damped regime

We rigorously derive multi-pulse interaction laws for the semi-strong interactions in a family of singularly-perturbed and weakly-damped reaction-diffusion systems in one space dimension. Most significantly, we show the existence of a manifold of quasi-steady N-pulse solutions and identify a "normal-hyperbolicity" condition which balances the asymptotic weakness of the linear damping against the algebraic evolution rate of the multi-pulses. Our main result is the adiabatic stability of the manifolds subject to this normal hyperbolicity condition. More specifically, the spectrum of the linearization about a fixed N-pulse configuration contains essential spectrum that is asymptotically close to the origin as well as semi-strong eigenvalues which move at leading order as the pulse positions evolve. We characterize the semi-strong eigenvalues in terms of the spectrum of an explicit N by N matrix, and rigorously bound the error between the N-pulse manifold and the evolution of the full system, in a polynomially weighted space, so long as the semi-strong spectrum remains strictly in the left-half complex plane, and the essential spectrum is not too close to the origin

Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer-Meinhardt model

We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semi-strong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semi-strong limit the strongly localized activator pulses interact through the weakly localized inhibitor, the interaction is not tail-tail as in the weak interaction limit, and the pulses change amplitude and even stability as their separation distance evolves on algebraically slow time scales. The RG approach employed here validates the interaction laws of quasi-steady 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 the large difference between the quasi-steady NLEP operator and the operator arising from linearization about the pulse is controlled by the resolven

Slow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems

In this paper, we study a class of singularly perturbed reaction-diffusion systems, which exhibit under certain conditions slowly varying multi-pulse solutions. This class contains among others the Gray-Scott and several versions of the Gierer-Meinhardt model. We first use a classical singular perturbation approach for the stationary problem and determine in this way a manifold of quasi-stationary $N$-pulse solutions. Then, in the context of the time-dependent problem, we derive an equation for the leading order approximation of the slow motion along this manifold. We apply this technique to study 1-pulse and 2-pulse solutions for classical and modified Gierer-Meinhardt system. In particular, we are able to treat different types of boundary conditions, calculate folds of the slow manifold, leading to slow-fast motion, and to identify symmetry breaking singularities in the manifold of 2-pulse solutions

Slow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems

In this paper, we study a class of singularly perturbed reaction-diffusion systems, which exhibit under certain conditions slowly varying multi-pulse solutions. This class contains among others the Gray-Scott and several versions of the Gierer-Meinhardt model. We first use a classical singular perturbation approach for the stationary problem and determine in this way a manifold of quasi-stationary $N$-pulse solutions. Then, in the context of the time-dependent problem, we derive an equation for the leading order approximation of the slow motion along this manifold. We apply this technique to study 1-pulse and 2-pulse solutions for classical and modified Gierer-Meinhardt system. In particular, we are able to treat different types of boundary conditions, calculate folds of the slow manifold, leading to slow-fast motion, and to identify symmetry breaking singularities in the manifold of 2-pulse solutions

Stationary Multiple Spots for Reaction-Diffusion Systems

In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reaction-diffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activator-substrate systems (such as the Gray-Scott system or the Schnakenberg model). The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature. We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator. Existence of multi-spots is proved using tools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis. Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots. Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis. The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations

Oscillatory translational instabilities of localized spot patterns in the Schnakenberg reaction-diffusion system on general 2-D domains

For a bounded 2-D planar domain $\Omega$, we investigate the impact of domain geometry on oscillatory translational instabilities of $N$-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with \mO(\eps^2) \ll \mO(1) activator-inhibitor diffusivity ratio. An $N$-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in $\Omega$ except in $N$ well-separated localized regions of \mO(\eps) extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in \mO(\eps^2)-frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a $2N$-vector that characterizes the possible modes (directions) of oscillation. The $2N\times 2N$ matrix contains terms associated with a certain Green's function on $\Omega$, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates $r = 1 + \sigma f(\theta)$ with \red{$0< \varepsilon \ll \sigma \ll 1$}, $\theta \in [0,2\pi)$, and $f(\theta)$ $2\pi$-periodic, we show that only the mode-$2$ coefficients of the Fourier series of $f$ impact the bifurcation threshold at leading order in $\sigma$. We further show that when $f(\theta) = \cos2\theta$, the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains $\Omega$ and $N$-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes

Turing Instabilities are Not Enough to Ensure Pattern Formation

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing’s reaction–diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation