Moving and jumping spot in a two-dimensional reaction–diffusion model

2018

Nonlinearity

Localized spatial patterns commonly occur for various classes of linear and nonlinear diffusive processes. In particular, localized spot patterns, where the solution concentrates at discrete points in the domain, occur in the nonlinear reactiondiffusion (RD) modeling of diverse phenomena such as chemical patterns, biological morphogenesis, and the spatial distribution of urban crime. In a 2-D spatial domain we survey some recent and new results for the existence, linear stability, and slow dynamics of localized spot patterns by using the Brusselator RD model as the prototypical example. In the context of linear diffusive systems with localized solution behavior, we will discuss some previous results for the determination of the mean first capture time for a Brownian particle in a 2-D domain with localized traps, and the determination of the persistence threshold of a species in a 2-D landscape with patchy food resources. Common features in the analysis of all of these spatially localized patterns are emphasized, including the key role of certain matrices involving various Green's functions, and the derivation and study of new classes of interacting particle systems and discrete variational problems arising from various asymptotic reductions. The mathematical tools include matched asymptotic analysis based on strong localized perturbation theory, spectral analysis, the analysis of nonlocal eigenvalue problems, and bifurcation theory. Some specific open problems are highlighted and, more broadly, we will discuss a few new research frontiers for the analysis of localized patterns in multi-dimensional domains.

Section: Introductionmentioning

confidence: 99%

Spots, traps, and patches: asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems

2018

Nonlinearity

“…5 and 6 for surveys). In a spatially homogeneous 2‐D medium, and for various specific RD systems, the slow dynamical behavior of quasi‐equilibrium spot patterns, together with their various types of bifurcations that trigger a range of different instabilities of the pattern such as spot‐annihilation, spot‐replication, and temporal oscillations of the spot amplitude, has been well studied 7–15 . The primary focus of this article is to investigate, for one prototypical RD system, how certain spatial heterogeneities in the model affect the dynamics and instabilities of quasi‐equilibrium spot patterns, and lead to new dynamical phenomena that would otherwise not occur in a medium free of defects.…”

Section: Introductionmentioning

confidence: 99%

Spot patterns in the 2‐D Schnakenberg model with localized heterogeneities

Wong

2021

Stud Appl Math

A hybrid asymptotic‐numerical theory is developed to analyze the effect of different types of localized heterogeneities on the existence, linear stability, and slow dynamics of localized spot patterns for the two‐component Schnakenberg reaction‐diffusion model in a 2‐D domain. Two distinct types of localized heterogeneities are considered: a strong localized perturbation of a spatially uniform feed rate and the effect of removing a small hole in the domain, through which the chemical species can leak out. Our hybrid theory reveals a wide range of novel phenomena such as saddle‐node bifurcations for quasi‐equilibrium spot patterns that otherwise would not occur for a homogeneous medium, a new type of spot solution pinned at the concentration point of the feed rate, spot self‐replication behavior leading to the creation of more than two new spots, and the existence of a creation‐annihilation attractor with at most three spots. Depending on the type of localized heterogeneity introduced, localized spots are either repelled or attracted toward the localized defect on asymptotically long time scales. Results for slow spot dynamics and detailed predictions of various instabilities of quasi‐equilibrium spot patterns, all based on our hybrid asymptotic‐numerical theory, are illustrated and confirmed through extensive full PDE numerical simulations.

“…These localized patterns have been shown to exhibit a wide variety of phenomena such as slow spot dynamics, spotpinning behavior, spot self-replication, and two types of O(1) time-scale spot amplitude instabilities that occur in certain parameter regimes (cf. [3], [7], [10], [11], [13], [17], [18], [20], [22], [24], [25], [26], [27], [32], [33], [34], [37]). Localized spot patterns have also been observed in diverse experimental settings [16], [8], and [2].…”

Section: Introductionmentioning

confidence: 99%

Refined stability thresholds for localized spot patterns for the Brusselator model in

Chang

Tzou²,

et al. 2018

Eur. J. Appl. Math

In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in $\mathbb{R}^2$, for the Brusselator reaction–diffusion model $$ \begin{equation*} v_t = \epsilon^2 \Delta v + \epsilon^2 - v + fu v^2 \,, \qquad \tau u_t = D \Delta u + \frac{1}{\epsilon^2}\left(v - u v^2\right) \,, \end{equation*} $$ where the parameters satisfy 0 < f < 1, τ > 0 and D > 0. A previous leading-order linear stability theory characterizing the onset of spot amplitude instabilities for the parameter regime D = ${\mathcal O}$(ν−1), where ν = −1/log ϵ, based on a rigorous analysis of a non-local eigenvalue problem (NLEP), predicts that zero-eigenvalue crossings are degenerate. To unfold this degeneracy, the conventional leading-order-in-ν NLEP linear stability theory for spot amplitude instabilities is extended to one higher order in the logarithmic gauge ν. For a multi-spot pattern on a finite domain under a certain symmetry condition on the spot configuration, or for a periodic pattern of spots centred at the lattice points of a Bravais lattice in $\mathbb{R}^2$, our extended NLEP theory provides explicit and improved analytical predictions for the critical value of the inhibitor diffusivity D at which a competition instability, due to a zero-eigenvalue crossing, will occur. Finally, when D is below the competition stability threshold, a different extension of conventional NLEP theory is used to determine an explicit scaling law, with anomalous dependence on ϵ, for the Hopf bifurcation threshold value of τ that characterizes temporal oscillations in the spot amplitudes.