Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

Moyles, Iain; Ward, Michael J.

doi:10.1137/16m1060327

Cited by 10 publications

(8 citation statements)

References 41 publications

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“…As the saturation parameter increases towards a critical value associated with a homoclinic bifurcation point, it has been shown that the principal eigenvalue of the local part of the linearized operator decreases to zero. This mechanism has been shown for straight stripes in [14], and more generally in [19], to eliminate the band of unstable breakup modes associated with the underlying NLEP, leading to a stabilization of the homoclinic stripe. Since the effect of increasing the time-delay in the activator kinetics also decreases the principal eigenvalue of the local part of the linearized operator to zero (see the left panel of Fig.…”

Section: 2mentioning

confidence: 84%

“…For the GM model, such homoclinic stripe solutions, formed from the localization of a spike on a one-dimensional curve in a 2-D domain, are known to be unconditionally unstable to breakup into localized spots unless one includes a strong saturation mechanism for the autocatalysis term (cf. [14], [19]). As the saturation parameter increases towards a critical value associated with a homoclinic bifurcation point, it has been shown that the principal eigenvalue of the local part of the linearized operator decreases to zero.…”

Section: 2mentioning

confidence: 99%

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A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model

Fadai¹,

Ward²,

Wei³

2018

Discrete &Amp; Continuous Dynamical Systems - B

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We study the spectrum of a new class of nonlocal eigenvalue problems (NLEPs) that characterize the linear stability properties of localized spike solutions to the singularly perturbed two-component Gierer-Meinhardt (GM) reaction-diffusion (RD) system with a fixed time-delay T in only the nonlinear autocatalytic activator kinetics. Our analysis of this model is motivated by the computational study of Seirin Lee et al. [Bull. Math. Bio., 72(8), (2010)] on the effect of gene expression time delays on spatial patterning for both the GM and some related RD models. For various limiting forms of the GM model, we show from a numerical study of the associated NLEP, together with an analytical scaling law analysis valid for large delay T , that a time-delay in only the activator kinetics is stabilizing in the sense that there is a wider region of parameter space where the spike solution is linearly stable than when there is no time delay. This enhanced stability behavior with a delayed activator kinetics is in marked contrast to the de-stabilizing effect on spike solutions of having a time-delay in both the activator and inhibitor kinetics. Numerical results computed from the RD system with delayed activator kinetics are used to validate the theory for the 1-D case. Introduction. For activator-inhibitor two-component reaction-diffusion (RD)systems, it is a well-known result, originating from Turing [21], that a small perturbation of a spatially uniform steady-state solution can become unstable when the diffusivity ratio is large enough. This initial instability then leads to the generation of large-amplitude stable spatial patterns. Although this mechanism for the development of spatially inhomogeneous patterns is well-understood, and has been applied to a broad range of specific RD systems (cf. [7]) and modeling scenarios on various spatial scales, what is less well-understood is the effect on pattern development of any time-delays in the reaction kinetics. Although there are now general results for the linear stability of spatially uniform steady-states under the effect of a

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Section: 2mentioning

confidence: 84%

Section: 2mentioning

confidence: 99%

A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model

Fadai¹,

Ward²,

Wei³

2018

Discrete &Amp; Continuous Dynamical Systems - B

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“…We further remark that 1-D NLEP analysis has been used to study the transverse stability of a 1-D homoclinic stripe (cf. [28], [41], [60]). The breakup of the stripe, as characterized by unstable eigenvalues of the NLEP for a particular transverse wavenumber, is a key mechanism through which localized spots in 2-D are created.…”

Section: Introductionmentioning

confidence: 99%

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

Ward

2018

Nonlinearity

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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.

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“…Unlike in the extended Klausmeier model [60,64], localized stripes are of 2-front type and may thus be expected to possibly be stable -see [1] for a rigorous treatment in a generalized Klausmeier type model (posed on a sloped terrain without a diffusion term for the water component -like the original Klausmeier model [42]). Naturally, the interfaces will evolve and their curvature driven dynamics may be studied analytically along the lines of [51]. Especially in the above discussed multi-front transition region between bare soil and homogeneous vegetation, the ecosystem dynamics generated by the model may be very rich and complex -see for instance [35,36,37].…”

Section: Discussionmentioning

confidence: 99%

The existence of localized vegetation patterns in a systematically reduced model for dryland vegetation

Jaibi,

Doelman,

Chirilus-Bruckner

et al. 2020

Preprint

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In this paper we consider the 2-component reaction-diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics [28]. The nonlinear structure of this model is more involved than other more conceptual models, such as the extended Klausmeier model, and the analysis a priori is more complicated. However, the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations. Moreover, we find that the model exhibits a richness of analytically tractable patterns that exceeds that of Klausmeier-type models. Our study focuses on the 4-dimensional dynamical system associated with the reaction-diffusion model by considering traveling waves in 1 spatial dimension. We use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic/homoclinic/periodic orbits that 'jump' between (normally hyperbolic) slow manifolds, representing various kinds of localized vegetation patterns. The basic 1-front invasion patterns and 2-front spot/gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model. By exploiting the rich nonlinear structure of the model, we construct many multi-front patterns that are novel, both from the ecological and the mathematical point of view. In fact, we argue that these orbits/patterns are not specific for the model considered here, but will also occur in a much more general (singularly perturbed reaction-diffusion) setting. We conclude with a discussion of the ecological and mathematical implications of our findings.

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Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

Cited by 10 publications

References 41 publications

A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model

A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model

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

The existence of localized vegetation patterns in a systematically reduced model for dryland vegetation

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