Stability of pulse solutions for the discrete FitzHugh–Nagumo system

Hupkes, Hermen Jan; Sandstede, Björn

doi:10.1090/s0002-9947-2012-05567-x

Cited by 36 publications

(60 citation statements)

References 41 publications

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“…However, using formal asymptotic computations this possibility is excluded: all described patterns to (1.2) -stripes, gaps and fronts -are thus (always) stable against two-dimensional perturbations. These formal arguments are also verified rigorously by carefully constructing eigenfunctions using techniques previously employed to prove stability of traveling pulses in the FitzHugh-Nagumo system in [6]; similar arguments were also used in [30,31]. However, in those previous works, only stability with respect to perturbations in one spatial dimension was considered.…”

Section: Introductionmentioning

confidence: 78%

Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems

2019

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In water-limited regions, competition for water resources results in the formation of vegetation patterns; on sloped terrain, one finds that the vegetation typically aligns in stripes or arcs. We consider a twocomponent reaction-diffusion-advection model of Klausmeier type describing the interplay of vegetation and water resources and the resulting dynamics of these patterns. We focus on the large advection limit on constantly sloped terrain, in which the diffusion of water is neglected in favor of advection of water downslope. Planar vegetation pattern solutions are shown to satisfy an associated singularly perturbed traveling wave equation, and we construct a variety of traveling stripe and front solutions using methods of geometric singular perturbation theory. In contrast to prior studies of similar models, we show that the resulting patterns are spectrally stable to perturbations in two spatial dimensions using exponential dichotomies and Lin's method. We also discuss implications for the appearance of curved stripe patterns on slopes in the absence of terrain curvature.

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Section: Introductionmentioning

confidence: 78%

Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems

2019

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“…Indeed, when considering LDEs posed on one-dimensional lattices, the Fredholm properties of similar operators have been used to study the stability of waves [24], glue waves together [26], and analyze singular perturbations [23]. In addition, a recent result [20] provides a set of spectral conditions on Λ c,0 that are sufficient Downloaded 06/30/14 to 129.237.46.100.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

“…Bates, Chen, and Chmaj [2] used implicit function theorem arguments to obtain the existence of traveling waves for LDEs with longrange interactions that can be both attracting and repelling. In [23] Hupkes and Sandstede developed a version of singular perturbation theory to construct traveling waves for the two-component discrete FitzHugh-Nagumo system. In [22] modulated traveling waves were constructed using a global center manifold analysis for (1.1) with γ > 0.…”

Section: Lattice Differential Equationsmentioning

confidence: 99%

Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems

Hupkes¹,

Vleck²

2013

SIAM J. Math. Anal.

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Abstract. We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have positive spatially periodic coefficients, and the two spatially periodic equilibria are required to be well ordered. We establish the existence of traveling wave solutions to such pure lattice systems that connect the two stable equilibria. In addition, we show that these waves can be approximated by traveling wave solutions to systems that incorporate both local and nonlocal diffusion. In certain special situations our results can also be applied to reaction diffusion systems that include (potentially large) negative coefficients. Indeed, upon splitting the lattice suitably and applying separate coordinate transformations to each sublattice, such systems can sometimes be transformed into a periodic diffusion problem that fits within our framework. In such cases, the resulting traveling structure for the original system has a separate wave profile for each sublattice and connects spatially periodic patterns that need not be well ordered. There is no direct analogue of this procedure that can be applied to reaction diffusion systems with continuous spatial variables.

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“…(5.4) Then the strategy would be to obtain spectral properties forv(t) = DF(u * (t))v(t) from those of the operator L in order to be able to close a nonlinear stability argument. This method was introduced and successfully implemented by Benzoni-Gavage and coauthors in [6] by analyzing associated Green's functions for the nonlinear stability analysis of semidiscrete shock waves and more recently reused in the context of nonlinear stability analysis of traveling pulses in the discrete FitzHugh-Nagumo equations with finite and infinite range interactions [20,25]. We conjecture the following result with perturbations measured in the Banach spaces p (R), which are defined by…”

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confidence: 88%

Traveling fronts for lattice neural field equations

Faye

2018

Physica D: Nonlinear Phenomena

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We show existence and uniqueness of traveling front solutions to a class of neural field equations set on a lattice with infinite range interactions in the regime where the kinetics of each individual neuron is of bistable type. The existence proof relies on a regularization of the traveling wave problem allowing us to use well-known existence results for traveling front solutions of continuous neural field equations. We then show that the traveling front solutions which have nonzero wave speed are unique (up to translation) by constructing appropriate sub and super solutions. The spectral properties of the traveling fronts are also investigated via a careful study of the linear operator around a traveling front in co-moving frame where we crucially use Fredholm properties of nonlocal differential operators previously obtained by the author in an earlier work. For the spectral analysis, we need to impose an extra exponential localization condition on the interactions.

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Stability of pulse solutions for the discrete FitzHugh–Nagumo system

Cited by 36 publications

References 41 publications

Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems

Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems

Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems

Traveling fronts for lattice neural field equations

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