Bifurcation of Solutions to Reaction-Diffusion Systems with Jumping Nonlinearities

Eisner, Jan; Kučera, Milan

doi:10.1007/0-306-47096-9_7

Cited by 5 publications

(8 citation statements)

References 13 publications

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“…There are also earlier theoretical studies, e.g. [10,13,14,15,29] and references therein, predicting new and interesting features of systems with various unilateral conditions or terms. Let us note that the original necessary condition of one fast and one slow diffusion can be removed or relaxed also by other approaches.…”

Section: Introductionmentioning

confidence: 90%

“…We assume that (6) withĝ(v) = τ n ε (v) has a unique solutionv ε for any ε > 0. Thisv ε together withū ε given by (5) forms the ground state of (14).…”

Section: Significance Of the Nonsmooth Unilateral Termmentioning

confidence: 99%

“…Lemma 2. Let (ū ε ,v ε ) be the ground state of system (14). For any ε > 0, let the function n ε be smooth in a neighbourhood ofv ε and let n ε (v) denote the derivative of n ε (v).…”

Section: Significance Of the Nonsmooth Unilateral Termmentioning

confidence: 99%

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Unilateral regulation breaks regularity of Turing patterns

et al. 2017

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We consider a reaction-diffusion system undergoing Turing instability and augment it by an additional unilateral source term. We investigate its influence on the Turing instability and on the character of resulting patterns. The nonsmooth positively homogeneous unilateral term τv^{-} has favorable properties, but the standard linear stability analysis cannot be performed. We illustrate the importance of the nonsmoothness by a numerical case study, which shows that the Turing instability can considerably change if we replace this term by its arbitrarily precise smooth approximation. However, the nonsmooth unilateral term and all its approximations yield qualitatively similar patterns although not necessarily developing from small disturbances of the spatially homogeneous steady state. Further, we show that the unilateral source breaks the approximate symmetry and regularity of the classical patterns and yields asymmetric and irregular patterns. Moreover, a given system with a unilateral source produces spatial patterns even for diffusion parameters with ratios closer to 1 than the same system without any unilateral term.

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Section: Introductionmentioning

confidence: 90%

“…We assume that (6) withĝ(v) = τ n ε (v) has a unique solutionv ε for any ε > 0. Thisv ε together withū ε given by (5) forms the ground state of (14).…”

Section: Significance Of the Nonsmooth Unilateral Termmentioning

confidence: 99%

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Unilateral regulation breaks regularity of Turing patterns

et al. 2017

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“…Indeed, study of bifurcating solutions of RD systems have been the focus of various studies in recent years (see, for example Scheel, 1998;Eisner and Kucera, 2000). Here, to be more specific, suppose that solution U o (r, t) of (5 )-(7 ) remains bounded for all t .…”

Section: Analysis Of Weak Interactionsmentioning

confidence: 99%

Dynamics of Notch Activity in a Model of Interacting Signaling Pathways

Bani-Yaghoub

Amundsen

2010

Bull. Math. Biol.

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Networks of interacting signaling pathways are formulated with systems of reaction-diffusion (RD) equations. We show that weak interactions between signaling pathways have negligible effects on formation of spatial patterns of signaling molecules. In particular, a weak interaction between Retinoic Acid (RA) and Notch signaling pathways does not change dynamics of Notch activity in the spatial domain. Conversely, large interactions of signaling pathways can influence effects of each signaling pathway. When the RD system is largely perturbed by RA-Notch interactions, new spatial patterns of Notch activity are obtained. Moreover, analysis of the perturbed Homogeneous System (HS) indicates that the system admits bifurcating periodic orbits near a Hopf bifurcation point. Starting from a neighborhood of the Hopf bifurcation, oscillatory standing waves of Notch activity are numerically observed. This is of particular interest since recent laboratory experiments confirm oscillatory dynamics of Notch activity.

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“…In [5], the corresponding result for reaction-diffusion systems with jumping nonlinearities is given. This paper belongs to a series of results concerning bifurcation of spatial patterns in reaction-diffusion systems with unilateral boundary conditionssee e.g.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations for a problem with jumping nonlinearities

Kárná¹,

Kučera²

2002

Math. Bohem.

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A bifurcation problem for the equation ∆u + λu − αu + + βu − + g(λ, u) = 0 in a bounded domain in Ê N with mixed boundary conditions, given nonnegative functions α, β ∈ L∞ and a small perturbation g is considered. The existence of a global bifurcation between two given simple eigenvalues λ (1) , λ (2) of the Laplacian is proved under some assumptions about the supports of the functions α, β. These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to λ (1) , λ (2) .

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Bifurcation of Solutions to Reaction-Diffusion Systems with Jumping Nonlinearities

Cited by 5 publications

References 13 publications

Unilateral regulation breaks regularity of Turing patterns

Unilateral regulation breaks regularity of Turing patterns

Dynamics of Notch Activity in a Model of Interacting Signaling Pathways

Bifurcations for a problem with jumping nonlinearities

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