Dynamics of self-replicating patterns in the one-dimensional Gray-Scott model

Ueyama, Daishin

doi:10.14492/hokmj/1351001084

Cited by 23 publications

(36 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This leads to the eigenvalue problem 34) it follows from (2.33) and D ′ (u c0 ) = 0 that (2.34) is an eigenfunction pair corresponding to λ = 0. We now construct an asymptotic approximation to this eigenpair φ, ψ of (2.33) corresponding to λ = 0.…”

Section: The Dimple Eigenfunctionmentioning

confidence: 99%

“…In recent years, many theoretical and numerical studies have been made in both one and two spatial dimensions to analyze self-replication behavior for the Gray-Scott model in different parameter regimes (cf. [33], [32], [26], [27], [34], [20], [2], [1], [16]). In addition to the GrayScott model, many other reaction-diffusion systems have been found to exhibit self-replication behavior.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Self-replication of mesa patterns in reaction–diffusion systems

Kolokolnikov

Ward

Wei

2007

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

Certain two-component reaction-diffusion systems on a finite interval are known to possess mesa (box-like) steadystate patterns in the singularly perturbed limit of small diffusivity for one of the two solution components. As the diffusivity D of the second component is decreased below some critical value Dc, with Dc = O(1), the existence of a steady-state mesa pattern is lost, triggering the onset of a mesa self-replication event that ultimately leads to the creation of additional mesas. The initiation of this phenomena is studied in detail for a particular scaling limit of the Brusselator model. Near the existence threshold Dc of a single steady-state mesa, it is shown that an internal layer forms in the center of the mesa. The structure of the solution within this internal layer is shown to be governed by a certain core problem, comprised of a single non-autonomous second-order ODE. By analyzing this core problem using rigorous and formal asymptotic methods, and by using the Singular Limit Eigenvalue Problem (SLEP) method to asymptotically calculate small eigenvalues, an analytical verification of the conditions of Nishiura and Ueyema [Physica D, 130, No. 1, (1999), pp. 73-104], believed to be responsible for self-replication, is given. These conditions include: (1) The existence of a saddle-node threshold at which the steady-state mesa pattern disappears; (2) the dimple-shaped eigenfunction at the threshold, believed to be responsible for the initiation of the replication process; and (3) the stability of the mesa pattern above the existence threshold. Finally, we show that the core problem is universal in the sense that it pertains to a class of reaction-diffusion systems, including the Gierer-Meinhardt model with saturation, where mesa self-replication also occurs.

show abstract

Section: The Dimple Eigenfunctionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Self-replication of mesa patterns in reaction–diffusion systems

Kolokolnikov

Ward

Wei

2007

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…For certain singularly perturbed two-component reaction-diffusion models in one space dimension, such as the Gray-Scott and Gierer-Meinhardt models, there has been considerable analytical progress in understanding both the dynamics and the various types of instabilities of spike patterns, including self-replicating instabilities (see [37], [38], [43], [15], [12], [25], [41] and many of the references therein). In contrast, in a two-dimensional spatial domain there are only a few analytical results characterizing spot dynamics, such as [9], [24], and [46], for a one-spot solution of the Gierer-Meinhardt model, and the studies of [16], [17], and [18], for exponentially weakly interacting spots in various contexts.…”

Section: Introductionmentioning

confidence: 99%

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

2008

View full text Add to dashboard Cite

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction-diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε → 0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength Sj for j = 1, . . . , K at unknown spot locations xj ∈ Ω for j = 1, . . . , K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/ log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates Sj and xj for j = 1, . . . , K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green's function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths Sj , for j = 1, . . . , K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.

show abstract

“…These numerical and experimental studies have stimulated much theoretical work to classify steady-state and time-dependent spike behavior in the simpler case of one spatial dimension, including: spike-replication and dynamics in the weak-interaction regime (cf. [39], [41], [36], [44]); spatio-temporal chaos in the weak-interaction regime (cf. [37]); the existence and stability of equilibrium solutions in the semi-strong interaction regime (cf.…”

Section: Introductionmentioning

confidence: 99%

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

Wei

2006

Interfaces Free Bound.

View full text Add to dashboard Cite

Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric kspike pattern, simultaneously cross through zero at precisely the same parameter value at which k −1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB . . . BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε → 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.

show abstract

Dynamics of self-replicating patterns in the one-dimensional Gray-Scott model

Cited by 23 publications

References 0 publications

Self-replication of mesa patterns in reaction–diffusion systems

Self-replication of mesa patterns in reaction–diffusion systems

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Contact Info

Product

Resources

About