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The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime. (English) Zbl 1136.35003

Summary: The existence, stability, and pulse-splitting behavior of spike patterns in the one-dimensional Gray-Scott model on a finite domain is analyzed in the semi-strong spike-interaction regime. This regime is characterized by a localization of one of the components of the reaction near certain spike locations, while the other component exhibits a more global spatial variation across the domain. The method of matched asymptotic expansions is then used to construct \(k\)-spike equilibria in terms of a certain core problem. This core problem is studied numerically and asymptotically. For each integer \(k\geq 1\), it is shown that there are two branches of \(k\)-spike equilibria that meet at a saddle-node bifurcation value. For small values of the diffusivity \(D\) of the second component, these saddle-node bifurcation points occur at approximately the same value. A combination of asymptotic and numerical methods is used to analyze the stability of these branches of \(k\)-spike equilibria with respect to both drift instabilities associated with the small eigenvalues and oscillatory instabilities of the spike profile. In this way, the key bifurcation and spectral conditions of S. Ei, Y. Nishiura and K. Ueda, Japan J. Ind. Appl. Math. 18, No. 2, 181–205 (2001; Zbl 0983.35061)] believed to be essential for pulse-splitting behavior in a reaction-diffusion system are verified. By having verified these conditions, a simple analytical criterion for the occurrence of pulse-splitting is then formulated and confirmed with full numerical simulations of the Gray-Scott model. This criterion verifies a conjecture based on numerics and topological arguments reported in [A. Doelman, R. A. Gardner and T. J. Kaper, Physica D 122, No. 1–4, 1–36 (1998; Zbl 0943.34039)]. The analytical results are compared with previously obtained results for pulse-splitting behavior.

MSC:

35B25 Singular perturbations in context of PDEs
35B35 Stability in context of PDEs
35B32 Bifurcations in context of PDEs

Software:

COLSYS; LAPACK; NAG; d03pcf; nag
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References:

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