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Pulse-splitting for some reaction-diffusion systems in one-space dimension. (English) Zbl 1145.35401

Summary: Pulse-splitting, or self-replication, behavior is studied for some two-component singularly perturbed reaction-diffusion systems on a one-dimensional spatial domain. For the Gierer-Meinhardt model in the weak interaction regime, characterized by asymptotically small activator and inhibitor diffusivities, a numerical approach is used to verify 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)] that are believed to be essential for the occurrence of pulse-splitting in a reaction-diffusion system. The pulse-splitting that is observed here is edge-splitting, where only the spikes that are closest to the boundary are able to replicate. For the Gray-Scott model, it is shown numerically that there are two types of pulse-splitting behavior depending on the parameter regime: edge-splitting in the weak interaction regime, and a simultaneous splitting in the semi-strong interaction regime. For the semi-strong spike interaction regime, where only one of the solution components is localized, we construct several model reaction-diffusion systems where all of the pulse-splitting conditions of Ei et al. can be verified analytically, yet no pulse-splitting is observed. These examples suggest that an extra condition, referred to here as the multi-bump transition condition, is also required for pulse-splitting behavior. This condition is in fact satisfied by the Gierer-Meinhardt and Gray-Scott systems in their pulse-splitting parameter regimes.

MSC:

35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
92D25 Population dynamics (general)

Citations:

Zbl 0983.35061

Software:

NAG; d03pcf; LAPACK; nag
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