Pulse Dynamics in a Three-Component System: Existence Analysis

  • Arjen Doelman
  • Peter van Heijster
  • Tasso J. Kaper
Open Access
Article

Abstract

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781–3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

Keywords

Three-component reaction-diffusion systems One-pulse solutions Travelling pulse solutions Two-pulse solutions Geometric singular perturbation theory Melnikov function 

AMS (MOS) Subject Classifications

Primary: 35K55 35B32 34C37 Secondary: 35K40 

Notes

Acknowledgments

The authors thank Y. Nishiura for introducing us to the three-component model and for stimulating conversations. We thank P. Zegeling for valuable assistance with the software [1] used in the numerical simulations. A.D., P.v.H. and T.K. gratefully acknowledge support from the Netherlands Organization for Scientific Research (NWO). T. J. Kaper gratefully acknowledges support from the National Science Foundation through grant DMS-0606343, and thanks the CWI for its hospitality.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2008

Authors and Affiliations

  • Arjen Doelman
    • 1
    • 2
  • Peter van Heijster
    • 1
  • Tasso J. Kaper
    • 3
  1. 1.Centrum voor Wiskunde en Informatica (CWI)AmsterdamThe Netherlands
  2. 2.Korteweg-de Vries InstituutUniversiteit van AmsterdamAmsterdamThe Netherlands
  3. 3.Department of Mathematics & Center for BioDynamicsBoston UniversityBostonUSA

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