Bifurcation analysis of the Gierer–Meinhardt system with a saturation in the activator production

Abstract

The reaction–diffusion Gierer–Meinhardt system with a saturation in the activator production is considered. Stability of the unique positive constant steady state solution is analysed, and associated Hopf bifurcations and steady state bifurcations are obtained. A global bifurcation diagram of non-trivial periodic orbits and steady state solutions with respect to key system parameters is obtained, which improves the understanding of dynamics of Gierer–Meinhardt system with a saturation in different parameter regimes.

Keywords:

• Gierer–Meinhardt system with saturation
• Hopf bifurcation
• Steady state bifurcation
• Heterogeneous case

AMS Subject Classifications:

• 35K57; 92E20

Acknowledgments

This research is partially supported by the National Natural Science Foundation of China No. 11031002, Research Fund for the Doctoral Program of Higher Education of China No. 20122302110044, Natural Scientific Research Innovation Foundation in Harbin Institute of Technology(HIT.NSRIF.2014124), and NSF grant DMS-1022648 and Shanxi 100 talent program (Shi).