Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model

Gierer-Meinhardt system as a molecularly plausible model has been proposed to formalize the observation for pattern formation. In this paper, the Gierer-Meinhardt model without the saturating term is considered. By the linear stability analysis, we not only give out the conditions ensuring the stability and Turing instability of the positive equilibrium but also find the parameter values where possible Turing-Hopf and spatial resonance bifurcation can occur. Then we develop the general algorithm for the calculations of normal form associated with codimension-2 spatial resonance bifurcation to better understand the dynamics neighboring of the bifurcating point. The spatial resonance bifurcation reveals the interaction of two steady state solutions with different modes. Numerical simulations are employed to illustrate the theoretical results for both the Turing-Hopf bifurcation and spatial resonance bifurcation. Some expected solutions including stable spatially inhomogeneous periodic solutions and coexisting stable spatially steady state solutions evolve from Turing-Hopf bifurcation and spatial resonance bifurcation respectively.