Turing-Turing bifurcation and multi-stable patterns in a Gierer-Meinhardt system

The classical Gierer-Meinhardt system portrays the formation process of a self-organizing pattern of vascular mesenchymal cells. In this paper, the coexistence of multi-stable patterns with different spatial responses and the superposition for patterns have been explored in theory from the perspective of Turing-Turing bifurcation. On the one-dimensional region, the system is simplified near the Turing-Turing singularity to obtain a third-order ordinary differential equation employing center manifold and normal form theory, which is locally topologically equivalent to the primitive system and its coefficients can be represented by the parameters of original equation. Especially, considering the simplified system, it is theoretically revealed that the system supports semi-stable patterns superimposed by two different spatial resonances and the coexistence of four stable steady states with different single characteristic wavelengths, indicating that different initial conditions may tent to completely different spatial patterns. Finally, some numerical simulations are given, which are consistent with the theoretical analysis. The multi-stable and superimposed modes of the system are also studied on a two-dimensional region, which shows that some experimental patterns of vascular mesenchymal cells in vitro can be interpreted as the superposition of different spatial modal patterns to a certain extent.