Bifurcations and Turing patterns in a diffusive Gierer-Meinhardt model

In this paper, the Hopf bifurcations and Turing bifurcations of the Gierer- Meinhardt activator-inhibitor model are studied. The very interesting and complex spatially periodic solutions and patterns induced by bifurcations are analyzed from both theoretical and numerical aspects respectively. Firstly, the conditions for the existence of Hopf bifurcation and Turing bifurcation are established in turn. Then, the Turing instability region caused by diffusion is obtained. In addition, to uncover the diffusion mechanics of Turing patterns, the dynamic behaviors are studied near the Turing bifurcation by using weakly nonlinear analysis techniques, and the type of spatial pattern was predicted by the amplitude equation. And our results show that the spatial patterns in the Turing instability region change from the spot, spot-stripe to stripe in order. Finally, the results of the analysis are verified by numerical simulations.

Automated summary

This paper studies the Hopf bifurcations and Turing bifurcations of the Gierer-Meinhardt activator-inhibitor model. The interesting and complex spatially periodic solutions and patterns induced by bifurcations are analyzed theoretically and numerically. The conditions for the existence of Hopf bifurcation and Turing bifurcation are established, and the Turing instability region caused by diffusion is obtained. Moreover, the dynamic behaviors near the Turing bifurcation are studied using weakly nonlinear analysis techniques, and the spatial pattern types are predicted by the amplitude equation. Numerical simulations are conducted to verify the results of the analysis.