Turing instability of periodic solutions for the Gierer-Meinhardt model with cross-diffusion

A B S T R A C T In this paper, we establish the Gierer-Meinhardt model with cross-diffusion, and study Turing instability of its periodic solutions. Firstly, the stability of periodic solutions for the zero-dimensional system is studied by using the center manifold theory and normal form method. Secondly, according to Hopf bifurcation theorem, the diffusion rate formula for determining Turing instability of periodic solutions is established. Thirdly, by using the implicit function existence theorem and Floquet theory, the conditions of Turing instability of periodic solutions are derived, and it is proved that the periodic solutions of the model will undergo Turing instability. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is actually induced by cross-diffusion.(c) 2021 Elsevier Ltd. All rights reserved.

Automated summary

In this paper, the Gierer-Meinhardt model with cross-diffusion is established, and the Turing instability of its periodic solutions is studied. The conditions of Turing instability are derived and verified through theoretical analysis and numerical simulations, showing that the Turing instability of periodic solutions is induced by cross-diffusion.