The stability study of a Gierer-Meinhardt system

Abstract

In this work, we study the Turing patterns appearing in a Gierer-Meinhardt model of the activator-inhibitor type with di§erent sources. First, we investigate the corresponding kinetic equations and derive the conditions for the stability of the equilibrium and then, we turn our attention to the Hopf bifurcation of the system. In certain parameter range, the equilibrium experiences a Hopf bifurcation; the bifurcation is supercritical and the bifurcated periodic solution is stable. With added di§usions, we show that both the equilibrium and the stable Hopf periodic solution experience Turing instability, if the di§usion coe¢ cients of the two species are su¢ ciently di§erent. And we prove the global existence in time of the solutions of this system.