Résumé:
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.