Regulating spatiotemporal dynamics for a delay Gierer-Meinhardt model

As a classical activator-inhibitor system with diffusion effects, the Gierer-Meinhardt (GM) model has received a considerable attention in recent years. Discussions on dynamic behaviors of the GM model are well underway. However, we still lack the means to further detect the bifurcation direction and the stability of bifurcated periodic solutions as a Hopf bifurcation appears in the space-time evolution. Besides, control strategies for regulating the space-time dynamics to achieve the desired biological patterns are also neglected. These are the tasks that this paper attempts to tackle. Through Hopf bifurcation and Turing instability theorems, it is demonstrated that by adjusting the controller parameters appropriately, the occurrence of the Hopf bifurcation can be delayed and the threshold of Turing instability can also be broadened. Go a step further, we have deliberated the bifurcation direction of the controlled GM model under the influence of self-diffusion coefficients. As a result, the control effect is verified by comparison in numerical simulations.