<p>Currently, there is limited research addressing the Turing instability of spatially homogeneous periodic solutions. In this paper, we investigate the spatiotemporal dynamics of a Gierer-Meinhardt system with an emphasis on the interplay between Hopf and Turing bifurcations. We analyze the linear stability of the positive equilibrium and characterize the Hopf bifurcation of the associated kinetic system. Utilizing the Lyapunov-coefficient criterion, the diffusion-free system admits a locally stable spatially homogeneous periodic solution. We then demonstrate that diffusion can destabilize this periodic solution through a Turing mechanism, leading to the emergence of spatially inhomogeneous periodic patterns. Numerical simulations illustrate the bifurcation from spatially homogeneous oscillations to spatially heterogeneous structures. In particular, spotted and mixed patterns are observed within the Turing instability region, while spiral-wave patterns arise outside the Turing domain but near the Hopf bifurcation threshold. Chemically, our results not only reveal the competitive temporal evolution of the activator and inhibitor driven by the Hopf bifurcation but also demonstrate their complex spatiotemporal dynamics under the stimulation of diffusion.</p>