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The diffusive Schnakenberg model with gene expression time delay is considered. In this paper, the stability, diffusion-driven instability, and time delay-induced Hopf bifurcation have been investigated. By linear stability analysis, we find the parameter areas where the unique positive equilibrium is stable and Turing instability can occur for a certain relationship of diffusion rates. Then we obtain a series of critical values for the time delay at which the spatially homogeneous and inhomogeneous periodic solutions may emerge. Based on the explicit formula determining the properties of the Hopf bifurcation, we employ numerical simulations for parameters both in the stable region and Turing instability region. The numerical simulations show that delay can destabilize the stability of the positive equilibrium solution and eventually induce spatially homogeneous and inhomogeneous periodic solutions. Furthermore, the spatiotemporal patterns in the two spaces dimension from the Turing instability regime provide an indication of the wealth of patterns that the delayed system can exhibit.

Bifurcation analysis and spatiotemporal patterns in delayed Schnakenberg reaction-diffusion model