Bifurcation analysis and control on the delayed gene regulatory models

A delayed gene regulatory network model is established to describe biological discrete-time systems. Using stability analysis, the influence of time delay on discrete systems is illustrated. The direction and stability of Neimark–Sacker bifurcation are explored by using the normal form method and central manifold theorem. The hybrid control scheme is proposed and the bifurcation properties of controlled system are exhibited. Theoretical analysis and numerical simulation demonstrate that time delay causes oscillations in the system within a certain range and reveal that different feedback intensities affect the amplitude and period of the system. These results enrich our understanding of the impact of the feedback coupled loops with time delays on discrete system dynamics.