Stability analysis for genetic regulatory networks with delays: The continuous-time case and the discrete-time case

Convergent dynamics are investigated in this paper for the genetic regulatory networks (GRNs) with delays in both the continuous-time case and the discrete-time form. First, by utilizing the Lyapunov functional method and some inequality techniques, sufficient criterion is derived to ensure the global exponential stability of the continuous-time GRNs. Second, by using the semi-discretization technique and the IMEX-θ method as sources of reference, two discrete-time analogues of the continuous-time GRN model are formulated respectively. Moreover, it is shown that under the same sufficient conditions derived earlier, these two discrete-time GRN systems are also globally exponentially stable, and the explicit exponential convergent rates are also given. Finally, a numerical example is illustrated to show the validity of the obtained theoretical results, and comparisons of the convergent rates for these two different discrete-time analogues are also demonstrated.