Stability and oscillation of genetic regulatory networks with time delays

From biochemical reaction principles, a genetic regulatory network can be described by a group of nonlinear differential equations with time delays. Previous studies have investigated delay-independent stability of genetic regulatory networks with time delays. However, if it is delay-independently stable, a genetic regulatory network loses other interesting properties such as oscillation. In this paper, we provide a computational method for computing the maximal delay interval over which the genetic regulatory network maintains stability, and beyond which the network will not be stable. Furthermore we prove that when its delay is exactly the maximal delay the network is oscillated. In addition, the formula for calculating the oscillation period is presented. The autoregulatory genetic network in zebrafish is used as an example to illustrate the presented method. The oscillation period calculated from our method is very close to that observed from the real-life of a zebrafish, which indicates the effectiveness of our method.