Multistability and Bifurcation Analysis of Inhibitory Coupled Cyclic Genetic Regulatory Networks With Delays

Many biological systems have the conspicuous property to present more than one stable state and diverse rhythmic behaviors. A closed relationship between these complex dynamic behaviors and cyclic genetic structures has been witnessed by pioneering works. In this paper, a typical structure of inhibitory coupled cyclic genetic networks is introduced to further enlighten this mechanism of stability and biological rhythms of living cells. The coupled networks consist of two identical cyclic genetic subnetworks, which inhibit each other directly. Each subnetwork can be regarded as a genetic unit at the cellular level. Multiple time delays, including both internal and coupling delays, are considered. The existence of positive equilibriums for this kind of coupled systems is proved, and the stability for each equilibrium is analyzed without or with delays. It is shown that the coupled networks with positive cyclic genetic units have an ability to show multistability, while the coupled networks with negative units may present a series of Hopf bifurcations with the variation of time delays. Several numerical simulations are made to prove our results.