Monostability and Bistability of Probabilistic Boolean Networks

This paper is concerned with the monostability and bistability problem of probabilistic Boolean networks (PBNs). First, the concepts of the cycle and fixed point for the PBNs are put forward. Second, the definitions of (strictly) monostability and (strictly) bistability are introduced for the PBNs. Subsequently, a sufficient and necessary condition is derived for the monostability and bistability of the PBNs. In the light of the obtained sufficient and necessary condition, we consider the state feedback stabilization problem of probabilistic Boolean control networks (PBCNs). Specifically, we propose a method to determine whether the PBCN is (strictly) monostable or (strictly) bistable. Moreover, the state feedback controller has been designed for the state feedback stabilization of PBCNs. Finally, two examples, including the apoptosis network model, are given to illustrate the effectiveness of the obtained theoretical results.