Global monostability of a genetic toggle switch system

Abstract:

In this paper the global qualitative properties of a system which models a genetic toggle switch with cooperativties $1$ are given. Firstly it is proved that the system has a unique equilibrium, which is a stable node. Then it is shown that there are no periodic orbits by the Poincare-Bendixson Theorem, and the system has exactly two equilibria at infinity, which are both saddle-nodes. Consequently, the global phase portrait indicates the system is globally monostable.

Key words:genetic toggle switch; Poincare-Bendixson Theorem; periodic orbit; Poincare transformation; equilibria at infinity