Necessary conditions for multistationarity in discrete dynamical systems

R. Thomas conjectured, twenty years ago, that the presence of a positive circuit in the interaction graph of a dynamical system is a necessary condition for the presence of several stable states. Recently, E. Remy et al. stated and proved the conjecture for Boolean dynamical systems. Using a similar approach, we generalize the result to discrete dynamical systems, and by focusing on the asynchronous dynamics that R. Thomas used in the course of his analysis of genetic networks, we obtain a more general variant of the R. Thomas’ conjecture. In this way, we get a necessary condition for genetic networks to lead to differentiation. Key words: Discrete dynamical system, Discrete Jacobian matrix, Interaction graph, Positive circuit, Multistationarity.

• Type: Journal
• Year: 2007
• Where: DAM
• Authors: Adrien Richard, Jean-Paul Comet
• Keywords: DAM 2007, Dynamical System, Interaction Graph, Necessary Condition