A new necessary condition on interaction graphs for multistationarity

We consider a dynamical system, described by a system of ordinary differential equations, and the associated interaction graphs, which are defined using the matrix of signs of the Jacobian matrix. After stating a few conjectures about the role of circuits in these graphs, we prove two new results relating them to the dynamic behaviour of the system: a sufficient condition for qualitative unstability, and a necessary condition for the existence of several stationary states. These results are illustrated by examples of regulatory modules in two variables, such as those occurring in biological networks.