Positive circuits and maximal number of fixed points in discrete dynamical systems Original Research Article

We consider a product image of image finite intervals of integers, a map image from image to itself, the asynchronous state transition graph image on image that Thomas proposed as a model for the dynamics of a network of image genes, and the interaction graph image that describes the topology of the system in terms of positive and negative interactions between its image components. Then, we establish an upper bound on the number of fixed points for image, and more generally on the number of attractors in image, which only depends on image and on the topology of the positive circuits of image. This result generalizes the following discrete version of Thomas’ conjecture recently proved by Richard and Comet: If image has no positive circuit, then image has a unique attractor. This result also generalizes a result on the maximal number of fixed points in Boolean networks obtained by Aracena, Demongeot and Goles. The interest of this work in the context of gene network modeling is briefly discussed.