Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks

We represent interactions among biochemical species using a directed multigraph, which is a generalization of a more commonly used digraph. we show that network properties that are known to lead to multistability or oscillations, such as the existence of a positive feedback cycle, can be generalized to critical subnetworks that can contain several cycles. we also derive corresponding graph-theoretic conditions for pattern formation for the respective reaction-diffusion models. We present as an example a model for cell cycle and apoptosis along with bifurcation diagrams and sample solutions that confirm the predictions obtained with the help of the multigraph, network conditions.