Global uniform stability analysis of biological networks with different time-scales under perturbations

We establish stability results for a class of biological networks with different time-scales under parameter perturbations and determine conditions that ensure the existence of exponentially stable equilibria of the perturbed biological system. The perturbed system is modelled as nonlinear perturbations to a known nonlinear idealized system and is represented by two time-scale subsystems. We derive a Lyapunov function for the coupled system and a maximal upper bound for the fast time scale associated with the fast state. Resulting stability conditions for two important biological networks, a neural network and a gene regulatory network are derived.