Stability analysis of quasi-polynomial dynamical systems with applications to biological network models

We study asymptotic stability properties of a class of quasi-polynomial dynamical systems. This class of nonlinear systems is a special class of interconnected systems arising in several biochemical and biological system applications and can be represented using quasi-polynomial dynamical systems. It is known that a special class of such systems can be embedded into a higher dimensional space and cast in Lotka–Volterra canonical form. We characterize a class of quasi-polynomial dynamical systems with asymptotic stability properties for all initial conditions in the positive orthant. The key advantage of the proposed method is that it is algebraic such that asymptotic stability conditions can be derived in terms of (as they are usually in biological network models) parameters of the system. We apply our results to parameterized models of three different biological systems: the generalized mass action (GMA) model, an oscillating biochemical network, and a reduced order model of the glycolysis pathway, and show that one can apply our proposed method to verify asymptotic stability for each case in terms of underlying parameters.