Model reduction of polynomial dynamical systems using differential algebra

In this paper, we propose a model reduction scheme for a special class of polynomial dynamical systems. The biochemical processes and networks are our main motivation for this study. It is well known that many biochemical processes can be represented using quasi-polynomial systems. We show that a special class of quasi-polynomial systems can be cast in the Lotka-Volterra canonical form. For a given polynomial dynamical system, we propose a procedure to verify whether a given set of polynomials represents an invariant manifold of the system. Then, we study under what algebraic conditions a Lotka-Volterra system admits invariant manifolds. Finally, we combine our results with tools from differential algebra to propose a model reduction procedure for polynomial dynamical systems.