Lorena Cid-Montiel , Applied Mathematics, University of Waterloo
Higher dimensional slow manifolds in chemical reaction networks
In most reaction networks it is common to have several time scales involved. In such case some reactions may quickly reach a steady-state and dominate the initial dynamics, while other reactions occur slowly thus dominating in a long time scale. We are interested in the slow dynamics of the system and we intend to use qualitative methods to describe them.
An interesting example is given by considering the Michaelis-Menten mechanism of enzyme kinetics; in the substrate/complex plane, the slow, asymptotic motion of the system is bounded above by the EA (Equilibrium Approximation) and below by the SSA (steady-state approximation). It has been shown there is a unique trajectory contained entirely between the two approximations which attracts the flow, such a trajectory is called the slow manifold. For planar systems, the description of the slow dynamics has been well studied, it is actually possible to compute the location of the slow manifold.
In this talk we will briefly review the theory related to fences, funnels and antifunnels in planar systems needed to ensure existence and uniqueness of the slow manifold. Based on these ideas for planar systems, we will go over some analogous conditions needed in higher dimensional systems to guarantee existence and uniqueness of the slow manifold. Finally, we outline future goals and possible extensions of our current work.