Analysis of metabolic systems with complex slow and fast dynamics (Q1124547)

Tikhonov's theorem gives sufficient conditions to ensure that the solutions of the differential equation \[ \dot x=f\circ (x,y),\quad \epsilon \dot y=g\circ (x,y)\quad (\epsilon \ll 1) \] be approximated with the solutions of the system \[ \dot x=f\circ (x,y),\quad 0=g\circ (x,y). \] Here the system may verbally be depicted as having two time scales and the fast variable x is governed by a one-point attractor. A theorem recently proved by the authors extends the above theorem for the case when the slow variable tends to a more complicated attractor, e.g. to a limit cycle. Applying this theorem, an example of a simple enzymatic reaction with product activation and slow (genetically driven) enzyme turnover is rigorously studied. The dimensionless equations of this model are: \[ \dot e=1-e(1-p^ 2/(p^ 2+1)), \] \[ \epsilon \dot s=v_ s-e s p^ 2/(p^ 2+1),\quad \epsilon \dot p=e s p^ 2/(p^ 2+1)-p+p_ 0. \] It is shown that even in such a simple model there exist certain regions of the parameter space for which the fast variables oscillate. The model is also investigated numerically and the two approaches are compared.