Linear stability analysis and periodic behaviour

Equations (2.7) admit a unique steady-state solution obtained by solving the equations da/dt - dy/dt = 0. The product concentration at steady state, yo is given by eqn (2.12), while the corresponding concentration of substrate, Oq, is obtained by solving eqn (2.13)  

The latter equation admits at most a single real, positive root when O V. In the opposite case, the substrate injection rate exceeds the maximal capacity of the enzyme to transform the substrate into product, so that the former accumulates in the course of time. 

Linearization of the kinetic equations and condition of instability of the steady state 

The stability properties of the steady state can be analysed by slightly perturbing the concentrations of substrate and product aroimd their steady-state values, and by determining the time evolution of the infinitesimal perturbations 8a, 8y defined by relations (2.14) (see Nicolis Prigogine, 1977)  

Inserting these relations into eqns (2.7) leads to the linearized evolution equations for the perturbations  

---