First-Order Equations with Full, Three-Variable Model

COMPARISON OF FIRST-ORDER EQUATIONS WITH FULL, THREE-VARIABLE MODEL 

The magnitude of the correction to the simple (zero-order) autocatalator provided by the first-order equations, (14) and (15) for the time dependent behaviour and (17) for the stationary-states, can be seen quantitatively for a typical system from the first column of numerical results in Table 1. The corrections in ass and /3SS appear always to be an order of magnitude less than e and k, even when these parameters are of order unity. The second column, which gives the solutions of the full set of three equations (9a)-(9c), shows that the first-order equations actually overestimate the deviation from the e = k = 0 autocatalator. Returning to column I, we see that the stationary-state concentration of X is adequately given by ss = kol, even for e = k = 0.1, and is very accurately given by ss = Kass/3ss(l - e ) even for e = k = 1. Notice that for row (d) there are three stationary-state solutions at Tres = 225. 

The value of e and k at which the mushroom bifurcation diagram becomes 

FIGURE 3 Sustaining oscillations in the concentrations a, 13 and for the full, three-variable model. The concentration is shown X20. 

Additional computations to be reported elsewhere, confirm that the parameter k = a0K3 plays a more influential role than e = (k4lk3)a0K3, the behaviour of the system being relatively insensitive to small variations in the latter. 

---