Quadratic autocatalysis with non-linear decay

Abstract

We provide a detailed, and thorough, investigation into the concentration multiplicity and dynamic stability of a prototype non-linear chemical mechanism: quadratic autocatalysis subject to non-linear decay in a continuously stirred tank reactor. This model was previously investigated in the literature using numerical path-following techniques. The contribution of this study is the application of singularity theory and degenerate Hopf-bifurcation theory to obtain analytical representations of many of the features of interest in this system. In particular, we use these presentations to identify critical values of an unfolding parameter below which specified phenomenon are no longer exhibited.

Appendix: Globally attracting invariant region

Here we show that the region

$$\begin{aligned} 0&\le \alpha _1 \le 1, \\ 0&\le \beta _1 \le 1 +\beta _0 \end{aligned}$$

is both (positively) invariant and exponentially attracting for any solution with physically meaningful initial conditions outside it.

We first demonstrate that solutions with non-negative initial conditions can not become negative. We have

$$\begin{aligned} \left. \frac{\mathrm{d }\alpha _1}{\mathrm{d }t}\right| _{\alpha _{1}=0}&= \frac{1}{\tau } > 0, \\ \left. \frac{\mathrm{d }\beta _1}{\mathrm{d }t}\right| _{\alpha _{1}=0}&= \frac{\beta _0}{\tau } \ge 0. \end{aligned}$$

(Observe that when $\beta _0=0$ that the line $\beta =0$ is itself invariant).

We now show that solutions with (physically meaningful) initial conditions outside the region $0\le \alpha _1\le 1$ are exponentially attracted to it. From Eq. (8) we have

$$\begin{aligned} \frac{d\alpha _1}{dt^*}&= \frac{1-\alpha _1}{\tau } -\alpha _1 \beta _1, \\&\le \frac{1-\alpha _1}{\tau }\quad \text {as } \alpha _1\ge 0 \,\,\text {and } \beta _1\ge 0. \end{aligned}$$

It follows that

$$\begin{aligned} \alpha _1\left( t^*\right) \le 1 -\left[ 1-\alpha _1\left( 0\right) \right] \exp \left[ -\frac{t^*}{\tau }\right] , \end{aligned}$$

i.e. the solution trajectory is attracted into the invariant region. This inequality also demonstrates that the region $0\le \alpha _1\le 1$ is positively invariant.

We have shown that the reactant concentration ($\alpha _1$) is bounded. We now show that the autocatalyst concentration ($\beta _1$) is bounded. Let $Z_1\left( t^*\right) = \alpha _1\left( t^*\right) +\beta _1\left( t^*\right)$ (As $\alpha _1$ and $\beta _1$ are both non-negative so it $Z_1$.) Adding Eqs. (8) and (9) we have

$$\begin{aligned} \frac{\mathrm{d }Z}{\mathrm{d }t^*}&= \frac{1+\beta _0-Z_1}{\tau } -\frac{\kappa _2\beta _1}{1+\rho \beta _1}\\&\le \frac{1+\beta _0-Z_1}{\tau }, \quad \text {as}\quad \beta _1(t^*) \ge 0. \end{aligned}$$

It follows that

$$\begin{aligned} Z\left( t^*\right)&\le 1+\beta _0 -\left[ 1+\beta _0 -\left( \alpha _1\left( 0\right) +\beta _1\left( 0\right) \right) \right] \exp \left[ -\frac{t^*}{\tau }\right] . \end{aligned}$$

This inequality demonstrates that if the initial condition is within the invariant region ($1+\beta _0 -\left( \alpha _1\left( 0\right) +\beta _1\left( 0\right) \right) >0$) then the corresponding solution remains in it for all time. Furthermore, if the initial condition is outside the invariant region ($1+\beta _0 -\left( \alpha _1\left( 0\right) +\beta _1\left( 0\right) \right) <0$) then the solution is attracted into the invariant region exponentially quickly.

In the limit $t^*\rightarrow \infty$

$$\begin{aligned} Z\left( t^*\right)&\le 1 +\beta _0 \\ \Rightarrow \alpha _1 +\beta _1&\le 1 +\beta _0. \end{aligned}$$

As the reactant $\alpha _1$ is bounded so is the autocatalyst. The use of simple differential inequalities to establish solutions boundedness in these types of systems stems from [15] (though was possibly known much earlier).