The graph above shows the concentration of $X$ over time for various initial conditions $x_0$.

• When $x_0 = 0$, the concentration remains zero, but this is an unstable equilibrium.
• For small initial values ($x_0 = 0.1$, $x_0 = 0.5$), the concentration of $X$ increases and approaches the stable fixed point at $x = \frac{k_1 a}{k_{-1}} = 1$.
• For larger initial values ($x_0 = 2$), the concentration decreases towards the same stable fixed point.

The system stabilizes at $x = 1$, demonstrating the behavior of autocatalysis where the production of $X$ is balanced by the back reaction limiting its growth.

Do you want further details or have any questions?

Here are some related questions to explore:

1. How would the dynamics change if the concentration of $A$ were not constant?
2. What happens to the system if $k_1$ or $k_{-1}$ are increased or decreased?
3. How would the system behave if there was a delay in the autocatalytic reaction?
4. What is the physical interpretation of the stability of the fixed points in a real chemical system?
5. How could you modify this model to include other possible reactions involving $A$ and $X$?

Tip: The stability of fixed points can often be determined by considering the signs of the derivatives at those points, which gives insight into how small perturbations affect the system.