Consider the reaction system

$$A \stackrel{k_{1}}{\longrightarrow} X, \quad B+X \stackrel{k_{2}}{\longrightarrow} Y, \quad 2 X+Y \stackrel{k_{3}}{\longrightarrow} 3 X, \quad X \stackrel{k_{4}}{\longrightarrow} E,$$

where the $k$ s are the rate constants, and the reactant concentrations of $A$ and $B$ are kept constant. Write down the governing differential equation system for the concentrations of $X$ and $Y$ and nondimensionalise the equations by setting $u=\alpha X$ and $v=\alpha Y, \tau=k_{4} t$ so that they become

$$\frac{d u}{d \tau}=1-(b+1) u+a u^{2} v, \quad \frac{d v}{d \tau}=b u-a u^{2} v$$

by suitable choice of $\alpha$. Thus find $a$ and $b$. Determine the positive steady state and show that there is a bifurcation value $b=b_{c}=1+a$ at which the steady state becomes unstable to a Hopf bifurcation. Find the period of the oscillations in the neighbourhood of $b_{c}$.