$[$Solve by hand$-$writing$]$

A reaction system has three steps: $($i$)$ A enters the system with a contant rate of $k_0,($ii$)A$ is converted to $B$ with the help of an enzyme, with parameters $v_{\text{m}\text{a}\text{x}}$ and $K_m,($iii$)B$ is converted to $C$ chemically where no enzyme is involved, where two B molecules have to be available in the same place for the conversion to occur, with a coefficient of $k_1.$

$A$

$AB$

$2BC$

The parameter values are given as follows: $k_0=0\mathrm{.}5\frac{M}{m}in,v_{\text{m}\text{a}\text{x}}=2\frac{M}{m}in,K_m=1M,k_1=$ M$.$min$)$ with M designating molar concentrations.

$($a$)$ Write the ODE based models of changes in the concentrations of A and $B$ over time.

$($b$)$ Find the equilibrium point$($s$).$

$($c$)$ Check if the equilibrium point$($s$)$ are stable or not.