In a discrete-time model, a proportion $\mu$ of mature bacteria divides at each time step. When a mature bacterium divides it is destroyed and two new immature bacteria are produced. A proportion $\lambda$ of the immature bacteria matures at each time step, and a proportion $k$ of mature bacteria dies at each time step. Show that this model may be represented by the equations

$$\begin{aligned} &a_{t+1}=a_{t}+2 \mu b_{t}-\lambda a_{t} \\ &b_{t+1}=b_{t}-\mu b_{t}+\lambda a_{t}-k b_{t} \end{aligned}$$

Give an expression for the general solution to these equations and show that the population may grow if $\mu>k$.

At time $T$, the population is treated with an antibiotic that completely stops bacteria from maturing, but otherwise has no direct effects. Explain what will happen to the population of bacteria afterwards, and give expressions for $a_{t}$ and $b_{t}$ for $t>T$ in terms of $a_{T}, b_{T}, \mu$ and $k$.