Solution
= Solution
For $t\geq T$, set $\lambda=0$. Mature cells then decay geometrically:
$$
b_{T+j}=(1-\mu-k)^jb_T,
$$
while their divisions add $4\mu b_{T+j}$ to the permanently immature population. If $\mu+k>0$, mature cells disappear and
$$
N_\infty=a_T+4\mu b_T\sum_{j=0}^\infty(1-\mu-k)^j
=a_T+\frac{4\mu}{\mu+k}b_T.
$$
If $\mu=k=0$, no further change occurs and the <limit> is $a_T+b_T$.