= Solution
For this <linear birth process with catastrophes>, the off-diagonal entries of the <Q-matrix> are
$$
q_{0,1}=\lambda,\qquad
q_{n,n+1}=\lambda(n+1),\qquad q_{n,0}=\mu\quad(n\ge1),
$$
with all other off-diagonal entries zero. The diagonal entries are $q_{00}=-\lambda$ and $q_{nn}=-[\lambda(n+1)+\mu]$ for $n\ge1$. At zero there is no separate catastrophe transition, because it would not change the state.
The chain is irreducible: every positive state can jump to zero, and from zero any finite sequence of births has positive probability. To prove nonexplosion, couple it with the <linear birth process with immigration> having birth rate $\lambda(n+1)$ and no catastrophes. Its mean population is finite at each finite time, obtained from $m'=\lambda(m+1)$, or its exponential holding-time sum diverges since $\sum_n1/[\lambda(n+1)]=\infty$. Couple births monotonically and implement catastrophes with an independent rate-$\mu$ Poisson clock, resetting the smaller population. On a finite time interval the dominating pure-birth process has finitely many births, and the catastrophe clock finitely many rings. Thus the original chain is nonexplosive.
For its <jump chain>, the transitions are
$$
\boxed{P_{0,1}=1,\quad
P_{n,n+1}=\frac{\lambda(n+1)}{\lambda(n+1)+\mu},\quad
P_{n,0}=\frac{\mu}{\lambda(n+1)+\mu}\quad(n\ge1).}
$$
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