= Solution
Under the specified evolution rule, conditional on the entire history, the next event's <probability distribution> depends only on the current population size. This is the <time-homogeneous Markov property>. Birth and immigration both increase the size by one, while death decreases it by one. Writing $q=1-p$, the <transition probabilities> on $\mathbb Z_{\geq0}$ are
$$
\boxed{P(k,k+1)=u_k=\frac{kp+\theta}{k+\theta}\quad(k\geq0),\qquad P(k,k-1)=d_k=\frac{kq}{k+\theta}\quad(k\geq1),}
$$
and all other entries are zero. In particular $P(0,1)=1$ and $d_0=0$. The row sum is $[kp+\theta+kq]/(k+\theta)=1$. Every neighboring pair communicates, so this is an <irreducible Markov chain>, specifically a <birth-death-immigration jump chain>.
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