Consider the following model.

$\displaystyle X_{n+1}$ given $\displaystyle X_n, X_{n-1},...,X_0$ has a Poisson distribution with mean $\displaystyle \lambda=a+bX_n$ where $\displaystyle a>0,b\geq{0}$. Show that $\displaystyle X=(X_n)_{n\in\mathrm{N_0}}$ is an irreducible M.C & it is recurrent if $\displaystyle 0\leq b <1$. In addition, it is transient if $\displaystyle b\geq 1$.

How do we approach this question? I was thinking of using the theorem below.

Suppose $\displaystyle S$ is irreducible, and $\displaystyle \phi\geq 0$ with $\displaystyle E_x\phi(X_1) \leq \phi(x)$ for

$\displaystyle x\notin F$, a finite set, and $\displaystyle \phi(x)\rightarrow \infty$ as $\displaystyle x\rightarrow \infty$, i.e., $\displaystyle {\{x : \phi(x) \leq M}\}$ is finite for any $\displaystyle M < \infty$, then the chain is recurrent.

However I have no idea of how to start. Thanks in advance.