## condition for recurrence and transience

Consider the following model.

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

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

Suppose $S$ is irreducible, and $\phi\geq 0$ with $E_x\phi(X_1) \leq \phi(x)$ for
$x\notin F$, a finite set, and $\phi(x)\rightarrow \infty$ as $x\rightarrow \infty$, i.e., ${\{x : \phi(x) \leq M}\}$ is finite for any $M < \infty$, then the chain is recurrent.

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