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.