Suppose I have a family of countable state-space, discrete-time Markov chains, indexed by a parameter $r \in \mathbb{R}$. The state space is the same for all values of $r$; the transition probabilities depend on $r$ continuously. Is there some result like:

If the chain obtained for $r=r_0$ is positive recurrent, then the same is true for all values of $r$ in some open neighborhood of $r_0$; and as $r \to r_0$, the associated invariant distributions converge. (I assume that if the first statement is true, then so is the second one.)

In terms of simple linear algebra: I have a countable set of linear equations (defining the invariant distribution in terms of the transition probabilities of the Markov chain), which depend continuously on $r$. At some value $r=r_0$ the system has a unique positive solution. Is there a neighborhood of $r_0$ for which this will still be true? (What conditions do I need to impose on my set of equations to get this?)