Proportions in an Infinite Line

Let $c$ and $d$ be the long-run proportions of cats and dogs, so $c+d=1$. The transition probabilities are

Balance of flows. In a stationary sequence, the proportion of consecutive pairs of the form (cat, dog) is $\tfrac67c$. The proportion of pairs of the form (dog, cat) is $\tfrac14d$.

Along the line, cat-blocks and dog-blocks alternate. Hence every switch from cats to dogs is followed by a switch back, and the two kinds of switch occur equally often in the long run:

This is exactly the stationarity equation $\pi P=\pi$ for a two-state chain.

Substituting into $c+d=1$ gives $\tfrac{31}{7}c=1$. Hence $c=\tfrac{7}{31}$ and $d=\tfrac{24}{31}$.

Generalisation. Suppose a cat is followed by a dog with probability $p$, and a dog by a cat with probability $q$. Then $pc=qd$, so the proportion of dogs is $\dfrac{p}{p+q}$.