Gibbs measures for fertile hard-core models on the Cayley tree

We study fertile hard-core models with the activity parameter $\lambda>0$ and four states on the Cayley tree. It is known that there are three types of such models. For each of these models, we prove the uniqueness of the translation-invariant Gibbs measure for any value of the parameter $\lambda$ on the Cayley tree of order three. Moreover, for one of the models, we obtain critical values of $\lambda$ at which the translation-invariant Gibbs measure is nonunique on the Cayley tree of order five. In this case, we verify a sufficient condition (the Kesten–Stigum condition) for a measure not to be extreme.