The following two are very basic axioms put upon the successor function $S$:
\begin{align}
\neg Sx&=0\tag{1}\\\
Sx=Sy&\rightarrow x=y.\tag{2}
\end{align}

Let $PA^{(-1)}$ be the subtheory of $PA$ (first-order Peano Arithmetic) which has all other axioms except for (1), similarly $PA^{(-2)}$ let be the subtheory without (2). It is rather easy, but nevertheless interesting, result that both this theories have finite models, $PA^{(-1)}$ even has the degenerate one-element model.

My question is: has any research been made towards characterization of class of models of the theories above? If yes, could please someone provide me with suitable information? I am particularly interested in finite models.