Let $\mathfrak {R}_A = (\Bbb {N}; 0, S,<,+)$. What can we say about ${}^{\ast}\Bbb N$, the universe of non-standard structure of the first order theory of $\mathfrak {R}_A$?
Firstly, because of the property every non-zero element must have a predessor and a successor, ${}^{\ast}\Bbb N$ must be a union of $\Bbb N$ and multiple $\Bbb Z$-chain which is ${}^{\ast}\Bbb N = \Bbb{N} \cup (A \times \Bbb {Z})$. So the problem reduces to what should the set $A$ be like.
Secondly since for every element either itself or its successor must be some element's double, $A$ must dense ordered set without a supremum and a infimum.
But there's still a lot of candidates for $A$, say $\Bbb R$, $(0,1)$, $\Bbb Q$. Can we sharpen this result?
What is known is that if the model is countable, then $A=\mathbb Q$. Since the ultrapower construction does not produce a countable model, $^\star \mathbb N$ must be more complicated. It is probably hellishly complicated.
A good reference for this is Bovykin, Andrey; Kaye, Richard Order-types of models of Peano arithmetic. Logic and algebra, 275–285, Contemp. Math., 302, Amer. Math. Soc., Providence, RI, 2002. A review may be found at http://www.ams.org/mathscinet-getitem?mr=1928396