It's easy enough to show that if $\mathbb{N}_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}_1$, and we have a theorem that if $x \in \mathbb{N}_1$ and $y \in \mathbb{N}$ such that $x < y$, then $x \in \mathbb{N}$.

What if we had two non-standard models $\mathbb{N}_1 \subseteq \mathbb{N}_2$, ideally an elementary embedding? Must it be true that if $x \in \mathbb{N}_2$ and $y \in \mathbb{N}_1$ such that $x < y$, then $x \in \mathbb{N}_1$?

I'm also curious about the analogous questions for models of real analysis or of ZFC; e.g. comparing the set of integers in two nonstandard models of analysis.