Let us define a countable model $\cal{M}$ = $(M,+_M ,\cdot_M, <_M)$ of $Q$ (Robinson arithmetic) to have a (primitive) recursive presentation if $\cal{M}$ is isomorphic to $(\omega, \oplus, \otimes)$, where $\omega$ is the set of natural numbers, and $\oplus$, and $\otimes$ are (primitive) recursive functions.
Similarly we can define what it means for $\cal{M}$ to have a feasible presentation, by insisting that $\oplus$, and $\otimes$ are polynomial-time computable functions (and representing natural numbers by their base-2 representations).
Questions.
(1) Is there a nonstandard model of the fragment IOpen of $PA$ with a feasible presentation? IOpen is a subsystem of Peano arithmetic whose induction axioms are limited to open (quantifier free) formulae.
(2) Suppose a model $\cal{M}$ of IOpen has a recursive (i.e., computable) presentation, does $\cal{M}$ also a primitive recursive presentation? If so, and the answer to (1) is positive, does it have a feasible presentation?
Motivation for the Questions.
Shepherdson (1967) showed that (a) models of IOpen are precisely the integer parts of real closed fields; and (b) in contrast to $PA$, there are nonstandard models of IOpen that have a recursive presentation.
As shown in this paper (which appeared in Theor. Comp. Sci. 2017), in certain classes of structures the notion of having a recursive presentation coincides with the notion of having a primitive recursive presentation; and in some other classes of structures, these two notions diverge (but the paper does not address the status of models of arithmetic).
Shepherdson's work was extended by a number of researchers, including Berarducci, Otero, Moniri, and most recently (to my knowledge) Mohsenipour, who constructed a recursive nonstandard model for IOpen with the GCD property and cofinal primes (in the reference below, he informed me that his construction results in a primitive recursive presentation).
Mohsenipour, Shahram, A recursive nonstandard model for open induction with GCD property and cofinal primes. (English summary) Logic in Tehran, 227–238, Lect. Notes Log., 26, Assoc. Symbol. Logic, La Jolla, CA, 2006.