As @user6726 said, in New Zealand, in theory, anyone can bring a private prosecution.
Criminal Procedure Act 2011, section 26 states:
If a person who is proposing to commence a private prosecution seeks to file a charging document, the Registrar may—
(a) accept the charging document for filing; or
(b) refer the matter to a District Court Judge for a direction that the person proposing to commence the proceeding file formal statements, and the exhibits referred to in those statements, that form the evidence that the person proposes to call at trial or such part of that evidence that the person considers is sufficient to justify a trial.
(2) The Registrar must refer formal statements and exhibits that are filed in accordance with subsection (1)(b) to a District Court Judge, who must determine whether the charging document should be accepted for filing.
(3) A Judge may issue a direction that a charging document must not be accepted for filing if he or she considers that—
(a) the evidence provided by the proposed private prosecutor in accordance with subsection (1)(b) is insufficient to justify a trial; or
(b) the proposed prosecution is otherwise an abuse of process.
(4)If the Judge determines under subsection (2) that the charging document should not be accepted for filing, the Registrar must—
(a) notify the proposed private prosecutor that the charging document will not be accepted for filing; and
(b)retain a copy of the proposed charging document.
(5) Nothing in this section limits the power of a Registrar to refuse to accept a charging document for want of form.
Unfortunately, bringing a private prosecution is not easy and could become costly. Pressuring the police to investigate may be the most practical course of action, although not likely to proceed. I can vouch that courts do not like non-qualified people bringing private prosecutions (they can't easily stop it, but can make life difficult and expensive)
I suspect you would have difficulty winning an action bought under 269(2)(a) because that person has a "claim of right" as this started on his land. I wonder if 269(3) might give you a slightly better chance of prevailing - although again, I suspect you might have difficulty proving intent.
Note that the kind of animal, and the date this happened may also be relevant to an action taken by (b) or (c), and you may want to mention it. (Some birds are protected, others sometimes, others not at all for example) for thoughts on how to proceed.
You might want to get in touch with NZALA - They have an interesting article here. The unfortunate upshot is that its unlikely you can do much (as its unlikely that this was a companion animal)