So a pets by-law generally can’t easily prevent a pet being kept on the property initially, but once it is there, if it makes a nuisance of itself, then the by-law can allow for its removal.
You don’t need a by-law to remove a pet that proves to be a nuisance after it has been permitted. Section 158 (below) provides for that exact contingency.
However you could theoretically have a by-law that forbids pets that would be likely to create a nuisance, in the situations that you mention, such as cats in a bird sanctuary or large dogs in a scheme with substantial numbers of elderly people.
You might even consider a by-law where there are residents with proven phobias or deep-seated cultural or religious antipathy toward having animals around – although how that would be worded and enforced is another matter.
(1) The Tribunal may, on application by an interested person, make an order against a person who is keeping an animal on a lot or common property in accordance with the by-laws for a strata scheme, if the Tribunal considers that the animal causes a nuisance or hazard to the owner or occupier of another lot or unreasonably interferes with the use or enjoyment of another lot or of the common property.
(2) The Tribunal may order that the person—
(a) cause the animal to be removed from the parcel within a specified time, and be kept away from the parcel, or
(b) within a time specified in the order, take such action as, in the opinion of the Tribunal, will terminate the nuisance or hazard or unreasonable interference.