Word to describe pets that can be uncomfortable to live with?

I cannot figure out how to word this. I'm creating a rental system and have a question with regard to pets contained within a residence. But there are different types of pets.

For example, a dog who eats everything and 'litters' the backyard is very different from a turtle in a glass box.

Is there a way to differentiate between the two? I can only think of Uncaged pets but it's not really the right way to explain it.

Solution 1:

Troublesome pets will not be tolerated.

Solution 2:

You can say:

indoor pets only.

No noisy pets.

low maintenance dogs.

dogs welcome (some breeds only).

small dogs welcome.

Solution 3:

No pets that damage property or disturb other tenants.