You can ban smokers, pets (but not ESA/Service animals), multi-generational families (with two rooms, there isn't space for grandma and the kids to move in).

Animals? 

How many animals?

When you're asking questions of whether or not you can charge a pet deposit for a service animal....Or are you required to pay interest on security deposit and when is it enforced....?

By the way, if there is any way you can just rent to grad students - preferably med students or law students or engineering students - just not art or theater majors (party animals!)

While I sympathize with the plight of the animal the owner who left the animal behind should be prosecuted. 

In your case, I would justify the intra-lease charge by pointing to the impaired usage of the property for the other tenants and the pet addendum terms calling for immediate cleanup of animal waste.  

You said small animal.

Yeah, even the very best animals will leave odors, if no other damage.  

Win-win.At any rate, if you have law students or med students or other grad students (preferably not theater or music majors - they're flaky party animals), see if there is an off-campus housing listing for their grad students, or go talk to them about how to advertise to their grad students.You have to be careful how you word advertising on Craigslist or anywhere, though.