Probably depends on how long you have been there.
If you have recently moved in I would think they have a responsibility to ensure the place is vermin free. After a while pest control becomes your responsibility and the landlord can take the cost of pest control from your bind if you leave an infestation behind.
If the fence was broken when you moved in and the landlord didn't agree to fix it prior to you signing the lease then they don't have to fix it.