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No amount of time will cause the ownership of the property (the dog) to pass from its current owners to you.
If the owners abandon the animal, you can report the animal to local animal control, and then try to adopt the animal from their system. However, if you do report it to animal control, they are going to want to see what kind of notice you have given to the owners of record, and will be notifying the owners themselves as well (so be prepared with copies of letters, emails, etc. and have their last known contact information available).