You need a paper trail in case this happens again. Trust me, if you don't have evidence, they will claim it never happened. Been there and done that with a bad neighbor. If you are not wanting to file a report with animal control, the Sheriff's Dept or the police, write them a letter. Be concise and polite in the letter. Send one to them registered mail and one regular mail. That way, if they don't sign for the registered mail, they still get the letter. A lawyer advised me to do it this way a long time ago.
I sent such a letter to my neighbors that own the evil rhodesian ridgebacks that killed my birds. Before the massacre they were suspected of being out and killing one of my birds. It was a polite warning that their dogs were a problem and were not welcome on my property. The dogs came back at a later date and killed 13 birds. In court, the letter was very helpful. It showed the judge that they were irresponsible enough to allow the dogs out when they knew they were a problem. I would reccommend you write something like this:
Dear (neighbor)
On (insert date) your dog entered my property. He chased and attempted to kill my poultry. At least one bird was injured. I spoke with you that day, and I know that you are aware of the problem.
In the past, your dog has chased my children and gotten into our trash. I consider your dog dangerous and a menace. It is your responsibility to keep your animal on your property.
In most jurisdictions it is legal for property owners to shoot roaming dogs that are on their property harrassing, chasing or killing livestock. If your dog is on my property again, I will take any and all legal action necessary to remedy the situation. I will also hold you liable for any and all damages caused by your dog.
Sincerely,
You
Also send a nice letter and a copy of the letter sent to your neighbor to their landlord. Yes, a landlord might be held liable if they permit tenants to keep dangerous dogs on their property. Once the landlord has been made aware of the propensity of the dog to a problem he/she assumes some liability if they permit the animal to stay there in some jurisdictions.
What state are you in?