Presumably you have records of the incidents; copies of the AC reports, vet bills, killed chickens, etc.
Total the damages and send the neighbor a demand letter, detailing the damages for all harm related to their dog's activities. It should be sent Cert. Mail/Return Receipt.
If - more likely, when - they fail to pay, file an action in Small Claims Court, using the above details for evidence.
The CA and your county bar association can provide referrals, if you need counsel.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
Keep filing your reports. Also, do as my fellow attorney states. Add up your damages and send a letter demanding compensation (in a courteous tone.) If he does not pay, file suit in small claims. You may also want to seek the assistance of an attorney to write your demand letter.
This is not to be construed as legal advice. I do not have an attorney client relationship with you.
I agree with my colleagues and would also advise you to speak with an attorney regarding any other options you may have.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.