I think we need more information.
You mention dog parks and day care. How long has he been going to the dog park, and how often. How often does he go to day care? Have you discussed the behavior with people at the daycare to see if he is having issues there?
You mention a trainer. Is this one-on-one training or is it training classes? Has your dog been to classes? If so, how old was he and for how many weeks (I am guessing dog class once a week for a number of weeks)?
He is maturing now. The barking and lunging on leash is usually a sign of an insecure/fearful dog reacting to something to make it go away. I don't like it that it tried to bite the puppy though.
Your dog has two choices when it is confronted with something that makes it uneasy: it can fight with whatever it is and conquer, or it can run away from whatever it is. On leash, the dog knows it cannot run away. Lots of dogs that are tethered to something act like they are guarding it. Lots of bites happen when someone gets too close to a tethered dog. The dog cannot run away, so it needs to make whatever it is afraid of go away. Barking and lunging causes you to increase the distance between the dog and the object of his fear. And therefore, it is something that works, and he will do that again and again.
Next question: What do you do when he acts like this?
Without those answers, I would say skip the dog park and the day care. Read up on NILIF, and implement it. Take your dog to training classes, and if it makes you more comfortable, put a muzzle on him. It's not necessarily forever, but you do not want him connecting. Also, you need to either work below his threshold and build up his tolerance by decreasing the distance between objects that he reacts to slowly, or you need to give him a quick, meaningful correction the moment he starts his routine and move on very positively. Don't give him the change to get entrenched in the behavior.
The problem with the correction, is that it masks the symptom, and not necessarily deals with the problem. Sometimes symptoms do get a life of their own though, and snapping him out of his behavior, and then moving on quickly and confidently, very positive, can allow you to work within proximity of whatever he is reacting to, and without any negative problems from those joggers or dogs, he will become less concerned.
I think that the problem with day care and dog parks is that we condition dogs to interact with other dogs, when for some of our dogs it would be best if they interact with us, and ignore other dogs. The interaction may look like aggression and be play, it can be punished/discouraged, while other behaviors may be encouraged. The interaction may look like a good response and it could be subtle bullying or pack order stuff. And God knows what happens in day care when we are not there?
Dog classes are better in some ways, because our dogs learn to function, interacting with us, while other people interact with their dogs. They can be in the same proximity, but there is no racing about, play fighting, humping going on. They are learning to view other dogs as something of interest that is no concern, because they are working on getting that hunk of steak or hot dog out of their owners hand. And eventually it will be for just a bit of praise that the dog will find you much more entertaining than other dogs.
Dog parks and doggy daycare provides a situation where dogs are highly stimulated by other dogs, and lets face it, humans are pretty darned BORING.
When the dog can be running, chasing, body slamming, jumping up, locking teeth on each other's necks. Yayyy!!!!
How do you compete with that?
There are ways. But the best thing is to forgo dog parks and day cares, and take the part of the dog's best outlet for mental and physical stimulation. Get out there and play fetch, play tug, catch frisbees, teach him to run next to your bicycle (not while he is reacting to other dogs), be more exciting than anything out there, and then pull the training into line with the wonderful games. Use treats, use praise, use a highly stimulating happy voice, use a tug or toy for when he does really good.
This builds a bond between you and the dog, which a dog park or doggy daycare CANNOT do. Working in class training, helps to solidify that bond with the distractions of other people and dogs around.
I think I am rambling.
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