How Colonel Miles Quaritch and the military found it so necessary to continue to go after Jake and his family.
To me it logically doesnt make sense he would be that interested in finding Jake. He literally never met the guy. Even his video to himself was odd. Like why does Quaritch tell himself to go after Jake? If that was recorded before what happened in the first movie he would have had no way of knowing Jake would turn on him. For all Quaritch knew Jake was on his side. There could be a million reasons for his demise.
Also why is the military so fully invested to help? Jake is pretty much doing nothing except living on an island. No real threat at that point.