One example of Dr. Livesey’s presence of mind in a difficult situation is the way he responds to Billy Bones.
It is easy to be frightened of the pirate Billy Bones, especially when he hauls off singing loudly. However, Dr. Livesey holds his own. He is not afraid of the pirate captain, and stands up to him.
The voices stopped at once, all but Doctor Livesey's; he went on as before, speaking clear and kind, and drawing briskly at his pipe between every word or two. (ch 1)
This demonstrates that Dr. Livesey is a tough cookie. He is not easily intimidated, and not afraid to give orders the man around even if he is a scary looking pirate. He not only stands up to the guy when he is rude and everyone else is terrified of him, he threatens to see him hung if he does not put his knife away. This shows presence of mind in a difficult situation right off the bat.