The first point that we need to make clear in answering this question is that Peter does not want to be around Jerry. All Peter wanted was to sit in the park by himself, read his book, and relax. Jerry, however, will not be ignored and forces Peter into a conversation. Peter, being polite, reveals some facts about his life, such as that he is married, has two children and two parakeets, and works in textbook publishing.
Reading between the lines, it quickly becomes clear that Jerry has some mental health issues, and he tells Peter the story about a dog he started to feed. When the hamburgers he was giving the dog failed to make the dog like him, he attempted to poison the dog, but this did not work either, which Jerry sees as his complete failure to forge any kind of connection with another living being.
After hearing this story, Peter loses his temper, and Jerry does the same, turning the verbal altercation into a fist fight. Jerry then takes the fight to the next level by pulling out a knife, which Peter uses to defend himself, leading to Jerry ultimately being impaled on the knife. At this point, Peter is far too involved in the situation to have contemplated leaving, and when he does eventually leave, Jerry is dying from a stab wound.