(Ignoring the fact that I got very distracted-)
At some point during the hour, Shuichi and Rantaro felt a tension forming between them, a desire to have Kokichi. Well, it was mostly Rantaro since he was becoming protective of Kokichi. Despite Shuichi letting them both into his house, he didn't trust the detective by any means. He could easily arrest them if he desired, so what if he used a relationship to manipulate him until arresting him? It was a thought that he disliked, so he had to get him to know that Kokichi was his.
It was as he watched Kokichi sleep that memories of one of his other friends came flooding back to him. They were very similar to Kokichi, even in the aspect that they had fallen in love with a detective. That friend ended up being used by the detective, lured into his own arrest through a relationship. He had forgotten originally, hence his immediate trust of Shuichi, but as the memory returned, so did his distrust of detectives. It was what caused him to feel angry with him for possibly liking Kokichi.
"Leave him alone!" he yelled suddenly.
Shuichi stood there, frozen from the momentary surprise. That was a sudden shift in tone from Rantaro. He had typically been rather calm and quiet, allowing Kokichi to sleep peacefully. The sudden outburst was surprising, but he had to go about it carefully. Kokichi was still sleeping, as far as he knew, and he didn't want to wake him up. He needed the sleep, that was for sure.
"I'm not…doing anything…?" he responded, a genuine confusion in his tone.
"You're trying to take him from me!" Rantaro accused. He was still yelling as he did so. "Stop it!"
Shuichi couldn't hold back, and this ended up turning into a pretty big argument as he tried to convince Rantaro that he wasn't trying to do anything to Kokichi, but he never believed him. He only grew more suspicious as this went on, fueling the argument. It seemed that they had forgotten about Kokichi in the end.