In our previous studies, we suggested equations for describing the hunting and the escaping behaviors employed by predators and prey. The equations were successfully used to understand the stability of the predator-prey system and showed that they had a strong advantage simulating the dynamics of the system. However, in the studies, the equations were simply constituted based on the concepts of probability without a rigorous mathematical proof. We believe that the proof is necessary to not only further expand into other applications but also build more sophisticated models that can be used in characterizing the real animal’s behavior. For this reason, in this study, we show a rigorous derivation with two biological constraints.