As a generalization of Irregular Repetition Slotted ALOHA (IRSA), the probabilistic framework of coded Poisson receivers (CPR) offers a unified approach to analyze coded multiple access with successive interference cancellation (SIC). One crucial performance metric of CPRs is the stability region in which every packet can be successfully received with probability 1. In this paper, we use the potential function for convolutional Low Density Parity Check (LDPC) codes to derive the potential function for CPRs. Based on such a potential function, we derive three thresholds: the single-system threshold $G_s^*$ (for the original CPRs), the potential threshold $G_{conv}^*$ (for the convolutional CPRs), and the potential upper bound $G_{up}^*$. We prove that $G_s^* < G_{conv}^* < G_{up}^*$. Our numerical results show that these thresholds match very well with existing works for $D$-fold ALOHA.