Abstract: This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value $\lambda_1^D(\Omega_0)$, and demonstrate that the existence of the predator in $\overline{\Omega}_0$ only depends on the relationship of the growth rate $\mu$ of the predator and $\lambda_1^D(\Omega_0)$, not on the prey. Furthermore, when $\mu<\lambda_1^D(\Omega_0)$, we obtain the existence and uniqueness of its positive steady state solution, while when $\mu\geq \lambda_1^D(\Omega_0)$, the predator and the prey cannot coexist in $\overline{\Omega}_0$. Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region $\overline{\Omega}_0$, which is different from that of the classical Lotka-Volterra predator-prey model.

Key words: Lotka-Volterra predator-prey model, crowding term, critical value, coexistence