Abstract: We study the effect of cross-diffusion on the existence of coexistence solutions for a predator-prey model with Ivlev type functional response, in which cross-diffusion describes the situation of predator chasing its prey and prey avoiding its predator. Firstly, the sufficient conditions for the existence of coexistence solutions are obtained by using the topological degree theory and the fixed point index theory. Secondly, the sufficient conditions for the nonexistence of coexistence solutions are derived by using the energy estimation method.