Analysis of diffusive two-competing-prey and one-predator systems with Beddington–Deangelis functional response

In this paper, a diffusive two-competing-prey and one-predator system with Beddington–DeAngelis functional response is considered. The sufficient and necessary conditions for the existence of coexistence states are provided using the fixed point index theory developed. In addition, the stability and uniqueness of coexistence states are investigated. Finally, this paper discusses the sufficient conditions for extinction and permanence of the time-dependent system.