Cooperative predation on mutualistic prey communities.

•Predator cooperation can lead to extinction of the relatively more attacked prey.•Minimal mutualism strength is required to maintain coexistence equilibrium.•When cooperation is high, bistability between prey extinction and oscillatory coexistence increases survival chance of the otherwise extinct prey.•Interaction between prey mutualism and predator cooperation can lead to equilibrium-cycle bistability and cycle-cycle bistability.•The system demonstrates two different types of tristability.