Continuous and discontinuous stationary patterns and bifurcation of a predator–prey model with strong Allee effect

This paper concerns a diffusion equation coupled to an ordinary differential equation, which models the interaction between predator and prey with a strong Allee effect. First, we prove the existence of continuous steady states by applying the bifurcation theory and analyzing their stability. Moreover, the global behavior of these solutions is studied. Then we construct two types of steady states with jump discontinuity and verify that under a suitable topology, they are always stable. Finally, the asymptotic behavior of steady states with jump discontinuity is considered as the diffusion coefficient tends to infinity.