Pattern formation with jump discontinuity in a predator–prey model with Holling-II functional response

Abstract This paper is focused on the existence and uniqueness of nonconstant steady states in a reaction–diffusion–ODE system, which models the predator–prey interaction with Holling-II functional response. Firstly, we aim to study the occurrence of regular stationary solutions through the application of bifurcation theory. Subsequently, by a generalized mountain pass lemma, we successfully demonstrate the existence of steady states with jump discontinuity. Furthermore, the structure of stationary solutions within a one-dimensional domain is investigated and a variety of steady-state solutions are built, which may exhibit monotonicity or symmetry. In the end, we create heterogeneous equilibrium states close to a constant equilibrium state using bifurcation theory and examine their stability.