Cross-diffusion induced stationary patterns in a prey–predator system with parental care for predators

This paper is concerned about a strongly-coupled nonlinear reaction–diffusion system, which describes a prey–predator model with parental care for predators. The long time behaviors of the solution are discussed, stability and instability of the positive constant equilibrium are studied. Our results show that even though the unique positive constant steady-state is stable for the kinetic system and for the self-diffusion reaction system, cross-diffusion can generate the stationary patterns (nonconstant positive steady states). The numerical simulation is also given to illustrate the theoretical result.