Abstract: The nature of the reaction-diffusion system solution contains abundant information, which is of great significance to the study of population ecological phenomena. In this paper, the existence of positive solutions to the steady-state system of a predator-prey model with fear effect is studied. Firstly, some priori estimates of positive constant steady-state solution are obtained using the maximum principle, which lay the foundation for the subsequent research. Secondly, the sufficient condition of the unique existence for the equilibrium solution is given, and the stability of positive constant steady-state solution is discussed using the stability theory of linear operators. Finally, on the basis of the degree theory, the existence of non-constant steady-state solution is given.

Key words: predator-prey model, fear effect, stability, steady-state solution, degree theory