Abstract: In this paper, the positive solution of the steady-state system for the predator-prey model with a non-monotonous growth rate is studied. Firstly, through calculating the fixed point index of compact maps in cone, the sufficient conditions for the existence of any possible positive solutions are obtained. Secondly, by the perturbation theory and the topological degree theory, the influence of the parameter on the uniqueness and linear stability of positive solutions is discussed. Finally, some numerical simulations are carried out to complement the existence theorem of positive solutions in one dimension and two dimensions, respectively. That is, the predator and prey can co-exist under certain conditions.

Key words: predator-prey model, fixed point index, uniqueness, stability, numerical simulation