A qualitative analysis for a ratio-dependent generalized Holling-Tanner system.

*(Chinese. English summary)*Zbl 1324.92036Summary: The stability and existence of positive steady-state solutions for a predator-prey model are studied under a homogeneous Neumann boundary condition. Firstly, the global asymptotic stability of positive constant steady-state solutions is obtained by means of spectrum theory. Secondly, the priori estimates of positive steady-state solutions are given by applying the maximum principle and the Harnack inequality. Thirdly, the non-existence of the non-constant positive steady-state solutions is proved through the integral property, \(\varepsilon\)-Young inequality and PoincarĂ© inequality. Lastly, the existence of non-constant positive steady-state solutions is investigated with the help of the priori estimates and Leray-Schauder degree theory. Moreover, sufficient conditions for the existence of positive steady-state solutions are obtained. The results show that if the parameters satisfy certain conditions, two species will coexist.