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A qualitative analysis for a ratio-dependent generalized Holling-Tanner system. (Chinese. English summary) Zbl 1324.92036
Summary: 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 Poincaré 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.
##### MSC:
 92D25 Population dynamics (general) 34D20 Stability of solutions to ordinary differential equations
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