Stability criterion of a nonautonomous 3-species ratio-dependent diffusive predator-prey model

The global stability of a nonautonomous 3-species ratio-dependent diffusive predator-prey model is studied in this paper. Firstly, some easily verifiable sufficient conditions which guarantee the existence of the strictly positive space homogenous periodic solution (SHPS) of the ratio- dependent predator-prey model (RDPPM) with diffusive and variable coefficient are achieved by using a comparison theorem of differential equation and fixed point theorem. At the same time, some new analysis techniques are developed as a byproduct. Secondly, some sufficient conditions ensuring the globally asymptotically stability of the strictly positive SHPS of the diffusive nonautonomous predator-prey model are given by using the method of upper and lower solutions (UALS) for the parabolic partial differential equations and Lyapunov stability theory. In addition, two numerical examples are given to validate the theoretical results obtained in this paper.