Study On The Ratio-dependent Predator-prey Dynamical System With Stage-structure For Prey

In this paper, we establish some ratio-dependent predator-prey dynamical mod-els with stage-structure and completely study the effects of the stage-stucture, ratio-dependent and so on, on the permanence, extinction, stability of the species. We obtaina set of easly veried sufficent and necessary conditions. These will benefit the manegerplaning strategies to keep sustainable development of ecosystem.In subsection 1, a ratio-dependent predator-prey model with stage-structure for preyis poposed and analyed. By using the persistence theory for infinite dimensional systems,we obtain the uniform persistence of stage-structure system and stability of the boundaryequilibrium and the conditions of permananent and temporary exploiting for the maturepopution. In addition, we mainly discuss the stability of the positive equilibrium, byconstructing Lyapunov fuctions, applying Hurwitz theory and comparison thorem. Inaddition, by using the persistence theory for infinite dimensional systems, we obtainthe uniform persistence of stage-structure system and global asymptotic stability of theboundary equilibrium and the conditions of permananent and temporary exploiting forthe mature popution. In addition, we study the effect of nonautonomous stage-structuresystem on the ratio-dependent predator-prey. We obtain the global asymptotic conditionsof the nonegative equilibruim and postive equilibrium, by constructing the Lyapunovfuctions, applying Lasalle theory and comparson thorem.In subsecton 2, we consider a periodic ratio-dependent predator-prey model in whichimmature prey changes into mature prey with a proportionality, while the predator onlyfeeds on mature prey. A set of easily verifiable sufficient conditions is derived for the global existence of periodic solutions with strictly positive components by using the method ofcoincidence degree.