Dynamical analysis of a stage-structured predator-prey model with cannibalism

This paper concerns with a stage-structured predator-prey model with cannibalism in the predator, where the predator population is divided into two stages, a juvenile stage and a mature stage. The effects of cannibalism on the model have been partially discussed by Magnússon [12] and Mařík and Přibyloná [13] by means of numerical simulation and simple mathematical analysis. Here we reinvestigate the dynamics of this model. We obtain the global stability of the model, illustrate the effects of the two factors (one describes the cannibalization and the other the benefit from cannibalism) on dynamics of the model, and find that the positive equilibrium can undergo stability switch (from stable to unstable to stable, or from unstable to stable to unstable) with the change of the cannibalization rate. The obtained results indicate that large cannibalization rate can make the positive equilibrium globally stable although its stability would change with the increase of the rate. We also analyze theoretically Hopf bifurcation. It turns out that supercritical/subcritical Hopf bifurcation can occur with the rates of the above mentioned two factors as bifurcation parameters. Finally, we introduce some quantities with clear biological meanings to better understand our results from the point of biology.