Dynamics of a predator-prey model

We describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator-prey model proposed by R. M. May [Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1974]. In particular, we give a qualitative description of the bifurcation curve when two limit cycles collapse on a semistable limit cycle and disappear. Moreover, we show that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator-prey models.