Analysis of a Model Predator-Prey System with Refuges

We study a model of a predator-prey system with refuges proposed in McNair [Theoret. Population Biol.29 (1986), 38-63]. We show that all solutions are ultimately bounded and eventually enter a triangular region in the first quadrant. Moreover, two Hopf bifurcations occur and, using singularity theory, one can show that the two branches of periodic solutions join. Finally, we show that the periodic solutions produced by the Hopf bifurcation are unique and hence limit cycles which are globally asymptotically stable from the first quadrant with an equilibrium point deleted.