Global Analysis of a System of Predator–Prey Equations

We study a class of ODE'S modelling the interaction of one predator and one prey. The prey is assumed to be "asocial" in the sense that there is a threshold below which it will die even in the absence of predators. Knowledge of the separatrices of the flow and the local nature of the equilibrium points yields an analogue of the Kolmogorov theorem. A more detailed study of the direction of Hopf bifurcation and uniqueness of limit cycles, together with recent work on their continuation allows us to exhibit a very rich dynamics for both this system as well as for the more classical case of a prey without threshold. We briefly discuss the implications of this work for ecology.