Stability of the unfolding of the predator-prey model

Abstract

We prove a conjecture of Zeeman that any generic unfolding of the Volterra's original predator-prey model is stable. This well-known two-dimensional model has co-dimension one in the planar Lotka- Volterra system and all its orbits are closed in the region of physical interest. Any generic unfolding of the model locally induces a degenerate Hopf bifurcation, but the presence of a cycle of saddles makes the global stability analysis quite involved. We solve the problem by working in the equivalent replicator system. Our proof of stability uses a family of Lyapunov functions for the unfolding. There are two other co-dimension one bifurcations in the planar replicator (equivalently Lotka- Volterra) system, which involve cycles of saddles and are therefore non-trivial. In one case we prove the stability of the bifurcation and in the other we determine a topologically versa! unfolding of the co-dimension one flow. This then, together with previous work on the subject, completes the study of co-dimension one bi...