Spiral attractors as the root of a new type of “bursting activity” in the Rosenzweig–MacArthur model

Abstract

We study the peculiarities of spiral attractors in the Rosenzweig–MacArthur model, that relates to the life-science systems and describes dynamics in a food chain “prey–predator–superpredator”. It is well-known that spiral attractors having a “teacup” geometry are typical for this model at certain values of parameters for which the system can be considered as slow–fast system. We show that these attractors appear due to the Shilnikov scenario, the first step in which is associated with a supercritical Andronov–Hopf bifurcation and the last step leads to the appearance of a homoclinic attractor containing a homoclinic loop to a saddle-focus equilibrium with two-dimension unstable manifold. It is shown that the homoclinic spiral attractors together with the slow–fast behavior give rise to a new type of bursting activity in this system. Intervals of fast oscillations for such type of bursting alternate with slow motions of two types: small amplitude oscillations near a saddle-focus equilibrium and motions near a stable slow manifold of a fast subsystem. We demonstrate that such type of bursting activity can be either chaotic or regular.