Arnold tongues and quasiperiodicity in a prey–predator model

We consider a two-dimensional map in which one of the fixed points is destabilized via a supercritical Naimark–Sacker bifurcation. We investigate, via analytical and numerical simulations, phenomena associated with the appearance, in the parameter-space, of structures like Arnold tongues involved in the Naimark–Sacker bifurcation. We determine analytically the location of the parameter sets where Naimark–Sacker bifurcation occurs, and the location on this line where tongues of arbitrary period are born. Lyapunov exponents, bifurcation diagrams, parameter-space and phase-space diagrams are used to show the transition from quasiperiodic to chaotic states.