Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain

The asymptotic behavior of a tri-trophic food chain model is studied. The a nalysis is carried out numerically, by finding both local and global bifurc ations of equilibria and limit cycles. The existence of transversal homocli nic orbits to a limit cycle is shown. The appearance of homoclinic orbits, by moving through a homoclinic bifurcation point, is associated with the su dden disappearance of a chaotic attractor. A homoclinic bifurcation curve, which bounds a region of extinction, is continued through a two-dimensional parameter space. Heteroclinic orbits from an equilibrium to a limit cycle are computed. The existence of these heteroclinic orbits has important cons equences on the domains of attraction. Continuation of non-transversal hete roclinic orbits through parameter space shows the existence of two codimens ion-two bifurcations points, where the saddle cycle is non-hyperbolic. The results are summarized by dividing the parameter space in subregions with d ifferent asymptotic behavior.