Singular homoclinic bifurcations in tritrophic food chains

The Rosenzweig–MacArthur food chain model is proved to have homoclinic orbits. The proof is in two steps. First, we use a geometric approach based on singular perturbation and detect singular homoclinic orbits as well as parameter combinations for which these orbits exist. Second, we show, numerically, that for slightly different parameter values there exist also nonsingular homoclinic orbits that tend toward the singular ones when the time responses of the three trophic levels are extremely diversified. The analysis is performed without exploiting too deeply the mathematical structure of the Rosenzweig–MacArthur model. This is done intentionally, to assist readers interested more in the methodology than in the application to food chains.