Singular Homoclinic Bifurcations in Tri-trophic Food Chains

Feo, O. de & Rinaldi, S. (1996). Singular Homoclinic Bifurcations in Tri-trophic Food Chains. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-96-076

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Abstract

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. Then we show, numerically, that for slightly different parameter values there exist also non singular homoclinic orbits which tend toward the singular ones when the time responses of the three trophic levels are extremely diversified. The analysis is performed systematically, without exploiting too deeply the mathematical structure of the Rosenzweig-MacArthur model. This is done intentionally, in order to facilitate readers interested more in the methodology than in the application to food chains.

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