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Food chain chaos due to Shilnikov's orbit
1Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588.
Abstract:
Assume that the reproduction rate ratio zeta of the predator over the prey is sufficiently small in a basic tri-trophic food chain model. This assumption translates the model into a singularly perturbed system of two time scales. It is demonstrated, as a sequel to the earlier paper of Deng [Chaos 11, 514-525 (2001)], that at the singular limit zeta=0, a singular Shilnikov's saddle-focus homoclinic orbit can exist as the reproduction rate ratio epsilon of the top-predator over the predator is greater than a modest value epsilon(0). The additional conditions under which such a singular orbit may occur are also explicitly given. (c) 2002 American Institute of Physics.