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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 469072, 8 pages
http://dx.doi.org/10.1155/2013/469072
Research Article

Global Dynamics of the Hastings-Powell System

Departamento de Ingeniería Eléctrica y Electrónica, Instituto Tecnológico de Tijuana, Boulevard Alberto Limón Padilla s/n, Mesa de Otay, 22454 Tijuana, BCN, Mexico

Received 4 September 2013; Revised 16 November 2013; Accepted 17 November 2013

Academic Editor: Sebastian Anita

Copyright © 2013 Luis N. Coria. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the problem of bounding a domain that contains all compact invariant sets of the Hastings-Powell system. The results were obtained using the first-order extremum conditions and the iterative theorem to a biologically meaningful model. As a result, we calculate the bounds given by a tetrahedron with excisions, described by several inequalities of the state variables and system parameters. Therefore, a region is identified where all the system dynamics are located, that is, its compact invariant sets: equilibrium points, periodic-homoclinic-heteroclinic orbits, and chaotic attractors. It was also possible to formulate a nonexistence condition of the compact invariant sets. Additionally, numerical simulations provide examples of the calculated boundaries for the chaotic attractors or periodic orbits. The results provide insights regarding the global dynamics of the system.