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Chaos and crises in a model for cooperative hunting: A symbolic dynamics approach

Chaos 19, 043102 (2009); doi:10.1063/1.3243924

Published 12 October 2009

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Jorge Duarte,1 Cristina Januário,1 Nuno Martins,2 and Josep Sardanyés3
1Department of Chemistry, Mathematics Unit, ISEL—High Institute of Engineering of Lisbon, Rua Conselheiro Emídio Navarro 1, 1949-014 Lisboa, Portugal and CIMA-UE, Universidade de Évora, Rua Romão Ramalho 59, 7000-671 Évora, Portugal
2Centre of Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
3Complex Systems Laboratory, ICREA-UPF, Barcelona Biomedical Research Park (PRBB-GRIB), Dr. Aiguader 88, 08003 Barcelona, Spain and Instituto de Biología Molecular y Celular de Plantas, CSIC-UPV, Ingeniero Fausto Elio s/n, 46022 València, Spain
In this work we investigate the population dynamics of cooperative hunting extending the McCann and Yodzis model for a three-species food chain system with a predator, a prey, and a resource species. The new model considers that a given fraction sigma of predators cooperates in prey's hunting, while the rest of the population 1−sigma hunts without cooperation. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of the kneading sequences associated with one-dimensional maps that reproduce significant aspects of the dynamics of the species under several degrees of cooperative hunting. Our model also allows us to investigate the so-called deterministic extinction via chaotic crisis and transient chaos in the framework of cooperative hunting. The symbolic sequences allow us to identify a critical boundary in the parameter spaces (K,C0) and (K,sigma) which separates two scenarios: (i) all-species coexistence and (ii) predator's extinction via chaotic crisis. We show that the crisis value of the carrying capacity Kc decreases at increasing sigma, indicating that predator's populations with high degree of cooperative hunting are more sensitive to the chaotic crises. We also show that the control method of Dhamala and Lai [Phys. Rev. E 59, 1646 (1999)] can sustain the chaotic behavior after the crisis for systems with cooperative hunting. We finally analyze and quantify the inner structure of the target regions obtained with this control method for wider parameter values beyond the crisis, showing a power law dependence of the extinction transients on such critical parameters. ©2009 American Institute of Physics
History: Received 6 June 2009; accepted 15 September 2009; published 12 October 2009
Permalink: http://link.aip.org/link/?CHAOEH/19/043102/1
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ISSN:
1054-1500 (print)   1089-7682 (online)
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