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Chaos in perturbed Lotka-Volterra systems

Published online by Cambridge University Press:  17 February 2009

J. R. Christie ,
K. Gopalsamy  and
Jibin Li
J. R. Christie
Affiliation:
Department of Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, Adelaide, South Australia, 5001, Australia; e-mails: johnc and [email protected].
K. Gopalsamy
Affiliation:
Department of Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, Adelaide, South Australia, 5001, Australia; e-mails: johnc and [email protected].
Jibin Li
Affiliation:
institute of Applied Mathematics of Yunnan Province, Department of Mathematics, Kunming University of Technology, Yunnan 650093, People's Republic of China; e-mail: [email protected].
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Abstract

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Lotka-Volterra systems have been used extensively in modelling population dynamics. In this paper, it is shown that chaotic behaviour in the sense of Smale can exist in timeperiodically perturbed systems of Lotka-Volterra equations. First, a slowly varying threedimensional perturbed Lotka-Volterra system is considered and the corresponding unperturbed system is shown to possess a heteroclinic cycle. By using Melnikov's method, sufficient conditions are obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Then a special case involving a reduction to a two-dimensional Lotka-Volterra system is examined, leading finally to an application with a model for the self-organisation of macromolecules.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 3 , January 2001 , pp. 399 - 412
Copyright
Copyright © Australian Mathematical Society 2001

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