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Journal of Applied Mathematics
Volume 2014, Article ID 561958, 20 pages
http://dx.doi.org/10.1155/2014/561958
Research Article

Awakened Oscillations in Coupled Consumer-Resource Pairs

Department of General and Theoretical Physics, Kazakh National Technical University, 22 Satpayev Street, Almaty 050013, Kazakhstan

Received 9 June 2014; Revised 17 July 2014; Accepted 28 July 2014; Published 10 September 2014

Academic Editor: Qingdu Li

Copyright © 2014 Almaz Mustafin. 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

The paper concerns two interacting consumer-resource pairs based on chemostat-like equations under the assumption that the dynamics of the resource is considerably slower than that of the consumer. The presence of two different time scales enables to carry out a fairly complete analysis of the problem. This is done by treating consumers and resources in the coupled system as fast-scale and slow-scale variables, respectively, and subsequently considering developments in phase planes of these variables, fast and slow, as if they are independent. When uncoupled, each pair has unique asymptotically stable steady state and no self-sustained oscillatory behavior (although damped oscillations about the equilibrium are admitted). When the consumer-resource pairs are weakly coupled through direct reciprocal inhibition of consumers, the whole system exhibits self-sustained relaxation oscillations with a period that can be significantly longer than intrinsic relaxation time of either pair. It is shown that the model equations adequately describe locally linked consumer-resource systems of quite different nature: living populations under interspecific interference competition and lasers coupled via their cavity losses.