Journal of Applied Mathematics
Volume 2013 (2013), Article ID 167671, 12 pages
http://dx.doi.org/10.1155/2013/167671
Research Article
Dynamics of a Single Species in a Fluctuating Environment under Periodic Yield Harvesting
1Center for Theoretical Chemistry and Physics, New Zealand Institute for Advanced Study, Massey University Albany,
Auckland 0745, New Zealand
2Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey
3Department of Mathematical Sciences, University of Agder, Serviceboks 422, 4604 Kristiansand, Norway
4Department of Mathematics and Mathematical Statistics, Umeå University, 90187 Umeå, Sweden
Received 6 December 2012; Accepted 11 February 2013
Academic Editor: Theodore E. Simos
Copyright © 2013 Mustafa Hasanbulli et al. 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
We discuss the effect of a periodic yield harvesting on a single species population whose dynamics in a fluctuating environment is described by the logistic differential equation with periodic coefficients. This problem was
studied by Brauer and Sánchez (2003) who attempted the proof of the existence of two positive periodic solutions; the flaw in their argument is corrected. We obtain estimates for positive attracting and repelling periodic solutions and describe behavior of other solutions. Extinction and blow-up times are evaluated for solutions with small and large initial data; dependence of the number of periodic solutions on the parameter associated with the intensity of harvesting is explored. As grows, the number of periodic solutions drops from two to zero. We provide bounds for the bifurcation parameter whose value in practice can be efficiently approximated numerically.