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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 756315, 13 pages
http://dx.doi.org/10.1155/2015/756315
Research Article

On the Dynamical Behavior of Toxic-Phytoplankton-Zooplankton Model with Delay

College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China

Received 11 October 2014; Accepted 5 January 2015

Academic Editor: Gualberto Solís-Perales

Copyright © 2015 Mehbuba Rehim 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

A toxin producing phytoplankton-zooplankton model with inhibitory exponential substrate and time delay has been formulated and analyzed. Since the liberation of toxic substances by phytoplankton species is not an instantaneous process but is mediated by some time lag required for maturity of the species and the zooplankton mortality due to the toxic phytoplankton bloom occurs after some time laps of the bloom of toxic phytoplankton, we induced a discrete time delay to both of the consume response function and distribution of toxic substance term. Furthermore, based on the fact that the predation rate decreases at large toxic-phytoplankton density, the system is modelled via a Tissiet type functional response. We study the dynamical behaviour and investigate the conditions to guarantee the coexistence of two species. Analytical methods and numerical simulations are used to obtain information about the qualitative behaviour of the models.