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International Journal of Differential Equations
Volume 2011, Article ID 603183, 16 pages
Research Article

Role of Delay on Planktonic Ecosystem in the Presence of a Toxic Producing Phytoplankton

1School of Mathematics and Allied Sciences, Jiwaji University, MP Gwalior 474011, India
2Department of Applied Sciences, ABV-Indian Institute of Information Technology and Management, MP Gwalior 474010, India

Received 11 May 2011; Accepted 25 July 2011

Academic Editor: Gershon Wolansky

Copyright © 2011 Swati Khare 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.


A mathematical model is proposed to study the role of distributed delay on plankton ecosystem in the presence of a toxic producing phytoplankton. The model includes three state variables, namely, nutrient concentration, phytoplankton biomass, and zooplankton biomass. The release of toxic substance by phytoplankton species reduces the growth of zooplankton and this plays an important role in plankton dynamics. In this paper, we introduce a delay (time-lag) in the digestion of nutrient by phytoplankton. The stability analysis of all the feasible equilibria are studied and the existence of Hopf-bifurcation for the interior equilibrium of the system is explored. From the above analysis, we observe that the supply rate of nutrient and delay parameter play important role in changing the dynamical behaviour of the underlying system. Further, we have derived the explicit algorithm which determines the direction and the stability of Hopf-bifurcation solution. Finally, numerical simulation is carried out to support the theoretical result.