Hopf-Pitchfork Bifurcation in a Phytoplankton-Zooplankton Model with Delays

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Abstract

The purpose of this paper is to study the dynamics of a phytoplankton-zooplankton model with toxin delay. By studying the distribution of the eigenvalues of the associated characteristic equation, the pitchfork bifurcation curve of the system is obtained. Furthermore, on the pitchfork bifurcation curve, we find that the system can undergo a Hopf bifurcation at the positive equilibrium, and we derive the critical values where Hopf-Pitchfork bifurcation occurs.

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Dynamical analysis of toxin producing Phytoplankton–Zooplankton interactions

Tapan Saha, Malay Bandyopadhyay (2009)

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A delay differential equation model on harmful algal blooms in the presence of toxic substances.

J. Chattopadhyay, R. R. Sarkar, A El Abdllaoui (2002)

The periodic nature of blooms is the main characteristic in marine plankton ecology. Release of toxic substances by phytoplankton species or toxic phytoplankton reduce the growth of zooplankton by decreasing grazing pressure and have an important role in planktonic blooms. A simple mathematical model of phytoplankton-zooplankton systems with such characteristics is proposed and analysed. As the process of liberation of toxic substances by phytoplankton species is still not clear, we try to describe a suitable mechanism to explain the cyclic nature of bloom dynamics by using different forms of toxin liberation process. To substantiate our analytical findings numerical simulations are performed and these adequately resemble the results obtained in our field study.