A delay model for viral infection in toxin producing phytoplankton and zooplankton system Sunita Gakkhar * , Anuraj Singh Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India article info Article history: Received 20 July 2009 Received in revised form 6 January 2010 Accepted 7 January 2010 Available online 18 January 2010 Keywords: Local stability Hopf bifurcation Limit cycle Chaos abstract An eco-epidemiological delay model is proposed and analysed for virally infected, toxin producing phytoplankton (TPP) and zooplankton system. It is shown that time delay can destabilize the otherwise stable non-zero equilibrium state. The coexistence of all species is possible through periodic solutions due to Hopf bifurcation. In the absence of infection the delay model may have a complex dynamical behavior which can be controlled by infec- tion. Numerical simulation suggests that the proposed model displays a wide range of dynamical behaviors. Different parameters are identified that are responsible for chaos. Ó 2010 Published by Elsevier B.V. 1. Introduction Phytoplankton are microscopic algae found in marine and fresh water. They have an important role in aquatic ecology as it is the top prey in all food chains. They live near the surface of the ocean as they need sunlight like all green plants. Phy- toplankton use water and absorb CO 2 from the air to grow. Some scientists believe that the phytoplankton population can combat the global warming and forms the basis of several eco-restoration projects. The cycles of rapid increase and decrease of phytoplankton population is a common feature in marine plankton ecology. Phytoplankton are normally present in low concentrations but may proliferate to form dense concentrations of cells on water surfaces referred to as ‘‘blooms”. An array of complex factors and influences collectively form conditions favourable to algal blooms. Generally, highly nutrients and favorable conditions play a key role in rapid or massive growth of algae and low nutrient concentration as well as unfavor- able conditions inevitably limits their growth. The high concentrations of pigment-containing phytoplankton may impart colour to the water resulting in their description as ‘‘red tides,” ‘‘brown tides,” etc. Although the sudden appearance and dis- appearance of blooms is not well understood but the adverse effects of harmful algal blooms (HAB) on human health, aquatic population, fisheries business, and tourism are well established. Blooms can occur over wide geographic areas and may involve long distance transport to affected resources. These blooms adversely affect the aquatic vegetation. Thorough study regarding the pattern of blooms is required to understand its impact on water quality, fishing and related economical losses as well as on the ecosystem. The adverse effect of harmful algal blooms is clear but the control of such problems is under investigation. As harmful phytoplankton certainly plays an important role in the blooms and succession. Probably, the main reason behind population succession and bloom is due to the toxin produced by harmful phytoplankton. When a bloom of a particular harmful phy- toplankton occurs, the cumulative effect of all the toxins released may affect the other organisms, causing mass mortality. 1007-5704/$ - see front matter Ó 2010 Published by Elsevier B.V. doi:10.1016/j.cnsns.2010.01.010 * Corresponding author. Tel.: +91 01332 285171. E-mail addresses: sungkfma@iitr.ernet.in (S. Gakkhar), anurajiitr@gmail.com (A. Singh). Commun Nonlinear Sci Numer Simulat 15 (2010) 3607–3620 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns