Role of instant nutrient replenishment on plankton dynamics with diffusion in a closed system: A pattern formation Joydip Dhar a , Randhir Singh Baghel b,⇑ , Anuj Kumar Sharma c a ABV-Indian Institute of Information Technology & Management, Gwalior, M.P., India b School of Mathematics and Allied Science, Jiwaji University, Gwalior, M.P., India c Department of Mathematics, L.R.D.A.V. College, Jagraon, Punjab, India article info Keywords: Aquatic ecosystem Reaction–diffusion mechanism Diffusion driven instability Spatiotemporal chaos Spatial pattern formation Higher order stability analysis abstract A mathematical model is proposed to study the role of instantaneous nutrient recycling on the plankton ecosystem. In this model, we include three state variables namely, nutrient biomass, phytoplankton and zooplankton population with Holling type II response func- tion for the population density transformation from phytoplankton to zooplankton. It is obtained that the local stability of different equilibrium depends on the nutrient supply rate to the phytoplankton for the temporal system and also existence of the oscillatory behavior of the temporal system is established by using Bendixson–Dulac criteria. In the spatiotemporal model, we also determine the diffusion-driven instability condition, with the numerical support for the effect of diffusivity coefficients on chaotic behavior of the system. Furthermore, we obtained the instability condition for linear and no-linear system from the higher order stability analysis. Finally, we analyze the time evaluation pattern for- mation of the spatial system. This shows that it is useful to use the reaction–diffusion sys- tem to reveal the spatial dynamics in the real world. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Pollution of fresh water in marine system by anthropogenic sources has become a concern over the last decades [1–3]. Researchers have found out that each tea spoon of ocean water contains 10 million to 100 million of viruses. Viral infection is known to cause a cell lysis in phytoplankton [4]. The measurable level of toxin due to harmful species is responsible for the bloom dynamics. In coastal area, viral disease can infect bacteria and phytoplankton [5]. Virus like particles are found in nat- ural phytoplankton community [4,6]. These virus like particles have also been found in many eukaryotic algae [7]. The par- asite modifying behavior has also been exhibited by the infected individuals of host population. This may happen by reducing stamina, disorientation and altering responses in infected population [8]. Killifish (Fundulus parvipinnis) tends to come closer to the surface of the sea after being infected, which make them more vulnerable to predation by birds [9]. Viruses have been held responsible for the collapse of Emiliania huxleyi bloom in mesocosms [10,6]. Since, viruses have ma- jor role in shaping the dynamics of plankton, so many researchers have developed and analyzed different mathematical models [11–13]. The dynamics of rapid (or massive) increase or decrease of plankton populations is an important subject in marine plank- ton ecology [14]. Generally high nutrient levels and favorable conditions play a key role in rapid or massive growth of algae and low nutrient concentration as well as unfavorable conditions limits their growth. The water must contain high levels of 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.02.052 ⇑ Corresponding author. E-mail addresses: jdhar@iiitm.ac.in (J. Dhar), randhirsng@gmail.com (R.S. Baghel), anujsumati@gmail.com (A.K. Sharma). Applied Mathematics and Computation 218 (2012) 8925–8936 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc