ecological modelling 197 ( 2 0 0 6 ) 498–504 available at www.sciencedirect.com journal homepage: www.elsevier.com/locate/ecolmodel Short communication Order to chaos and vice versa in an aquatic ecosystem Sandip Mandal a , Santanu Ray b,* , Samar Roy a , Sven Erik Jørgensen c a Department of Physics, Visva Bharati University, Santiniketan 731 235, India b Department of Zoology, Visva Bharati University, Santiniketan 731 235, India c DFH, Institute A, Miljokemi, Universitesparken 2, 2100 Copenhagen O, Denmark article info Article history: Received 6 October 2005 Received in revised form 22 January 2006 Accepted 14 March 2006 Published on line 18 April 2006 Keywords: Phytoplankton Zooplankton Fish Allometric principle Order Cycle Chaos Toxin abstract Chaotic situation may arise from equilibrium state for different reasons in any ecosystem. But to overcome this chaotic situation sometimes system itself has some mechanism and self-adaptability. In aquatic system of such condition it is reported that toxins are produced by many phytoplankton and these toxins may turn the ecosystem in to ordered state from chaos by reducing the grazing pressure of zooplankton. In this paper a three species (phy- toplankton, zooplankton and fish) model is proposed. The model is run in two different conditions. First, it has been considered that there is no toxin production from phytoplank- ton and in this condition the system exhibits different states (equilibrium point – stable limit cycle – doubling and ultimately chaos) by gradual decreasing of zooplankton body size. Decrement of zooplankton body size is controlled by increasing zooplankton grazing rate and decreasing the half saturation constant of zooplankton following allometric principles. When the system is in chaos, a toxin production parameter is introduced in the model equation. This toxin, produced from phytoplankton reduces the grazing pressure of zoo- plankton on phytoplankton. Gradually toxin values of phytoplankton are increased in the model which turn the system dynamics from chaos to doubling state to limit cycle and the system finally settles down to an equilibrium point. © 2006 Elsevier B.V. All rights reserved. 1. Introduction The most important mathematical attribute of chaos is the absence of any stable equilibrium point or any stable limit cycle in system dynamics, for which the pattern never repeat themselves. Ecological systems have all the elements to pro- duce chaotic dynamics (May, 1987). Although chaos is com- monly predicted by mathematical models, evidence for its existence in the natural world is scarce and inconclusive. Even the characteristics of chaos and its presence in nature are much discussed in ecology (Godfray and Grenfell, 1993; ∗ Corresponding author. Tel.: +91 3463 261268; fax: +91 3463 261268. E-mail address: santanu 5@yahoo.com (S. Ray). Hastings et al., 1993; Perry et al., 1993; Jørgensen, 1995). Recent developments in dynamical system theory consider chaotic fluctuations of a dynamical system as highly desirable because fluctuations allow such a system to be easily con- trolled. A number of mathematical models have been devel- oped to detect chaotic system dynamics using time–density data (Hastings et al., 1993). To assess the ecological implica- tions of chaotic dynamics in different natural system, it is important to explore changes in the dynamics when struc- tural assumptions of the system are varied. One approach to the study of the dynamics of ecological community is its 0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.03.020