Short-term recurrent chaos and role of Toxin Producing Phytoplankton (TPP) on chaotic dynamics in aquatic systems Ranjit Kumar Upadhyay a, * , V. Sree Hari Rao b,1 a Department of Applied Mathematics, Indian School of Mines University, Dhanbad 826 004, Jharkhand, India b Department of Mathematics & Statistics, University of Missouri, Rolla, MO, 65409-0020, USA Accepted 1 June 2007 Abstract We propose a new mathematical model for aquatic populations. This model incorporates mutual interference in all the three populations and an extra mortality term in zooplankton population and also taking into account the toxin liberation process of TPP population. The proposed model generalizes several other known models in the literature. The principal interest in this paper is in a numerical study of the model’s behaviour. It is observed that both types of food chains display same type of chaotic behaviour, short-term recurrent chaos, with different generating mecha- nisms. Toxin producing phytoplankton (TPP) reduces the grazing pressure of zooplankton. To observe the role of TPP, we consider Holling types I, II and III functional forms for this process. Our study suggests that toxic substances released by TPP population may act as bio-control by changing the state of chaos to order and extinction. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Mathematical ecology has its roots in population ecology, which treats the increase and fluctuations of populations. Ecological model building takes many different forms, depending on the purpose for which it is done. These ecological models have been proposed and studied since the pioneering work of May [26,27]. In population ecology, the practice has been to formulate either the difference or differential equation models. The difference equation models describe the evolution of biological populations with non-overlapping generations. On the other hand, differential equation models correspond to populations with overlapping generations. The success of these models depends on the underlying ecological principles. The identification of general ecological principles in itself is a great challenge. Of late, there have been some thoughts on this issue [34,37,40]. Two ecological principles that form the skeleton of the model systems that we study in this paper are • a specialist predator population decays exponentially fast in the absence of its lone prey (i.e., favorite food); • the generalist predator switches to an alternative food option as and when it finds difficult to its favorite prey. The per capita growth of a generalist predator is limited by dependence on its favorite preys and severity of this limitation is inversely proportional to per capita availability of preys at any instant of time. 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.132 * Corresponding author. Tel.: +91 326 2200817; fax: +91 326 2210028. E-mail addresses: ranjit_ism@yahoo.com (R.K. Upadhyay), vshrao@yahoo.com (V. Sree Hari Rao). 1 On leave from Jawaharlal Nehru Technological University, Hyderabad - 500 072, India. Chaos, Solitons and Fractals 39 (2009) 1550–1564 www.elsevier.com/locate/chaos