Nonlinear Analysis: Hybrid Systems 3 (2009) 74–86 Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs Integration of mussel in fish farm: Mathematical model and analysis Nurul Huda Gazi a,∗ , Safiur Rahaman Khan b , Charu Gopal Chakrabarti b a St. Xavier’s College, 30, Mother Teresa Sarani, Kolkata - 700 016, India b Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata - 700 009, India article info Article history: Received 23 July 2008 Accepted 31 October 2008 Keywords: Fish farm Nonlinear differential equation model Eutrophication Stability Time delay Oscillation Hopf-bifurcation abstract The paper deals with the dynamical behavior of fish and mussel population in a fish farm where external food is supplied. The ecosystem of the fish farm is represented by a set of nonlinear differential equations involving the nutrient (food), fish and mussels. We have studied the boundedness, local stability and global stability of the model system. We have incorporated the discrete type gestational delay of fish and analyze effect of the delay on the dynamical behavior of the model system. The delay parameter complicates the dynamics depending on the external food from changing the stable state to unstable damped periodic trajectories leading to a limit cycle oscillation. We have studied the Hopf-bifurcation of the model system in the neighborhood of the coexisting equilibrium point considering delay as a variable bifurcation parameter. We have performed numerical simulation to verify the analytical results. The entire study reveals that the external food supply controls the dynamics of the system. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction There is a necessity for sustainable development in aquaculture for both individual livelihoods and the economy. Most of the European countries have taken some strategy towards the development of aquaculture. There are several constraints, one, the most important, being the environmental concern linked to the location of the fish farm and the impact of their effluents on the surrounding environment. Nutrient pollution from aquacultural waste exceeds the assimilation capacity of receiving water deteriorating water quality [1]. Past studies show several types of impact of fish farming in the Mediterranean region. [2–5] and the references cited therein. Karakassis et al. [6] studied the potential impact of fish farming on nutrient content and concluded that fish farm waste can cause 1% on nutrient concentration in contrast to other anthropogenic activities. There is a greater impact of detritivorous fish on the estuarian ecosystem [7]. Samanta et al. [8] investigated how the maximum amount of economically important species can be harvested from a fishery. Bandyopadhyay et al. [9] investigated the dynamics of an ecological system where the nutrient has a very important role in controlling the behavior of the system. Lakes Erie, Michigan, Huron, Superior, and Ontario are well known important sources of fresh water and are home to many species of wildlife. However, with the belief that water could dilute any substance, the lakes have become destinations of dumping grounds for many different types of pollutants. With so many different sources of pollutants, ranging from industrial waste, pesticide and fertilizer runoff, and fecal matter, it is not surprising the extent to which these contaminants have affected the wildlife and ecosystem surrounding the Great Lakes. As seen in the biomass of Lake Huron trout, pollution directly influences the wildlife and environment surrounding the lakes. Such influences include promoting the abundance ∗ Corresponding author. Tel.: +91 33 3296 7065; fax: +91 33 2287 9966. E-mail addresses: nursha@rediffmail.com (N. Huda Gazi), safiurkhan@gmail.com (S. Rahaman Khan), cgc_math@rediffmail.com (C. Gopal Chakrabarti). 1751-570X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2008.10.008