The role of top predator interference on the dynamics of a food chain model Ranjit Kumar Upadhyay a , Raid Kamel Naji b,⇑ , Sharada Nandn Raw a , Balram Dubey c a Department of Applied Mathematics, Indian School of Mines, Dhanbad 826 004, India b Department of Mathematics, College of Science, University of Baghdad, Iraq c Department of Mathematics, BITS, Pilani 333 031, India article info Article history: Received 10 February 2012 Received in revised form 6 August 2012 Accepted 19 August 2012 Available online 8 September 2012 Keywords: Food chain Beddington–DeAngelis type functional response Top predator interference Leslie–Gower abstract In this paper, the effects of top predator interference on the dynamics of a food chain model involving an intermediate and a top predator are considered. It is assumed that the inter- action between the prey and intermediate predator follows the Volterra scheme, while that between the top predator and its favorite food depends on Beddington–DeAngelis type of functional response. The boundedness of the system, existence of an attracting set, local and global stability of non-negative equilibrium points are established. Number of the bifurcation and Lyapunov exponent bifurcation diagrams is established. It is observed that, the model has different types of attracting sets including chaos. Moreover, increasing the top predator interference stabilizes the system, while increasing the normalization of the residual reduction in the top predator population destabilizes the system. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Understanding the dynamic relationship between prey and predator in a food-chain model has long been and will con- tinue to be one of the dominant themes in both ecology and mathematical biology. One of the significant components of the predator–prey relationship is the predator’s rate of feeding upon prey. Among the four basic interactions (predator–prey, competition, interference and mutualism), predator–prey interaction is the most common and is well known for generating oscillatory dynamics. Food-chain predator–prey system has been extensively studied by many researchers and the results are concerned with persistence and permanence properties, stability analysis, periodic solution and global dynamic behavior of the system [1–3] and the references cited therein. Recently Liu and Wang [4] studied the global stability of a nonlinear stochastic stage-structured predator–prey system with Beddington–DeAngelis functional response. They have considered the stage structure only in prey population. Their re- sults show that if the positive equilibrium of the deterministic system is globally stable then the stochastic model will pre- serve this provided the environmental noise is sufficiently small [5]. Guo and Chen [6] have studied the global attractivity of positive solution for a Volterra model with mutual interference and Beddington–DeAngelis functional response. Positive periodic solutions of neutral predator–prey model with Beddington–DeAngelis functional response have been studied by Liu and Yan [7]. Li and Takeuchi [8] have studied the dynamics of the density dependent predator–prey system. Bedding- ton–DeAngelis type functional response is also used to model a class of virus dynamics model with nonlinear infection rate [9–11]. It is well known that the Rasenzweig–Macarthur (RM) model describes the dynamics of a predator and its lone prey, how- ever, a model given by Holling and Tanner describes the dynamics of a predator and their preys. Although, in the last decade some experts have studied predator–prey system with the Beddington–DeAngelis functional response [12–17], here we have 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.08.020 ⇑ Corresponding author. Tel.: +964 7901868685. E-mail addresses: ranjit_ism@yahoo.com (R.K. Upadhyay), rknaji@gmail.com (R.K. Naji), bdubey@bits-pilani.ac.in (B. Dubey). Commun Nonlinear Sci Numer Simulat 18 (2013) 757–768 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns