International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 4, Issue 2, February 2016, PP 42-46 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org ©ARC Page | 42 New Insight of Discrete-Time Food Chain Interaction Model Alaa Hussein Lafta Science College / Mathematics Department Baghdad University Baghdad, Iraq alaah74@yahoo.com V. C. Borkar Yeshwant Mahavidyalaya / Mathematics & Statistics Department SRTM University Nanded, India Abstract: The discrete- time food chain interaction model is proposed. Steady state implies all possible equilibrium points. Stability conditions of arising equilibrium points are analyzed with numerical examples. Further, the dynamical behavior of the coexistence equilibrium point is obtained. Keywords: discrete- time model, food chain, Stability theory, equilibrium point. 1. INTRODUCTION Linear Lotka-Volterra model [1] consists of two interacting species vis. Prey and predator. Such mathematical models have long proved useful in describing how populations vary over time [2]. The other basic sequence of energy is the food chain which describes the movement from producer to consumer then to decomposer which called lowest-level, mid-level and top-level, respectively. The chain food is more important in understanding the food and energy relationship in the ecosystems. The continuous time food chain of three or more species arise in (Hastings et al.1991, McCann et al.1994, Kuznetsov et al.1996, El-Owaidy 2001, and Chauvet 2002) while the Lotka-Volterra discrete-time food chain model proposed in ( Abd-Elalim et al. 2012, Hari 2014, and Sohel 2015). In [10] we studied a non-linear discrete-time prey-predator model such that each population cans model by the logistic equation. For more insight and covering three species ecosystems, we extend our model in [10] by adding third population. The new model includes the lowest-level prey is preyed upon by a mid-level species , which, in turn, preyed upon by a top-level predator as shown in the following system: (1) Where and are all positive parameters. 2. STEADY STATE In common parlance the system (1) has steady state if it is equal to the vector where .Thus, this implies the following equilibrium points: i. Fully washed out state ii- States in which only one of the three species is survives while the other two are not