Dynamical Behaviors of Discrete-time Prey- Predator System Harkaran Singh Department of Applied Sciences, Khalsa College of Engineering and Technology, Amritsar-143001, Punjab, India H.S. Bhatti Department of Applied Sciences, B.B.S.B. Engineering College, Fatehgarh Sahib, Punjab, India Abstract—In the present study, the dynamical behaviors of discrete-time prey-predator system. Global stability of the model at the fixed points has been discussed. The specific conditions for existence of flip bifurcation and Hopf bifurcation have been derived by using center manifold theorem and bifurcation theory. To analyse our results, numerical simulations have been carried out. Keywords—Prey-predator system, Center manifold theorem, Flip bifurcation, Hopf bifurcation, Chaos. I. INTRODUCTION The Prey-Predator model is a topic of great interest for many mathematicians and biologists which starts with the pioneer work of Lotka [1] and Volterra [2]. The dynamic relationship between predators and prey living in the same environment will continue to be one of the important themes in mathematical ecology [3-4]. Many authors [5-11] have suggested that the discrete time models are more appropriate than the continuous ones and provide efficient results when the populations have non-overlapping generations. However, there are few articles [12-18] discussing the dynamical behaviors of discrete-time predator–prey models by involving bifurcations and chaos phenomena. In the present study, we investigated the dynamical behaviors of discrete-time prey-predator model by involving bifurcations and chaos phenomena. This paper is organized as follows: In Section 2, we obtained the fixed points of the discrete-time model and discussed the stability criterion of the discrete-time model at fixed points. In Section 3, the specific conditions of existence of flip bifurcation and Hopf bifurcation have been derived. In Section 4, to analyse our results, numerical simulations have been carried out and further discussion on the period doubling bifurcation and chaotic behavior has been carried out. The discrete-time prey-predator model [19] is { = ( − ) − , = (− − ) + , (1.1) where and represents the densities of prey and predator respectively; , , , denotes the intrinsic growth rate of prey, capture rate, death rate of predator and the conversion rate respectively; and , denotes the intra-specific competition coefficients of prey and predator respectively. Applying forward Euler’s scheme to the system of equations (1.1), we obtain the system as { → + [( − ) − ], → + [(− − ) + ]. (1.2) II. STABILITY OF THE FIXED POINTS The fixed points of the system (1.2) are (0,0), ( , 0) , ( ∗ , ∗ ), where ∗ = + + , ∗ = − + . The jacobian matrix of (1.2) at the fixed point (, ) is given by =[ 1 + ( − 2 − ) − 1 + (− − 2 + ) ]. The characteristic equation of the jacobian matrix can be written as 2 + (, ) + (, ) = 0, (2.1) where (, ) = − = −2 − + + (2 − ) + ( + 2), (, ) = = [1 + ( − 2 − )][1 + (− − 2 + )] + 2 Lemma 2.1: Let () = 2 + + . Suppose that (1) > 0, 1 and 2 are roots of () = 0. Then (i) | 1 |<1 and | 2 |<1 if and only if (−1) > 0 and <1; (ii) | 1 |<1 and | 2 |>1 (or | 1 |>1 and | 2 |<1) if and only if (−1) < 0; (iii) | 1 |>1 and | 2 |>1 if and only if (−1) > 0 and >1; International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Published by, www.ijert.org ESDST - 2017 Conference Proceedings Volume 5, Issue 05 Special Issue - 2017