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Complexity
Volume 2018, Article ID 7101505, 18 pages
https://doi.org/10.1155/2018/7101505
Research Article

Global Dynamics and Bifurcations Analysis of a Two-Dimensional Discrete-Time Lotka-Volterra Model

Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan

Correspondence should be addressed to A. Q. Khan; moc.liamg@1nahkreedaqludba

Received 23 August 2017; Revised 11 December 2017; Accepted 19 December 2017; Published 21 January 2018

Academic Editor: Abraham J. A. Tawil

Copyright © 2018 A. Q. Khan and M. N. Qureshi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, global dynamics and bifurcations of a two-dimensional discrete-time Lotka-Volterra model have been studied in the closed first quadrant . It is proved that the discrete model has three boundary equilibria and one unique positive equilibrium under certain parametric conditions. We have investigated the local stability of boundary equilibria , , and the unique positive equilibrium , by the method of linearization. It is proved that the discrete model undergoes a period-doubling bifurcation in a small neighborhood of boundary equilibria and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium Further it is shown that every positive solution of the discrete model is bounded and the set is an invariant rectangle. It is proved that if and , then equilibrium of the discrete model is a global attractor. Finally it is proved that the unique positive equilibrium is a global attractor. Some numerical simulations are presented to illustrate theoretical results.