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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.

Linked References

  1. P. Waltman, Competition Models in Population Biology, Society for Industrial and Applied Mathematics (SIAM), 1983. View at MathSciNet
  2. J. D. Murray, Mathematical Biology, vol. 3, Springer, New York, NY, USA, 2002.
  3. F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, NY, USA, 2001.
  4. D. Neal, Introduction to Population Biology, Cambridge University Press, Cambridge, UK, 2004.
  5. J. M. Smith, Models in Ecology, Cambridge Universty Press, 1974.
  6. H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, vol. 15, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  7. V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Application, Kluwer Academic, Dordrecht, The Netherlands, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  8. Y. Kuang, “Global stability of Gause-type predator-prey systems,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 463–474, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. S. Ahmad, “On the nonautonomous Volterra-Lotka competition equations,” Proceedings of the American Mathematical Society, vol. 117, no. 1, pp. 199–204, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. X. Tang and X. Zou, “On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments,” Proceedings of the American Mathematical Society, vol. 134, no. 10, pp. 2967–2974, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. Q. Din, “Dynamics of a discrete lotka-volterra model,” Advances in Difference Equations, vol. 2013, article no. 95, 2013. View at Publisher · View at Google Scholar · View at Scopus
  12. M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall, CRC, 2002. View at MathSciNet
  13. S. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 3rd edition, 2005. View at MathSciNet
  14. E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. View at MathSciNet
  15. J. F. Selgrade and M. Ziehe, “Convergence to equilibrium in a genetic model with differential viability between the sexes,” Journal of Mathematical Biology, vol. 25, no. 5, pp. 477–490, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. S. Kalabušić, M. R. S. Kulenović, and E. Pilav, “Dynamics of a two-dimensional system of rational difference equations of Leslie-Gower type,” Advances in Difference Equations, vol. 2011, article no. 29, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. Z. Zhou and X. Zou, “Stable periodic solutions in a discrete periodic logistic equation,” Applied Mathematics Letters, vol. 16, no. 2, pp. 165–171, 2003. View at Publisher · View at Google Scholar · View at Scopus
  18. X. Liu, “A note on the existence of periodic solutions in discrete predator-prey models,” Applied Mathematical Modelling, vol. 34, no. 9, pp. 2477–2483, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer, New York, NY, USA, 1983. View at MathSciNet
  20. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112 of Applied Mathematical Sciences, Springer, New York, NY, USA, 3rd edition, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  21. X. Liu and D. Xiao, “Complex dynamic behaviors of a discrete-time predator-prey system,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 80–94, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. Q. Khan, J. Ma, and D. Xiao, “Bifurcations of a two-dimensional discrete time plant-herbivore system,” Communications in Nonlinear Science and Numerical Simulation, vol. 39, pp. 185–198, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. A. Q. Khan, J. Ma, and D. Xiao, “Global dynamics and bifurcation analysis of a host--parasitoid model with strong Allee effect,” Journal of Biological Dynamics, vol. 11, no. 1, pp. 121–146, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus