Abstract and Applied Analysis

Bifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Terms

V. Hadžiabdić, M. R. S. Kulenović, and E. Pilav

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We investigate global dynamics of the following systems of difference equations x n + 1 = x n / A 1 + B 1 x n + C 1 y n , y n + 1 = y n 2 / A 2 + B 2 x n + C 2 y n 2 , n = 0,1 , , where the parameters A 1 , A 2 , B 1 , B 2 , C 1 , and C 2 are positive numbers and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.

Article information

Source
Abstr. Appl. Anal., Volume 2017 (2017), Article ID 3104512, 19 pages.

Dates
Received: 1 April 2017
Accepted: 4 June 2017
First available in Project Euclid: 19 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1505786568

Digital Object Identifier
doi:10.1155/2017/3104512

Mathematical Reviews number (MathSciNet)
MR3686354

Zentralblatt MATH identifier
06929547

Citation

Hadžiabdić, V.; Kulenović, M. R. S.; Pilav, E. Bifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Terms. Abstr. Appl. Anal. 2017 (2017), Article ID 3104512, 19 pages. doi:10.1155/2017/3104512. https://projecteuclid.org/euclid.aaa/1505786568


Export citation

References

  • M. R. Kulenović and O. Merino, “Invariant manifolds for competitive discrete systems in the plane,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 8, pp. 2471–2486, 2010.
  • J. M. Cushing, S. Levarge, N. Chitnis, and S. M. Henson, “Some discrete competition models and the competitive exclusion principle,” Journal of Difference Equations and Applications, vol. 10, no. 13–15, pp. 1139–1151, 2004.
  • M. R. Kulenović and O. Merino, “Competitive-exclusion versus competitive-coexistence for systems in the plane,” Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, vol. 6, no. 5, pp. 1141–1156, 2006.
  • A. Brett and M. R. Kulenovic, “Two species competitive model with the Allee effect,” Advances in Difference Equations, vol. 2014, article 307, 2014.
  • A. Brett and M. R. Kulenović, “Basins of attraction for two-species competitive model with quadratic terms and the singular Allee effect,” Discrete Dynamics in Nature and Society, Article ID 847360, 16 pages, 2015.
  • D. Burgić, M. R. Kulenović, and M. Nurkanović, “Global dynamics of a rational system of difference equations in the plane,” Communications on Applied Nonlinear Analysis, vol. 15, no. 1, pp. 71–84, 2008.
  • D. Clark and M. R. Kulenović, “A coupled system of rational difference equations,” Computers & Mathematics with Applications, vol. 43, no. 6-7, pp. 849–867, 2002.
  • J. E. Franke and A.-A. Yakubu, “Mutual exclusion versus coexistence for discrete competitive systems,” Journal of Mathematical Biology, vol. 30, no. 2, pp. 161–168, 1991.
  • J. E. Franke and A.-A. Yakubu, “Geometry of exclusion principles in discrete systems,” Journal of Mathematical Analysis and Applications, vol. 168, no. 2, pp. 385–400, 1992.
  • M. W. Hirsch and H. Smith, “Chapter 4 Monotone Dynamical Systems,” Handbook of Differential Equations: Ordinary Differential Equations, vol. 2, pp. 239–357, 2006.
  • S. Kalabušić, M. R. Kulenović, and E. Pilav, “Multiple attractors for a competitive system of rational difference equations in the plane,” Abstract and Applied Analysis, vol. 2011, Article ID 295308, 35 pages, 2011.
  • R. M. May and W. J. Leonard, “Nonlinear aspects of competition between three species,” SIAM Journal on Applied Mathematics, vol. 29, no. 2, pp. 243–253, 1975.
  • H. L. Smith, “Invariant curves for mappings,” SIAM Journal on Mathematical Analysis, vol. 17, no. 5, pp. 1053–1067, 1986.
  • H. L. Smith, “Periodic competitive differential equations and the discrete dynamics of competitive maps,” Journal of Differential Equations, vol. 64, no. 2, pp. 165–194, 1986.
  • H. L. Smith, “Periodic solutions of periodic competitive and cooperative systems,” SIAM Journal on Mathematical Analysis, vol. 17, no. 6, pp. 1289–1318, 1986.
  • H. L. Smith, “Planar competitive and cooperative difference equations,” Journal of Difference Equations and Applications, vol. 3, no. 5-6, pp. 335–357, 1998.
  • P. de Mottoni and A. Schiaffino, “Competition systems with periodic coefficients: a geometric approach,” Journal of Mathematical Biology, vol. 11, no. 3, pp. 319–335, 1981.
  • P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, vol. 247 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1991.
  • S. Basu and O. Merino, “On the global behavior of solutions to a planar system of difference equations,” Communications on Applied Nonlinear Analysis, vol. 16, no. 1, pp. 89–101, 2009.
  • V. Hadžiabdić, M. R. Kulenović, and E. Pilav, “Dynamics of a two-dimensional competitive system of rational difference equations with quadratic terms,” Advances in Difference Equations, vol. 301, p. 32, 2014. \endinput