Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 5963594, 15 pages
https://doi.org/10.1155/2017/5963594
Research Article

Global Asymptotic Stability for Discrete Single Species Population Models

Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA

Correspondence should be addressed to M. R. S. Kulenović; ude.iru.liam@civonelukm

Received 28 March 2017; Accepted 16 May 2017; Published 13 June 2017

Academic Editor: Rigoberto Medina

Copyright © 2017 A. Bilgin and M. R. S. Kulenović. 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. L. J. S. Allen, An Introduction to Mathematical Biology, Prentice Hall, 2006.
  2. S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005. View at MathSciNet
  3. M. P. Hassell, H. N. Comins, and R. M. May, “Spatial structure and chaos in insect population dynamics,” Nature, vol. 353, no. 6341, pp. 255–258, 1991. View at Publisher · View at Google Scholar · View at Scopus
  4. R. M. May, “Stability and Complexity in Model Ecosystems,” Princeton University Press Princeton, Princeton, NJ, USA, 2001. View at Google Scholar
  5. 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
  6. H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, USA, 2003. View at MathSciNet
  7. M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. View at MathSciNet
  8. M. R. Kulenović and O. Merino, Discrete dynamical systems and difference equations with Mathematica, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. E. Franke and J. F. Selgrade, “Attractors for discrete periodic dynamical systems,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 64–79, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. E. A. Grove, Y. Kostrov, G. Ladas, and S. W. Schultz, “Riccati difference equations with real period-2 coefficients,” Communications on Applied Nonlinear Analysis, vol. 14, no. 2, pp. 33–56, 2007. View at Google Scholar · View at MathSciNet
  11. J. M. Cushing, “Periodically forced nonlinear systems of difference equations,” Journal of Difference Equations and Applications, vol. 3, no. 5-6, pp. 547–561, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  12. D. A. Jillson, “Insect populations respond to fluctuating environments,” Nature, vol. 288, no. 5792, pp. 699-700, 1980. View at Publisher · View at Google Scholar · View at Scopus
  13. J. M. Cushing and S. M. Henson, “Global dynamics of some periodically forced, monotone difference equations,” Journal of Difference Equations and Applications, vol. 7, no. 6, pp. 859–872, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  14. S. M. Henson and J. M. Cushing, “The effect of periodic habitat fluctuations on a nonlinear insect population model,” Journal of Mathematical Biology, vol. 36, no. 2, pp. 201–226, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  15. J. E. Franke and A.-A. Yakubu, “Attenuant cycles in periodically forced discrete-time age-structured population models,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 69–86, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. J. E. Franke and A.-A. Yakubu, “Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models,” Mathematical Biosciences, vol. 204, no. 1, pp. 1–20, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. J. E. Franke and A.-A. Yakubu, “Signature function for predicting resonant and attenuant population 2-cycles,” Bulletin of Mathematical Biology, vol. 68, no. 8, pp. 2069–2104, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. J. M. Cushing, “The Allee effect in age—structured population dynamics,” in Mathematical ecology, T. G. Hallam, L. J. Gross, and S. A. Levin, Eds., pp. 479–505, World Scientific Publishing Co., Singapore, 1988. View at Google Scholar · View at MathSciNet
  19. C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 97, supplement 45, pp. 5–60, 1965. View at Publisher · View at Google Scholar
  20. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  21. E. N. Dancer and P. Hess, “Stability of fixed points for order-preserving discrete-time dynamical systems,” Journal fur die Reine und Angewandte Mathematik, vol. 1991, no. 419, pp. 125–140, 1991. View at Publisher · View at Google Scholar · View at Scopus
  22. M. W. Hirsch and H. Smith, Monotone Dynamical Systems, Handbook of Differential Equations, Ordinary Differential Equations (second volume), Elsevier, Amsterdam, the Netherlands, 2005. View at Publisher · View at Google Scholar · View at Scopus
  23. M. W. Hirsch and H. Smith, “Monotone maps: a review,” Journal of Difference Equations and Applications, vol. 11, no. 4-5, pp. 379–398, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. M. R. S. Kulenović and O. Merino, “A global attractivity result for maps with invariant boxes,” Discrete and Continuous Dynamical Systems - Series B, vol. 6, no. 1, pp. 97–110, 2006. View at Publisher · View at Google Scholar · View at Scopus
  25. R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY. USA, 2nd edition, 2000. View at MathSciNet
  26. V. Lakshmikantham and D. Trigiante, Theory of difference equations: numerical methods and applications, vol. 251 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, NY, USA, 2nd edition, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  27. M. R. Kulenović and A.-A. Yakubu, “Compensatory versus overcompensatory dynamics in density-dependent Leslie models,” Journal of Difference Equations and Applications, vol. 10, no. 13-15, pp. 1251–1265, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  28. E. A. Grove, C. M. Kent, G. Ladas, S. Valicenti, and R. Levins, “Global stability in some population models,” in Communications in difference equations (Poznan, 1998), pp. 149–176, Gordon and Breach, Amsterdam, 2000. View at Google Scholar · View at MathSciNet
  29. E. J. Janowski and M. R. Kulenović, “Attractivity and global stability for linearizable difference equations,” Computers & Mathematics with Applications. An International Journal, vol. 57, no. 9, pp. 1592–1607, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  30. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  31. H. L. Smith, “Planar competitive and cooperative difference equations,” Journal of Difference Equations and Applications, vol. 3, no. 5-6, pp. 335–357, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  32. H. L. Smith, “Invariant curves for mappings,” SIAM Journal on Mathematical Analysis, vol. 17, no. 5, pp. 1053–1067, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  33. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. M. P. Hassell and H. N. Comins, “Discrete time models for two-species competition,” Theoretical Population Biology, vol. 9, no. 2, pp. 202–221, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  35. P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, vol. 247 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1991. View at MathSciNet
  36. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  37. A. Bilgin, M. R. Kulenović, and E. Pilav, “Basins of attraction of period-two solutions of monotone difference equations,” Advances in Difference Equations, vol. 2016, no. 74, 25 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  38. A. M. Amleh, E. Camouzis, and G. Ladas, “On the Dynamics of a Rational Difference Equation, Part I,” Journal of Difference Equations and Applications, vol. 3, pp. 1–35, 2008. View at Publisher · View at Google Scholar · View at Scopus