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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 147926, 17 pages
http://dx.doi.org/10.1155/2011/147926
Research Article

On a Difference Equation with Exponentially Decreasing Nonlinearity

1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W, Calgary, AB, Canada T2N 1N4
2Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia
3Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 7 April 2011; Accepted 1 June 2011

Academic Editor: Antonia Vecchio

Copyright © 2011 E. Braverman and S. H. Saker. 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. J. Losson, M. C. Mackey, and A. Longtin, “Solution multistability in first-order nonlinear differential delay equations,” Chaos, vol. 3, no. 2, pp. 167–176, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197, no. 4300, pp. 287–289, 1977. View at Google Scholar
  3. M. Ważewska-Czyżewska and A. Lasota, “Mathematical problems of the dynamics of the red blood cells system,” Applied Mathematics, vol. 17, pp. 23–40, 1976, Annals of the Polish Mathematical Society Series III. View at Google Scholar
  4. M. R. S. Kulenović and G. Ladas, “Linearized oscillations in population dynamics,” Bulletin of Mathematical Biology, vol. 49, no. 5, pp. 615–627, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. K. Gopalsamy and S. I. Trofimchuk, “Almost periodic solutions of Lasota-Wazewska-type delay differential equation,” Journal of Mathematical Analysis and Applications, vol. 237, no. 1, pp. 106–127, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. G. Liu, A. Zhao, and J. Yan, “Existence and global attractivity of unique positive periodic solution for a Lasota-Wazewska model,” Nonlinear Analysis, vol. 64, no. 8, pp. 1737–1746, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. E. Liz, C. Martínez, and S. Trofimchuk, “Attractivity properties of infinite delay Mackey-Glass type equations,” Differential and Integral Equations, vol. 15, no. 7, pp. 875–896, 2002. View at Google Scholar · View at Zentralblatt MATH
  8. E. Liz, M. Pinto, V. Tkachenko, and S. Trofimchuk, “A global stability criterion for a family of delayed population models,” Quarterly of Applied Mathematics, vol. 63, no. 1, pp. 56–70, 2005. View at Google Scholar · View at Zentralblatt MATH
  9. I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, New York, NY, USA, 1991.
  10. L. Berezansky and E. Braverman, “Linearized oscillation theory for a nonlinear delay impulsive equation,” Journal of Computational and Applied Mathematics, vol. 161, no. 2, pp. 477–495, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. X. Liu and Y. Takeuchi, “Periodicity and global dynamics of an impulsive delay Lasota-Wazewska model,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 326–341, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. G. Tr. Stamov, “On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model,” Applied Mathematics Letters, vol. 22, no. 4, pp. 516–520, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. Yan, “Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model,” Journal of Mathematical Analysis and Applications, vol. 279, no. 1, pp. 111–120, 2003. View at Publisher · View at Google Scholar · View at Scopus
  14. L. Berezansky and E. Braverman, “Linearized oscillation theory for a nonlinear equation with a distributed delay,” Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 287–304, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. X. Wang and Z. Li, “Global dynamical behavior for discrete Lasota-Wazewska model with several delays and almost periodic coefficients,” International Journal of Biomathematics, vol. 1, no. 1, pp. 95–105, 2008. View at Publisher · View at Google Scholar
  16. I. Kubiaczyk and S. H. Saker, “Oscillation and global attractivity in a discrete survival red blood cells model,” Applicationes Mathematicae, vol. 30, no. 4, pp. 441–449, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. W.-T. Li and S. S. Cheng, “Asymptotic properties of the positive equilibrium of a discrete survival model,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 29–38, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.
  19. V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order, vol. 256 of Mathematics and Its Applications, Kluwer Academic, Dodrecht, The Netherlands, 1993.
  20. G. Karakostas, Ch. G. Philos, and Y. G. Sficas, “The dynamics of some discrete population models,” Nonlinear Analysis, vol. 17, no. 11, pp. 1064–1084, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. X. Y. Zheng, B. Shi et al., “Oscillation and global attractivity on a rational recursive sequence,” in Proceedings of the 7th International Conference on Difference Equations and Applications, A Satellite Conference of the 2002 Beiging International Congress of Mathematicians, Changsha, China, Augus 2002.
  22. M. Ma and J. Yu, “Global attractivity of xn+1=(1αxn)+βexp(γxnk),” Computers & Mathematics with Applications, vol. 49, no. 9-10, pp. 1397–1402, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. Q. Meng and J. R. Yan, “Global attractivity of delay difference equations,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 3, pp. 233–242, 1999. View at Google Scholar · View at Zentralblatt MATH
  24. A. F. Ivanov, “On global stability in a nonlinear discrete model,” Nonlinear Analysis, vol. 23, no. 11, pp. 1383–1389, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. I. Györi and S. I. Trofimchuk, “Global attractivity in x(t)=δx(t)+pf(x(tτ)),” Dynamic Systems and Applications, vol. 8, no. 2, pp. 197–210, 1999. View at Google Scholar
  26. H. A. El-Morshedy and E. Liz, “Convergence to equilibria in discrete population models,” Journal of Difference Equations and Applications, vol. 11, no. 2, pp. 117–131, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. S. H. Saker, “Qualitative analysis of discrete nonlinear delay survival red blood cells model,” Nonlinear Analysis, vol. 9, no. 2, pp. 471–489, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. P. Cull, “Local and global stability for population models,” Biological Cybernetics, vol. 54, no. 3, pp. 141–149, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. P. Cull, “Population models: stability in one dimension,” Bulletin of Mathematical Biology, vol. 69, no. 3, pp. 989–1017, 2007. View at Publisher · View at Google Scholar
  30. A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dynamics of One-Dimensional Maps, vol. 407, Kluwer Academic, Dodrecht, The Netherlands, 1997.
  31. D. Singer, “Stable orbits and bifurcation of maps of the interval,” SIAM Journal on Applied Mathematics, vol. 35, no. 2, pp. 260–267, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. W. A. Coppel, “The solution of equations by iteration,” Proceedings of the Cambridge Philosophical Society, vol. 51, pp. 41–43, 1955. View at Google Scholar · View at Zentralblatt MATH
  33. W. E. Ricker, “Stock and recruitment,” Journal of the Fisheries Research Board of Canada, vol. 11, pp. 559–623, 1954. View at Google Scholar
  34. T. S. Bellows, Jr., “The descriptive properties of some models for density dependence,” The Journal of Animal Ecology, vol. 50, no. 1, pp. 139–156, 1981. View at Publisher · View at Google Scholar
  35. H. I. McCallum, “Effects of immigration on chaotic population dynamics,” Journal of Theoretical Biology, vol. 154, no. 3, pp. 277–284, 1992. View at Google Scholar
  36. L. Stone, “Period-doubling reversals and chaos in simple ecological models,” Nature, vol. 365, no. 6447, pp. 617–620, 1993. View at Google Scholar
  37. E. Braverman and D. Kinzebulatov, “On linear perturbations of the Ricker model,” Mathematical Biosciences, vol. 202, no. 2, pp. 323–339, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. S. J. Schreiber, “Chaos and population disappearances in simple ecological models,” Journal of Mathematical Biology, vol. 42, no. 3, pp. 239–260, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. E. Braverman and S. H. Saker, “Permanence, oscillation and attractivity of the discrete hematopoiesis model with variable coefficients,” Nonlinear Analysis, vol. 67, no. 10, pp. 2955–2965, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  40. E. Liz, “Local stability implies global stability in some one-dimensional discrete single-species models,” Discrete and Continuous Dynamical Systems. Series B, vol. 7, no. 1, pp. 191–199, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH