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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 654619, 13 pages
http://dx.doi.org/10.1155/2013/654619
Research Article

Dynamic Analysis of a Two-Language Competitive Model with Control Strategies

1College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China
2Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
3Departamento de Análisis Matemático and Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
4Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 2 August 2013; Accepted 7 October 2013

Academic Editor: Bashir Ahmad

Copyright © 2013 Lin-Fei Nie et al. 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. W. J. Sutherland, “Parallel extinction risk and global distribution of languages and species,” Nature, vol. 423, no. 6937, pp. 276–279, 2003. View at Publisher · View at Google Scholar · View at Scopus
  2. D. M. Abrams and S. H. Strogatz, “Modelling the dynamics of language death,” Nature, vol. 424, no. 6951, p. 900, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. X. Castelló, V. M. Eguíluz, and M. San Miguel, “Ordering dynamics with two non-excluding options: Bilingualism in language competition,” New Journal of Physics, vol. 8, article 308, 2006. View at Publisher · View at Google Scholar · View at Scopus
  4. I. Dyen, J. B. Kruskal, and P. Black, “An indoeuropean classication: a lexicostatistical experiment,” American Philosophical Society, vol. 82, p. 5, 1992. View at Google Scholar
  5. T. Gong, L. Shuai, M. Tamariz, and G. Jäger, “Studying language change using Price equation and Pólya-urn dynamics,” PLoS ONE, vol. 7, no. 3, Article ID e33171, 2012. View at Publisher · View at Google Scholar · View at Scopus
  6. J. A. Hawkins, “Innateness and function in language universals,” in The Evolution of Human Languages, J. A. Hawkins and G. M. Murray, Eds., Addison-Wesley, Reading, Mass, USA, 1992. View at Google Scholar
  7. D. Helbing, A. Szolnoki, M. Perc, and G. Szabó, “Evolutionary establishment of moral and double moral standards through spatial interactions,” PLoS Computational Biology, vol. 6, no. 4, Article ID e1000758, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. J. Mira and A. Plaredes, “Interlinguistic similarity and language death dynamics,” Europhysics Letters, vol. 69, no. 6, pp. 1031–1034, 2005. View at Publisher · View at Google Scholar · View at Scopus
  9. M. A. Nowak, N. L. Komarova, and P. Niyogi, “Computational and evolutionary aspects of language,” Nature, vol. 417, no. 6889, pp. 611–617, 2002. View at Publisher · View at Google Scholar · View at Scopus
  10. F. Petroni and M. Serva, “Language distance and tree reconstruction,” Journal of Statistical Mechanics, vol. 2008, no. 8, Article ID P08012, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Schulze, D. Stauffer, and S. Wichmann, “Birth, survival and death of languages by Monte Carlo simulation,” Communications in Computational Physics, vol. 3, no. 2, pp. 271–294, 2008. View at Google Scholar · View at Scopus
  12. M. Patriarca and E. Heinsalu, “Influence of geography on language competition,” Physica A, vol. 388, no. 2-3, pp. 174–186, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. M. Patriarca and T. Leppänen, “Modeling language competition,” Physica A, vol. 338, no. 1-2, pp. 296–299, 2004. View at Publisher · View at Google Scholar · View at Scopus
  14. D. Stauffer, X. Castelló, V. M. Eguíluz, and M. San Miguel, “Microscopic Abrams-Strogatz model of language competition,” Physica A, vol. 374, no. 2, pp. 835–842, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. J. Mira, L. F. Seoane, and J. J. Nieto, “The importance of interlinguistic similarity and stable bilingualism when two languages compete,” New Journal of Physics, vol. 13, Article ID 033007, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. V. Otero-Espinar, L. F. Seoane, J. J. Nieto, and J. Mira, “An analytic solution of a model of language competition with bilingualism and interlinguistic similarity,” Physica D, vol. 264, pp. 17–26, 2013. View at Google Scholar
  17. K. Blayneh, Y. Cao, and H.-D. Kwon, “Optimal control of vector-borne diseases: treatment and prevention,” Discrete and Continuous Dynamical Systems B, vol. 11, no. 3, pp. 587–611, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. H. R. Joshi, “Optimal control of an HIV immunology model,” Optimal Control Applications and Methods, vol. 23, no. 4, pp. 199–213, 2002. View at Publisher · View at Google Scholar · View at Scopus
  19. T. K. Kar and A. Batabyal, “Stability analysis and optimal control of an SIR epidemic model with vaccination,” BioSystems, vol. 104, no. 2-3, pp. 127–135, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. D. Kirschner, S. Lenhart, and S. Serbin, “Optimal control of the chemotherapy of HIV,” Journal of Mathematical Biology, vol. 35, no. 7, pp. 775–792, 1997. View at Google Scholar · View at Scopus
  21. A. A. Lashari and G. Zaman, “Optimal control of a vector borne disease with horizontal transmission,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 203–212, 2012. View at Publisher · View at Google Scholar · View at Scopus
  22. M. Ozari, A. A. Lashari, I. H. Jung, and K. O. Okosun, “Stability analysis and optimalcontrol of a Vector-Borne disease with nonlinear incidence,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 595487, 21 pages, 2012. View at Publisher · View at Google Scholar
  23. G. Zaman, Y. Han Kang, and I. H. Jung, “Stability analysis and optimal vaccination of an SIR epidemic model,” BioSystems, vol. 93, no. 3, pp. 240–249, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. E. Jung, S. Lenhart, and Z. Feng, “Optimal control of treatments in a two-strain tuberculosis model,” Discrete and Continuous Dynamical Systems B, vol. 2, no. 4, pp. 473–482, 2002. View at Google Scholar · View at Scopus
  25. L. F. Nie, Z. D. Teng, and B. Z. Guo, “A state dependent pulse control strategy for a SIRS epidemic system,” Bulletin of Mathematical Biology, vol. 75, no. 10, pp. 1697–1715, 2013. View at Google Scholar
  26. L. Nie, Z. Teng, and A. Torres, “Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1621–1629, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC, London, UK, 2007.
  28. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Process, vol. 14, Gordon & Breach Science, New York, NY, USA, 1986.
  29. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  30. J. Hale and H. Kocak, Dynamics and Bifurcations, Springer, New York, NY, USA, 1991.
  31. P. S. Simeonov and D. D. Bainov, “Orbital stability of periodic solutions of autonomous systems with impulse effect,” International Journal of Systems Science, vol. 19, no. 7, pp. 2561–2585, 1988. View at Google Scholar · View at Scopus