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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 482459, 14 pages
Chaos in a Discrete Delay Population Model
Department of Mathematics, Shandong Jianzhu University, Shandong, Jinan 250101, China
Received 2 August 2012; Accepted 21 August 2012
Academic Editor: Hua Su
Copyright © 2012 Zongcheng Li 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.
- K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, The Netherlands, 1992.
- K. Gopalsamy and G. Ladas, “On the oscillation and asymptotic behavior of ,” Quarterly of Applied Mathematics, vol. 48, no. 3, pp. 433–440, 1990.
- Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, NY, USA, 1993.
- M. Jiang, Y. Shen, J. Jian, and X. Liao, “Stability, bifurcation and a new chaos in the logistic differential equation with delay,” Physics Letters A, vol. 350, no. 3-4, pp. 221–227, 2006.
- L. Fan, Z. K. Shi, and S. Y. Tang, “Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters,” Nonlinear Analysis, vol. 11, no. 1, pp. 341–355, 2010.
- H. Seno, “A paradox in discrete single species population dynamics with harvesting/thinning,” Mathematical Biosciences, vol. 214, no. 1-2, pp. 63–69, 2008.
- E. Liz and P. Pilarczyk, “Global dynamics in a stage-structured discrete-time population model with harvesting,” Journal of Theoretical Biology, vol. 297, pp. 148–165, 2012.
- I. W. Rodrigues, “Oscillation and attractivity in a discrete model with quadratic nonlinearity,” Applicable Analysis, vol. 47, no. 1, pp. 45–55, 1992.
- L. H. Huang and M. S. Peng, “Qualitative analysis of a discrete population model,” Acta Mathematica Scientia B, vol. 19, no. 1, pp. 45–52, 1999.
- Z. M. He and X. Lai, “Bifurcation and chaotic behavior of a discrete-time predator-prey system,” Nonlinear Analysis, vol. 12, no. 1, pp. 403–417, 2011.
- M. S. Peng, “Multiple bifurcations and periodic “bubbling” in a delay population model,” Chaos, Solitons and Fractals, vol. 25, no. 5, pp. 1123–1130, 2005.
- M. S. Peng and A. Uçar, “The use of the Euler method in identification of multiple bifurcations and chaotic behavior in numerical approximation of delay-differential equations,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 883–891, 2004.
- M. S. Peng, “Symmetry breaking, bifurcations, periodicity and chaos in the Euler method for a class of delay differential equations,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1287–1297, 2005.
- M. S. Peng and Y. Yuan, “Stability, symmetry-breaking bifurcations and chaos in discrete delayed models,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 18, no. 5, pp. 1477–1501, 2008.
- M. S. Peng, J. C. Yu, and X. J. Wang, “Complex dynamics in simple delayed two-parameterized models,” Nonlinear Analysis, vol. 13, no. 6, pp. 2530–2539, 2012.
- D. J. Fan and J. J. Wei, “Bifurcation analysis of discrete survival red blood cells model,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3358–3368, 2009.
- T. Y. Li and J. A. Yorke, “Period three implies chaos,” The American Mathematical Monthly, vol. 82, no. 10, pp. 985–992, 1975.
- J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, “On Devaney's definition of chaos,” The American Mathematical Monthly, vol. 99, no. 4, pp. 332–334, 1992.
- R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, New York, NY, USA, 1987.
- P. E. Kloeden and Z. Li, “Li-Yorke chaos in higher dimensions: a review,” Journal of Difference Equations and Applications, vol. 12, no. 3-4, pp. 247–269, 2006.
- M. Martelli, M. Dang, and T. Seph, “Defining chaos,” Mathematics Magazine, vol. 71, no. 2, pp. 112–122, 1998.
- C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Boca Raton, Fla, USA, 1995.
- S. Wiggins, Global Bifurcations and Chaos, vol. 73, Springer, New York, NY, USA, 1988.
- W. Huang and X. D. Ye, “Devaney's chaos or 2-scattering implies Li-Yorke's chaos,” Topology and its Applications, vol. 117, no. 3, pp. 259–272, 2002.
- Y. M. Shi and G. R. Chen, “Chaos of discrete dynamical systems in complete metric spaces,” Chaos, Solitons and Fractals, vol. 22, no. 3, pp. 555–571, 2004.
- F. R. Marotto, “Snap-back repellers imply chaos in ,” Journal of Mathematical Analysis and Applications, vol. 63, no. 1, pp. 199–223, 1978.
- W. Rudin, Functional Analysis, McGraw-Hill, New York, NY, USA, 1973.
- Y. M. Shi and G. R. Chen, “Discrete chaos in Banach spaces,” Science in China, vol. 48, no. 2, pp. 222–238, 2005, Chinese version, vol. 34, pp. 595–609, 2004.
- Y. M. Shi and P. Yu, “Chaos induced by regular snap-back repellers,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1480–1494, 2008.
- Y. M. Shi and P. Yu, “Study on chaos induced by turbulent maps in noncompact sets,” Chaos, Solitons and Fractals, vol. 28, no. 5, pp. 1165–1180, 2006.
- Y. M. Shi, P. Yu, and G. R. Chen, “Chaotification of discrete dynamical systems in banach spaces,” International Journal of Bifurcation and Chaos, vol. 16, no. 9, pp. 2615–2636, 2006.
- Y. Huang and X. F. Zou, “Co-existence of chaos and stable periodic orbits in a simple discrete neural network,” Journal of Nonlinear Science, vol. 15, no. 5, pp. 291–303, 2005.