About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 402194, 10 pages
http://dx.doi.org/10.1155/2014/402194
Research Article

Dynamical Analysis of a Plateau Pika with Cross-Diffusion under Contraception Control

1School of Information and Communication Engineering, North University of China, Shanxi, Taiyuan 030051, China
2Department of Applied Mathematics, Yuncheng University, Shanxi, Yuncheng 044000, China

Received 6 July 2013; Revised 25 December 2013; Accepted 26 December 2013; Published 18 February 2014

Academic Editor: Seenith Sivasundaram

Copyright © 2014 Xiaoyan Wang 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. A. T. Smith and J. M. Foggin, “The plateau pika (Ochotona curzoniae) is a keystone species for biodiversity on the Tibetan plateau,” Animal Conservation, vol. 2, no. 4, pp. 235–240, 1999. View at Publisher · View at Google Scholar · View at Scopus
  2. C. H. Lai and A. T. Smith, “Keystone status of plateau pikas (Ochotona curzoniae): effect of control on biodiversity of native birds,” Biodiversity and Conservation, vol. 12, no. 9, pp. 1901–1912, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. C. Wang, H. He, M. Li et al., “Parasite species associated with wild plateau pika (Ochotona curzoniae) in southeastern Qinghai Province, China,” Journal of Wildlife Diseases, vol. 45, no. 2, pp. 288–294, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. Z. L. Wang, Y. Chen, J. Yang, W. J. Chen, Y. M. Zhang, and X. Q. Zhao, “Behavioral mechanisms of male sterilization on plateau pika in the Qinghai-Tibet plateau,” Behavioural Processes, vol. 89, no. 3, pp. 278–285, 2012. View at Publisher · View at Google Scholar · View at Scopus
  5. F. S. Dobson, A. T. Smith, and W. X. Gao, “The mating system and gene dynamics of plateau pikas,” Behavioural Processes, vol. 51, no. 1–3, pp. 101–110, 2000. View at Publisher · View at Google Scholar · View at Scopus
  6. G. C. Smith and C. L. Cheeseman, “A mathematical model for the control of diseases in wildlife populations: culling, vaccination and fertility control,” Ecological Modelling, vol. 150, no. 1-2, pp. 45–53, 2002. View at Publisher · View at Google Scholar · View at Scopus
  7. T. V. Burkey and N. C. Stenseth, “Population dynamics of territorial species in seasonal and patchy environments,” Oikos, vol. 69, no. 1, pp. 47–53, 1994. View at Scopus
  8. H. W. Liu, Z. Jin, Y. M. Chen, and F. Q. Zhang, “Population dynamics of plateau pika under lethalcontrol and contraception control,” Advances in Difference Equations, vol. 2012, article 29, 13 pages, 2012. View at Publisher · View at Google Scholar
  9. H. W. Liu and Q. Y. Li, “Model of single-species population under contraceptive control and lethal control,” Mathematics in Practice and Theory, vol. 39, no. 15, pp. 104–107, 2009.
  10. H. W. Liu, L. Zhou, W. Liu, and H. K. Zhou, “Using a cellular-automata model to investigate the effects of grazing on plateau pika population dynamics,” International Journal of Biomathematics, vol. 4, no. 3, pp. 275–287, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. Z. Jessie and S. H. David, “Polygynandry and even-sexed dispersal in a population of collared pikas, Ochotona collaris,” Animal Behaviour, vol. 83, no. 4, pp. 1075–1082, 2012. View at Publisher · View at Google Scholar · View at Scopus
  12. F. S. Dobson, A. T. Smith, and W. X. Gao, “Social and ecological influences on dispersal and philopatry in the plateau pika,” Behavioral Ecology, vol. 9, no. 6, pp. 622–635, 1998. View at Scopus
  13. Y.-X. Wang and W.-T. Li, “Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey-predator system,” Nonlinear Analysis: Real World Applications, vol. 14, no. 2, pp. 1235–1246, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y.-X. Wang and W.-T. Li, “Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 224–245, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. P. Y. H. Pang and M. X. Wang, “Existence of global solutions for a three-species predator-prey model with cross-diffusion,” Mathematische Nachrichten, vol. 281, no. 4, pp. 555–560, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. Liu, H. Zhou, and L. Zhang, “Cross-diffusion induced Turing patterns in a sex-structured predator-prey model,” International Journal of Biomathematics, vol. 5, no. 4, Article ID 1250016, 23 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. G.-P. Hu and X.-L. Li, “Turing patterns of a predator-prey model with a modified Leslie-Gower term and cross-diffusions,” International Journal of Biomathematics, vol. 5, no. 6, Article ID 1250060, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  18. C. R. Tian, Z. G. Lin, and M. Pedersen, “Instability induced by cross-diffusion in reaction-diffusion systems,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 1036–1045, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. G. Gambino, M. C. Lombardo, and M. Sammartino, “Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion,” Mathematics and Computers in Simulation, vol. 82, no. 6, pp. 1112–1132, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Z. Xie, “Cross-diffusion induced Turing instability for a three species food chain model,” Journal of Mathematical Analysis and Applications, vol. 388, no. 1, pp. 539–547, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. R. Ruiz-Baier and C. Tian, “Mathematical analysis and numerical simulation of pattern formation under cross-diffusion,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 601–612, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. W. Ko and K. Ryu, “On a predator-prey system with cross diffusion representing the tendency of predators in the presence of prey species,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1133–1142, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. W.-M. Ni, “Diffusion, cross-diffusion, and their spike-layer steady states,” Notices of the American Mathematical Society, vol. 45, no. 1, pp. 9–18, 1998. View at Zentralblatt MATH · View at MathSciNet
  24. T. Kolokolnikov and J. Wei, “Stability of spiky solutions in a competition model with cross-diffusion,” SIAM Journal on Applied Mathematics, vol. 71, no. 4, pp. 1428–1457, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Y. Lou and W. M. Ni, “Diffusion, self-diffusion and cross-diffusion,” Journal of Differential Equations, vol. 131, no. 1, pp. 79–131, 1996. View at Publisher · View at Google Scholar
  26. Y. Lou and W.-M. Ni, “Diffusion vs cross-diffusion: an elliptic approach,” Journal of Differential Equations, vol. 154, no. 1, pp. 157–190, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. Y. Lou, W.-M. Ni, and Y. Wu, “On the global existence of a cross-diffusion system,” Discrete and Continuous Dynamical Systems, vol. 4, no. 2, pp. 193–203, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. P. V. Tuôc, “Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions,” Proceedings of the American Mathematical Society, vol. 135, no. 12, pp. 3933–3941, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  29. X. Zeng, “Non-constant positive steady states of a prey-predator system with cross-diffusions,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 989–1009, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. L. S. Lin, W. M. Ni, and I. Takagi, “Large amplitude stationary solutions to a chemotaxis system,” Journal of Differential Equations, vol. 72, no. 1, pp. 1–27, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet