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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 828219, 12 pages
http://dx.doi.org/10.1155/2012/828219
Research Article

Pattern Formation in a Cross-Diffusive Holling-Tanner Model

1College of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China
2Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826004, India

Received 29 November 2012; Accepted 13 December 2012

Academic Editor: Yonghui Xia

Copyright © 2012 Weiming 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. R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, UK, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. B. E. Kendall, “Nonlinear dynamics and chaos,” in Encyclopedia of Life Sciences, vol. 13, pp. 255–262, Nature Publishing Group, London, UK, 2001.
  3. R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, no. 5560, pp. 459–467, 1976. View at Scopus
  4. V. N. Biktashev, J. Brindley, A. V. Holden, and M. A. Tsyganov, “Pursuit-evasion predator-prey waves in two spatial dimensions,” Chaos, vol. 14, no. 4, pp. 988–994, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. B. Shukla and S. Verma, “Effects of convective and dispersive interactions on the stability of two species,” Bulletin of Mathematical Biology, vol. 43, no. 5, pp. 593–610, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. E. H. Kerner, “Further considerations on the statistical mechanics of biological associations,” vol. 21, pp. 217–255, 1959. View at MathSciNet
  7. N. Shigesada, K. Kawasaki, and E. Teramoto, “Spatial segregation of interacting species,” Journal of Theoretical Biology, vol. 79, no. 1, pp. 83–99, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  8. A. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society of London B, vol. 237, no. 1, pp. 37–72, 1952.
  9. Y. Huang and O. Diekmann, “Interspecific influence on mobility and Turing instability,” Bulletin of Mathematical Biology, vol. 65, no. 1, pp. 143–156, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. R. B. Hoyle, Pattern Formation: An Introduction to Methods, Cambridge University Press, Cambridge, UK, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  11. J. Chattopadhyay and P. K. Tapaswi, “Effect of cross-diffusion on pattern formation—a nonlinear analysis,” Acta Applicandae Mathematicae, vol. 48, no. 1, pp. 1–12, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Chen and A. Jüngel, “Analysis of a parabolic cross-diffusion population model without self-diffusion,” Journal of Differential Equations, vol. 224, no. 1, pp. 39–59, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. B. Dubey, B. Das, and J. Hussain, “A predator-prey interaction model with self and cross-diffusion,” Ecological Modelling, vol. 141, no. 1–3, pp. 67–76, 2001. View at Publisher · View at Google Scholar · View at Scopus
  14. B. Dubey, N. Kumari, and R. K. Upadhyay, “Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach,” Journal of Applied Mathematics and Computing, vol. 31, no. 1-2, pp. 413–432, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Iida, M. Mimura, and H. Ninomiya, “Diffusion, cross-diffusion and competitive interaction,” Journal of Mathematical Biology, vol. 53, no. 4, pp. 617–641, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. 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
  17. W. Ko and K. Ryu, “On a predator-prey system with cross-diffusion representing the tendency of prey to keep away from its predators,” Applied Mathematics Letters, vol. 21, no. 11, pp. 1177–1183, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. K. Kuto and Y. Yamada, “Multiple coexistence states for a prey-predator system with cross-diffusion,” Journal of Differential Equations, vol. 197, no. 2, pp. 315–348, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14, Springer, New York, NY, USA, 2nd edition, 2001. View at MathSciNet
  20. P. Y. H. Pang and M. Wang, “Strategy and stationary pattern in a three-species predator-prey model,” Journal of Differential Equations, vol. 200, no. 2, pp. 245–273, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. C. V. Pao, “Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion,” Nonlinear Analysis: Theory, Methods and Applications A, vol. 60, no. 7, pp. 1197–1217, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Raychaudhuri, D. K. Sinha, and J. Chattopadhyay, “Effect of time-varying cross-diffusivity in a two-species Lotka-Volterra competitive system,” Ecological Modelling, vol. 92, no. 1, pp. 55–64, 1996. View at Publisher · View at Google Scholar · View at Scopus
  23. G. Q. Sun, Z. Jin, Q. X. Liu, and L. Li, “Pattern formation induced by cross-diffusion in a predator-prey system,” Chinese Physics B, vol. 17, no. 11, pp. 3936–3941, 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. V. K. Vanag and I. R. Epstein, “Cross-diffusion and pattern formation in reaction-diffusion systems,” Physical Chemistry Chemical Physics, vol. 11, no. 6, pp. 897–912, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. R. K. Upadhyay, W. Wang, and N. K. Thakur, “Spatiotemporal dynamics in a spatial plankton system,” Mathematical Modelling of Natural Phenomena, vol. 5, no. 5, pp. 102–122, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. E. Sáez and E. González-Olivares, “Dynamics of a predator-prey model,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1867–1878, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. D. Murray, “Discussion: Turing's theory of morphogenesis—its influence on modelling biological pattern and form,” Bulletin of Mathematical Biology, vol. 52, no. 1-2, pp. 119–152, 1990. View at Publisher · View at Google Scholar · View at Scopus
  28. P. A. Braza, “The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing,” SIAM Journal on Applied Mathematics, vol. 63, no. 3, pp. 889–904, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. H. Malchow, S. V. Petrovskii, and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology-Theory, Models, and Simulation, Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2008. View at MathSciNet
  30. M. R. Garvie, “Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB,” Bulletin of Mathematical Biology, vol. 69, no. 3, pp. 931–956, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  31. A. Munteanu and R. V. Solé, “Pattern formation in noisy self-replicating spots,” International Journal of Bifurcation and Chaos, vol. 16, no. 12, pp. 3679–3683, 2006. View at Scopus
  32. M. Baurmann, T. Gross, and U. Feudel, “Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations,” Journal of Theoretical Biology, vol. 245, no. 2, pp. 220–229, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  33. J. von Hardenberg, E. Meron, M. Shachak, and Y. Zarmi, “Diversity of vegetation patterns and desertification,” Physical Review Letters, vol. 87, no. 19, Article ID 198101, 4 pages, 2001. View at Publisher · View at Google Scholar · View at Scopus