- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 828219, 12 pages
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.
- R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, UK, 2003.
- B. E. Kendall, “Nonlinear dynamics and chaos,” in Encyclopedia of Life Sciences, vol. 13, pp. 255–262, Nature Publishing Group, London, UK, 2001.
- R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, no. 5560, pp. 459–467, 1976.
- 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.
- 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.
- E. H. Kerner, “Further considerations on the statistical mechanics of biological associations,” vol. 21, pp. 217–255, 1959.
- N. Shigesada, K. Kawasaki, and E. Teramoto, “Spatial segregation of interacting species,” Journal of Theoretical Biology, vol. 79, no. 1, pp. 83–99, 1979.
- A. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society of London B, vol. 237, no. 1, pp. 37–72, 1952.
- Y. Huang and O. Diekmann, “Interspecific influence on mobility and Turing instability,” Bulletin of Mathematical Biology, vol. 65, no. 1, pp. 143–156, 2003.
- R. B. Hoyle, Pattern Formation: An Introduction to Methods, Cambridge University Press, Cambridge, UK, 2006.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14, Springer, New York, NY, USA, 2nd edition, 2001.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.