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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 891738, 10 pages
Turing Patterns in a Predator-Prey System with Self-Diffusion
School of Science, Nanchang University, Nanchang 330031, China
Received 24 June 2013; Revised 17 September 2013; Accepted 2 October 2013
Academic Editor: Francisco Solís Lozano
Copyright © 2013 Hongwei Yin 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.
- V. Volterra, “Variazioni e fluttuazioni del numero d’individui in specie conviventi,” Memorie Accademia dei Lincei, vol. 2, pp. 31–113, 1926.
- J. D. Murray, Mathematical Biology: Spatial Models and Biomedical Applications, Springer, New York, NY, USA, 2003.
- E. A. McGehee and E. Peacock-López, “Turing patterns in a modified Lotka-Volterra model,” Physics Letters A, vol. 342, no. 1-2, pp. 90–98, 2005.
- J. D. Murray, Mathematical Biology I: An introduction, vol. 17, Springer, New York, NY, USA, 3rd edition, 2002.
- E. H. Colombo and C. Anteneodo, “Nonlinear diffusion effects on biological population spatial patterns,” Physical Review E, vol. 86, Article ID 036215, 2012.
- W. Wang, Q.-X. Liu, and Z. Jin, “Spatiotemporal complexity of a ratio-dependent predator-prey system,” Physical Review E, vol. 75, no. 5, Article ID 051913, 9 pages, 2007.
- M. A. Fuentes, M. N. Kuperman, and V. M. Kenkre, “Nonlocal interaction effects on pattern formation in population dynamics,” Physical Review Letters, vol. 91, Article ID 158104, 2003.
- L. Xue, “Pattern formation in a predator-prey model with spatial effect,” Physica A, vol. 391, pp. 5987–5996, 2012.
- J. A. R. Da Cunha, A. A. Penna, F. A. Oliveira, and Pattern formation a, “nd coexistence domains for a nonlocal population dynamics,” Physical Review E, vol. 83, Article ID 015201, 2011.
- D. Alonso, F. Bartumeus, and J. Catalan, “Mutual interference between predators can give rise to Turing spatial patterns,” Ecology, vol. 83, pp. 28–34, 2002.
- W. Wang, Y. Lin, F. Rao, L. Zhang, and Y. Tan, “Pattern selection in a ratio-dependent predator-prey model,” Journal of Statistical Mechanics, vol. 2010, Article ID P11036, 2010.
- A. M. de Roos, E. McCauley, W. G. Wilson, and Pattern formation a, “nd the spatial scale of interaction between predators and their prey,” Theoretical Population Biology, vol. 53, pp. 108–130, 1998.
- 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.
- V. K. Vanag, I. R. Epstein, and Others, “Out-of-phase oscillatory Turing patterns in a bistable reaction-diffusion system,” Physical Review E, vol. 71, Article ID 66212, 2005.
- B. I. Henry, T. A. M. Langlands, and S. L. Wearne, “Turing pattern formation in fractional activator-inhibitor systems,” Physical Review E, vol. 72, no. 2, Article ID 026101, 14 pages, 2005.
- A. M. Turing, “The chemical basis of mokphogenesis,” Philosophical Transactions of the Royal Society of London B, vol. 237, pp. 37–72, 1952.
- 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.
- Y. Li and C. Zhao, “Global existence of solutions to a cross-diffusion system in higher dimensional domains,” Discrete and Continuous Dynamical Systems A, vol. 12, no. 2, pp. 185–192, 2005.
- Y. S. Choi, R. Lui, and Y. Yamada, “Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,” Discrete and Continuous Dynamical Systems A, vol. 10, no. 3, pp. 719–730, 2004.
- S.-A. Shim, “Uniform boundedness and convergence of solutions to the systems with cross-diffusions dominated by self-diffusions,” Nonlinear Analysis: Real World Applications, vol. 4, no. 1, pp. 65–86, 2003.
- Y. Cai and W. Wang, “Spatiotemporal dynamics of a reaction-diffusion epidemic model with nonlinear incidence rate,” Journal of Statistical Mechanics, vol. 2011, Article ID P02025, 2011.
- C. Tian, Z. Ling, and Z. Lin, “Turing pattern formation in a predator-prey-mutualist system,” Nonlinear Analysis: Real World Applications, vol. 12, no. 6, pp. 3224–3237, 2011.
- W. Wang, Y. Cai, M. Wu, K. Wang, and Z. Li, “Complex dynamics of a reaction-diffusion epidemic model,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2240–2258, 2012.
- 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.
- X. Guan, W. Wang, and Y. Cai, “Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 2385–2395, 2011.
- C. Tian, Z. 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.
- G. Galiano and J. Velasco, “Competing through altering the environment: a cross-diffusion population model coupled to transport-Darcy flow equations,” Nonlinear Analysis: Real World Applications, vol. 12, no. 5, pp. 2826–2838, 2011.
- J.-F. Zhang, W.-T. Li, and Y.-X. Wang, “Turing patterns of a strongly coupled predator-prey system with diffusion effects,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 74, no. 3, pp. 847–858, 2011.
- A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14, Springer, New York, NY, USA, 2002.
- S.-A. Shim, “Long-time properties of prey-predator system with cross-diffusion,” Communications of the Korean Mathematical Society, vol. 21, no. 2, pp. 293–320, 2006.
- S. Xu, “Existence of global solutions for a predator-prey model with cross-diffusion,” Electronic Journal of Differential Equations, vol. 2008, pp. 1–14, 2008.