Research ArticleOpen AccessGoogle Scholar indexed
Pattern Formation in Tri-Trophic Ratio-Dependent Food Chain Model
- 1
- 2
Applied Mathematics·Volume 02 (2011)·Pages 1507–1514·Published 5 December 2011·DOI10.4236/am.2011.212213
Copy link · social · email
Abstract
In this paper, a spatial tri-trophic food chain model with ratio-dependent Michaelis-Menten type functional response under homogeneous Neumann boundary conditions is studied. Conditions for Hopf and Turing bifurcation are derived. Sufficient conditions for the emergence of spatial patterns are obtained. The results of numerical simulations reveal the formation of labyrinth patterns and the coexistence of spotted and stripe-like patterns.
KeywordsReaction-Diffusion EquationsHopf BifurcationTuring InstabilityTuring PatternFood Chain
- A. Al-Khedhairi, “The Chaos and Control of Food Chain Model Using Nonlinear Feedback,” Applied Mathematical Sciences, Vol. 3, No. 12, 2009, pp. 591-604.
- N. J. Gotelli and A. M. Ellison, “Food-Web Models Predict Species Abundances in Response to Habitat Change,” PLoS Biology, Vol. 4, No. 10, 2006, pp. 1869-1873. doi:10.1371/journal.pbio.0040324
- S. Gakkhar and R. K. Naji, “Chaos in Three Species Ratio Dependent Food Chain,” Chaos, Solitons and Fractals, Vol. 14, No. 5, 2002, pp. 771-778. doi:10.1016/S0960-0779(02)00038-3
- S. B. Hsu, T. W. Hwang and Y. Kuang, “A Ratio-Dependent Food Chain Model and Its Applications to Biological Control,” Mathematical Biosciences, Vol. 181, No. 1, 2003, pp. 55-83. doi:10.1016/S0025-5564(02)00127-X
- K. A. J. White and C. A. Gilligan, “Spatial Heterogeneity in Three-Species, Plant-Parasite-Hyperparasite, Systems,” Philosophical Transactions of Royal Society B, Vol. 353, No. 1368, 1998, pp. 543-557. doi:10.1098/rstb.1998.0226
- C. Neuhauser and S. W. Pacala, “An Explicitly Spatial Version of Lotka-Voltera Model with Inter-Specific Competition,” The Annals of Applied Probability, Vol. 9, No. 4, 1999, pp. 1226-1259. doi:10.1214/aoap/1029962871
- K. S. McCann, J. B. Rasmussen and J. Umbanhowar, “The Dynamics of Spatially Coupled Food Webs,” Ecology Letters, Vol. 8, No. 5, 2005, pp. 513-523. doi:10.1111/j.1461-0248.2005.00742.x
- S. H. Lee, H. K. Pak, H. S. Wi, T. S. Chon and T. Matsumoto, “Growth Dynamics of Domain Pattern in a Three-Trophic Population Model,” Physica A, Vol. 334, No. 1-2, 2004, pp. 233-242. doi:10.1016/j.physa.2003.11.017
- D. O. Maionchi, S. F. dos Reis and M. A. M. de Aguiar, “Chaos and Pattern Formation in a Spatial Tritrophic Food Chain,” Ecological Modelling, Vol. 191, No. 2, 2006, pp. 291-303. doi:10.1016/j.ecolmodel.2005.04.028
- M. Wang, “Stationary Patterns for a Prey-Predator Model with Prey-Dependent and Ratio-Dependent Functional Responses and Diffusion,” Physica D, Vol. 196, No. 1-2, 2004, pp. 172-192. doi:10.1016/j.physd.2004.05.007
- M. R. Garvie, “Finite-Difference Schemes for Reaction-Diffusion Equations Modeling Predator-Prey Interactions in MATLAB,” Bulletin of Mathematical Biology, Vol. 69, No. 3, 2007, pp. 931-956. doi:10.1007/s11538-006-9062-3
- A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonov, H. Malchow and B. L. Li, “Spatiotemporal Complexity of Plankton and Fish Dynamics,” SIAM Review, Vol. 44, No. 3, 2002, pp. 311-370. doi:10.1137/S0036144502404442