Pattern Formation in Tri-Trophic Ratio-Dependent Food Chain Model

DOI: 10.4236/am.2011.212213   PDF   HTML     4,055 Downloads   7,359 Views   Citations


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.

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D. Melese and S. Gakkhar, "Pattern Formation in Tri-Trophic Ratio-Dependent Food Chain Model," Applied Mathematics, Vol. 2 No. 12, 2011, pp. 1507-1514. doi: 10.4236/am.2011.212213.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] 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.
[2] 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
[3] 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
[4] 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
[5] 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
[6] 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
[7] 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
[8] 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
[9] 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
[10] 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
[11] 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
[12] 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
[13] S. V. Petrovskii, B. L. Li and H. Malchow, “Transition to Spatiotemporal Chaos Can Resolve the Paradox of Enrichment,” Ecological Complexity, Vol. 1, No. 1, 2004, pp. 37-47. doi:10.1016/j.ecocom.2003.10.001
[14] H. Shen and Z. Jin, “Two Dimensional Pattern Formation of Prey-Predator System,” Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, Qingdao, 30 July - 1 August 2007, pp. 343-346. doi:10.1109/SNPD.2007.215
[15] A.M.Turing, “The Chemical Basis of Morphogenesis,” Philosophical Transactions of Royal Society B, Vol. 237, No. 641, 1952, pp. 37-72. doi:10.1098/rstb.1952.0012
[16] M. Banerjee and S.V. Petrovskii, “Self-Organized Spatial Patterns and Chaos in a Ratio-Dependent Predator Prey System,” Theoretical Ecology, Vol. 4, No. 1, 2011, pp. 37-53. doi:10.1007/s12080-010-0073-1
[17] W. Wang, L. Zhang, Y. Xue and Z. Jin, “Spatiotemporal Pattern Formation of Beddington-DeAngelis-Type Predator-Prey Model,” arXiv: 0801.0797v1 [q-bio.PE], January 2008.
[18] L. Zhang, W. Wang and Y. Xue, “Spatiotemporal Complexity of a Predator-Prey System with Constant Harvest Rate,” Chaos Solitons Fractals, Vol. 41, No. 1, 2009, pp. 38-46. doi:10.1016/j.chaos.2007.11.009
[19] W. Wang, Q. X. Liu and Z. Jin, “Spatiotemporal Complexity of a Ratio-Dependent Predator-Prey System,” Physical Review E, Vol. 75, No. 5, 2007, Article ID 051913. doi:10.1103/PhysRevE.75.051913
[20] J.D. Murray, “Mathematical Biology II: Spatial Models and Biomedical Applications,” Springer, Berlin, 2003.
[21] G. Sun, Z. Jin, Q.X. Liu and L. Li, “Pattern Formation in a Spatial S-I model with Nonlinear Incidence Rates,” Journal of Statistical Mechanics: Theory and Experiment, Vol. 2007, 2007, P11011. doi:10.1088/1742-5468/2007/11/P11011
[22] H. Malchow, “Spatio-Temporal Pattern Formation in Coupled Models of Plankton Dynamics and Fish School Motion,” Nonlinear Analysis: Real World Applications, Vol. 1, No. 1, 2000, pp. 53-67. doi:10.1016/S0362-546X(99)00393-4
[23] S. B. L. Araújo and M. A. M. de Aguiar, “Pattern Formation, Outbreaks, and Synchronization in Food Chains with Two and Three Species,” Physical Review E, Vol. 75, No. 6, 2007, Article ID 061908. doi:10.1103/PhysRevE.75.061908
[24] W. Ko and I. Ahn, “Analysis of Ratio-Dependent Food Chain Model,” Journal of Mathematical Analysis Applications, Vol. 335, No. 1, 2007, pp. 498-523. doi:10.1016/j.jmaa.2007.01.089

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