Abstract and Applied Analysis

Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion

Lei Zhang

Full-text: Open access

Abstract

Predator-prey models describe biological phenomena of pursuit-evasion interaction. And this interaction exists widely in the world for the necessary energy supplement of species. In this paper, we have investigated a ratio-dependent spatially extended food chain model. Based on the bifurcation analysis (Hopf and Turing), we give the spatial pattern formation via numerical simulation, that is, the evolution process of the system near the coexistence equilibrium point ( u 2 * , v 2 * , w 2 * ) , and find that the model dynamics exhibits complex pattern replication. For fixed parameters, on increasing the control parameter c 1 , the sequence “holes holes-stripe mixtures stripes spots-stripe mixtures spots” pattern is observed. And in the case of pure Hopf instability, the model exhibits chaotic wave pattern replication. Furthermore, we consider the pattern formation in the case of which the top predator is extinct, that is, the evolution process of the system near the equilibrium point ( u 1 * , v 1 * , 0 ) , and find that the model dynamics exhibits stripes-spots pattern replication. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 130851, 9 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412607533

Digital Object Identifier
doi:10.1155/2014/130851

Mathematical Reviews number (MathSciNet)
MR3198145

Citation

Zhang, Lei. Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 130851, 9 pages. doi:10.1155/2014/130851. https://projecteuclid.org/euclid.aaa/1412607533


Export citation

References

  • P. A. Abrams and L. R. Ginzburg, “The nature of predation: prey dependent, ratio dependent or neither?” Trends in Ecology and Evolution, vol. 15, no. 8, pp. 337–341, 2000.
  • R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989.
  • C. Jost, Comparing predator-prey models qualitatively and quantitatively with ecological timeseries data [Ph.D. thesis], Institute National Agronomique, Paris, France, 1998.
  • C. Jost, O. Arino, and R. Arditi, “About deterministic extinction in ratio-dependent predator-prey models,” Bulletin of Mathematical Biology, vol. 61, no. 1, pp. 19–32, 1999.
  • Y. Kuang and E. Beretta, “Global qualitative analysis of a ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 36, no. 4, pp. 389–406, 1998.
  • F. Rao and W. Wang, “Dynamics of a Michaelis-Menten-type predation model incorporating a prey refuge with noise and external forces,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2012, no. 3, Article ID P03014, 2012.
  • S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445–1472, 2001.
  • W. Wang, Y. Cai, Y. Zhu, and Z. Guo, “Allee-effect-induced instability in a reaction-diffusion predator-prey model,” Abstract and Applied Analysis, vol. 2013, Article ID 487810, 10 pages, 2013.
  • A. M. Turing, “The chemical basis of morphogenisis,” Philosophical Transactions of the Royal Society B, vol. 237, pp. 7–72, 1952.
  • A. Aotani, M. Mimura, and T. Mollee, “A model aided understanding of spot pattern formation in chemotactic E. Coli colonies,” Japan Journal of Industrial and Applied Mathematics, vol. 27, no. 1, pp. 5–22, 2010.
  • D. Alonso, F. Bartumeus, and J. Catalan, “Mutual interference between predators can give rise to turing spatial patterns,” Ecology, vol. 83, no. 1, pp. 28–34, 2002.
  • S. A. Levin, “The problem of pattern and scale in ecology,” Ecology, vol. 73, no. 6, pp. 1943–1967, 1992.
  • M. Li, B. Han, L. Xu, and G. Zhang, “Spiral patterns near Turing instability in a discrete reaction diffusion system,” Chaos, Solitons & Fractals, vol. 49, pp. 1–6, 2013.
  • L. A. Díaz Rodrigues, D. C. Mistro, and S. Petrovskii, “Pattern formation in a space- and time-discrete predator-prey system with a strong Allee effect,” Theoretical Ecology, vol. 5, no. 3, pp. 341–362, 2012.
  • P. K. Maini, “Using mathematical models to help understand biological pattern formation,” Comptes Rendus–-Biologies, vol. 327, no. 3, pp. 225–234, 2004.
  • Z. Mei, Numerical Bifurcation Analysis for Reaction-Diffusion Equations, Springer, Berlin, Germany, 2000.
  • J. D. Murray, Mathematical Biology II, Spatial Models and Biomedical Applications, vol. 18 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 3rd edition, 2003.
  • N. Sapoukhina, Y. Tyutyunov, and R. Arditi, “The role of prey taxis in biological control: a spatial theoretical model,” American Naturalist, vol. 162, no. 1, pp. 61–76, 2003.
  • W. Wang, Q.-X. Liu, and Z. Jin, “Spatiotemporal complexity of a ratio-dependent predator-prey system,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 75, no. 5, Article ID 051913, 2007.
  • W. Wang, L. Zhang, H. Wang, and Z. Li, “Pattern formation of a predator-prey system with Ivlev-type functional response,” Ecological Modelling, vol. 221, no. 2, pp. 131–140, 2010.
  • A. Klebanoff and A. Hastings, “Chaos in three-species food chains,” Journal of Mathematical Biology, vol. 32, no. 5, pp. 427–451, 1994.
  • M. P. Boer, B. W. Kooi, and S. A. L. M. Kooijman, “Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain,” Journal of Mathematical Biology, vol. 39, no. 1, pp. 19–38, 1999.
  • 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, pp. 291–303, 2006.
  • S. Gakkhar and B. Singh, “The dynamics of a food web consisting of two preys and a harvesting predator,” Chaos, Solitons and Fractals, vol. 34, no. 4, pp. 1346–1356, 2007.
  • J. P. Keener, “Oscillatory coexistence in a food chain model with competing predators,” Journal of Mathematical Biology, vol. 22, no. 2, pp. 123–135, 1985.
  • B. W. Kooi, M. P. Boer, and S. A. L. M. Kooijman, “Complex dynamic behaviour of autonomous microbial food chains,” Journal of Mathematical Biology, vol. 36, no. 1, pp. 24–40, 1997.
  • J. López-Gómez and R. Pardo San Gil, “Coexistence in a simple food chain with diffusion,” Journal of Mathematical Biology, vol. 30, no. 7, pp. 655–668, 1992.
  • 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, pp. 55–83, 2003.
  • T. Lindström, “On the dynamics of discrete food chains: low- and high-frequency behavior and optimality of chaos,” Journal of Mathematical Biology, vol. 45, no. 5, pp. 396–418, 2002.
  • C. Neuhauser, “Mathematical challenges in spatial ecology,” Notices of the American Mathematical Society, vol. 48, no. 11, pp. 1304–1314, 2001.
  • R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, New York, NY, USA, 2003.
  • S. V. Petrovskii and H. Malchow, “Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics,” Theoretical Population Biology, vol. 59, no. 2, pp. 157–174, 2001.
  • 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, 2001.
  • C.-H. Chiu and S.-B. Hsu, “Extinction of top-predator in a three-level food-chain model,” Journal of Mathematical Biology, vol. 37, no. 4, pp. 372–380, 1998. \endinput