Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 675378, 15 pages
http://dx.doi.org/10.1155/2014/675378
Research Article

Spatial Complexity of a Predator-Prey Model with Holling-Type Response

1Computer Science and Technology Department, East China Normal University, Shanghai 200241, China
2Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received 21 January 2014; Accepted 1 February 2014; Published 1 June 2014

Academic Editor: Weiming Wang

Copyright © 2014 Lei Zhang and Zhibin Li. 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.

Linked References

  1. R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  2. B. E. Kendall, “Cycles, chaos, and noise in predator-prey dynamics,” Chaos, Solitons & Fractals, vol. 12, no. 2, pp. 321–332, 2001. View at Publisher · View at Google Scholar · View at Scopus
  3. 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. View at MathSciNet
  4. A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992. View at Google Scholar · View at Scopus
  5. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. 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. View at Google Scholar · View at Scopus
  7. 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. View at Google Scholar · View at Scopus
  8. 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. View at Publisher · View at Google Scholar · View at Scopus
  9. 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, 2000/01. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Y. Chen, “Multiple periodic solutions of delayed predator-prey systems with type IV functional responses,” Nonlinear Analysis. Real World Applications, vol. 5, no. 1, pp. 45–53, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. D. Estep and D. Neckels, “Fast methods for determining the evolution of uncertain parameters in reaction-diffusion equations,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 37-40, pp. 3967–3979, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Gakkhar and B. Singh, “The dynamics of a food web consisting of two preys and a harvesting predator,” Chaos, Solitons & Fractals, vol. 34, no. 4, pp. 1346–1356, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J.-C. Huang and D.-M. Xiao, “Analyses of bifurcations and stability in a predator-prey system with Holling type-IV functional response,” Acta Mathematicae Applicatae Sinica, vol. 20, no. 1, pp. 167–178, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W. Ko and K. Ryu, “Coexistence states of a predator-prey system with non-monotonic functional response,” Nonlinear Analysis: Real World Applications, vol. 8, no. 3, pp. 769–786, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. Leppänen, Coputational studies of pattern formation in Turing systems [Ph.D. thesis], Helsinki University of Technology, 2004.
  16. H. Malchow, F. M. Hilker, and S. V. Petrovskii, “Noise and productivity dependence of spatiotemporal pattern formation in a prey-predator system,” Discrete and Continuous Dynamical Systems. Series B, vol. 4, no. 3, pp. 705–711, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. K. Page, P. K. Maini, and N. A. M. Monk, “Pattern formation in spatially heterogeneous Turing reaction-diffusion models,” Physica D, vol. 181, no. 1-2, pp. 80–101, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. T. Skalski and J. F. Gilliam, “Functional responses with predator interference: viable alternatives to the Holling type II model,” Ecology, vol. 82, no. 11, pp. 3083–3092, 2001. View at Google Scholar · View at Scopus
  19. S. Zhang, D. Tan, and L. Chen, “Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations,” Chaos, Solitons & Fractals, vol. 27, no. 4, pp. 980–990, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. W. Zhang, D. Zhu, and P. Bi, “Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses,” Applied Mathematics Letters, vol. 20, no. 10, pp. 1031–1038, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. H. Zhu, S. A. Campbell, and G. S. K. Wolkowicz, “Bifurcation analysis of a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 63, no. 2, pp. 636–682, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. C. S. Holling, “The components of predation as revealed by a study of small mammal predation of the european pine sawfly,” The Canadian Entomologist, vol. 91, pp. 293–320, 1959. View at Google Scholar
  23. C. S. Holling, “Some characteristics of simple types of predation and parasitism,” The Canadian Entomologist, vol. 91, pp. 385–395, 1959. View at Google Scholar
  24. J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, pp. 707–723, 1968. View at Publisher · View at Google Scholar
  25. P. Y. H. Pang and M. Wang, “Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion,” Proceedings of the London Mathematical Society, vol. 88, no. 1, pp. 135–157, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. W. Sokol, “Oxidation of an inhibitory substrate by washed cells,” Biotechnology and Bioengineering, vol. 30, no. 8, pp. 921–927, 1987. View at Google Scholar · View at Scopus
  27. C. Neuhauser, “Mathematical challenges in spatial ecology,” Notices of the American Mathematical Society, vol. 48, no. 11, pp. 1304–1314, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. A. M. Turing, “The chemical basis of morphogenisis,” Philosophical Transactions of the Royal Society B, vol. 237, pp. 7–72, 1952. View at Google Scholar
  29. L. A. Segel and J. L. Jackson, “Dissipative structure: an explanation and an ecological example,” Journal of Theoretical Biology, vol. 37, no. 3, pp. 545–559, 1972. View at Google Scholar · View at Scopus
  30. S. A. Levin, “The problem of pattern and scale in ecology,” Ecology, vol. 73, no. 6, pp. 1943–1967, 1992. View at Google Scholar · View at Scopus
  31. 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. View at Publisher · View at Google Scholar · View at Scopus
  32. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  33. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Reviews of Modern Physics, vol. 65, no. 3, pp. 851–1112, 1993. View at Publisher · View at Google Scholar · View at Scopus
  34. J. García-Ojalvo and L. Schimansky-Geier, “Noise-induced spiral dynamics in excitable media,” Europhysics Letters, vol. 47, no. 3, pp. 298–303, 1999. View at Publisher · View at Google Scholar · View at Scopus
  35. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. A. Gierer and H. Meinhardt, “A theory of biological pattern formation,” Kybernetik, vol. 12, no. 1, pp. 30–39, 1972. View at Publisher · View at Google Scholar · View at Scopus
  37. D. A. Griffith and P. R. Peres-Neto, “Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses,” Ecology, vol. 87, no. 10, pp. 2603–2613, 2006. View at Publisher · View at Google Scholar · View at Scopus
  38. K. A. Hawick, H. A. James, and C. J. Scogings, “A zoology of emergent patterns in a predator-prey simulation model,” in Proceedings of the 6th IASTED International Conference on Modelling, Simulation, and Optimizatiom (MSO '06), pp. 84–89, Gabarone, Botswana, September 2006. View at Scopus
  39. H. Katsuragi, “Diffusion-induced spontaneous pattern formation on gelation surfaces,” Europhysics Letters, vol. 73, no. 5, pp. 793–799, 2006. View at Publisher · View at Google Scholar · View at Scopus
  40. C. A. Klausmeier, “Regular and irregular patterns in semiarid vegetation,” Science, vol. 284, no. 5421, pp. 1826–1828, 1999. View at Publisher · View at Google Scholar · View at Scopus
  41. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  42. Z.-Z. Li, M. Gao, C. Hui, X.-Z. Han, and H. Shi, “Impact of predator pursuit and prey evasion on synchrony and spatial patterns in metapopulation,” Ecological Modelling, vol. 185, no. 2–4, pp. 245–254, 2005. View at Publisher · View at Google Scholar · View at Scopus
  43. A. Luiz, C. Diomar, and P. Sergei, “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. View at Publisher · View at Google Scholar
  44. A. Madzvamuse, R. D. K. Thomas, P. K. Maini, and A. J. Wathen, “A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves,” Bulletin of Mathematical Biology, vol. 64, no. 3, pp. 501–530, 2002. View at Publisher · View at Google Scholar · View at Scopus
  45. P. K. Maini, R. E. Baker, and C.-M. Chuong, “The turing model comes of molecular age,” Science, vol. 314, no. 5804, pp. 1397–1398, 2006. View at Publisher · View at Google Scholar · View at Scopus
  46. 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. View at Publisher · View at Google Scholar · View at Scopus
  47. A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow, and B.-L. Li, “Spatiotemporal complexity of plankton and fish dynamics,” SIAM Review, vol. 44, no. 3, pp. 311–370, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. S. S. Riaz, S. Dutta, S. Kar, and D. S. Ray, “Pattern formation induced by additive noise: a moment-based analysis,” European Physical Journal B, vol. 47, no. 2, pp. 255–263, 2005. View at Publisher · View at Google Scholar · View at Scopus
  49. S. S. Riaz, S. Banarjee, S. Kar, and D. S. Ray, “Pattern formation in reaction-diffusion system in crossed electric and magnetic fields,” European Physical Journal B, vol. 53, no. 4, pp. 509–515, 2006. View at Publisher · View at Google Scholar · View at Scopus
  50. K. Uriu and Y. Iwasa, “Turing pattern formation with two kinds of cells and a diffusive chemical,” Bulletin of Mathematical Biology, vol. 69, no. 8, pp. 2515–2536, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  52. W. Wang, Y. Lin, F. Yang, L. Zhang, and Y. Tan, “Numerical study of pattern formation in an extended Gray-Scott model,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2016–2026, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  53. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  54. L. Yang, M. Dolnik, A. M. Zhabotinsky, and I. R. Epstein, “Pattern formation arising from interactions between turing and wave instabilities,” Journal of Chemical Physics, vol. 117, no. 15, pp. 7259–7265, 2002. View at Publisher · View at Google Scholar · View at Scopus
  55. L. Yang, M. Dolnik, A. M. Zhabotinsky, and I. R. Epstein, “Spatial resonances and superposition patterns in a reaction-diffusion model with interacting turing modes,” Physical Review Letters, vol. 88, no. 20, pp. 2083031–2083034, 2002. View at Google Scholar · View at Scopus
  56. C. Zhou and J. Kurths, “Noise-sustained and controlled synchronization of stirred excitable media by external forcing,” New Journal of Physics, vol. 7, article 18, 2005. View at Publisher · View at Google Scholar · View at Scopus
  57. J. M. Cushing, “Periodic time-dependent predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 32, no. 1, pp. 82–95, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  58. 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. View at Publisher · View at Google Scholar · View at Scopus
  59. Q. Liu, Z. Jin, and B. Li, “Resonance and frequency-locking phenomena in spatially extended phytoplankton-zooplankton system with additive noise and periodic forces,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2008, no. 5, Article ID P05011, 2008. View at Google Scholar
  60. R. Mankin, T. Laas, A. Sauga, and A. Ainsaar, “Colored-noise-induced Hopf bifurcations in predator-prey communities,” Physical Review E, vol. 74, no. 2, Article ID 021101, 10 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  61. F. N. Si, Q. X. Liu, J. Z. Zhang, and L. Q. Zhou, “Propagation of travelling waves in sub-excitable systems driven by noise and periodic forcing,” European Physical Journal B, vol. 60, no. 4, pp. 507–513, 2007. View at Publisher · View at Google Scholar · View at Scopus
  62. B. Spagnolo, D. Valenti, and A. Fiasconaro, “Noise in ecosystems: a short review,” Mathematical Biosciences and Engineering, vol. 1, no. 1, pp. 185–211, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  63. J. M. G. Vilar and R. V. Solé, “Effects of noise in symmetric two-species competition,” Physical Review Letters, vol. 80, no. 18, pp. 4099–4102, 1998. View at Google Scholar · View at Scopus