Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 278457, 11 pages
http://dx.doi.org/10.1155/2014/278457
Research Article

Nonlinear Analysis in a Nutrient-Algae-Zooplankton System with Sinking of Algae

Chuanjun Dai1,2,3 and Min Zhao2,3

1Key Laboratory of Saline-Alkali Vegetation Ecology Restoration in Oil Field, Ministry of Education, Alkali Soil Natural Environmental Science Center, Northeast Forestry University, Harbin 150040, China
2Zhejiang Provincial Key Laboratory for Water Environment and Marine Biological Resources Protection, Wenzhou University, Wenzhou, Zhejiang 325035, China
3School of Life and Environmental Science, Wenzhou University, Wenzhou, Zhejiang 325035, China

Received 2 April 2014; Revised 12 June 2014; Accepted 12 June 2014; Published 7 July 2014

Academic Editor: Mariano Torrisi

Copyright © 2014 Chuanjun Dai and Min Zhao. 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. H. Malchow, “Spatio-temporal pattern formation in nonlinear non-equilibrium plankton dynamics,” Proceedings of the Royal Society B: Biological Sciences, vol. 251, no. 1331, pp. 103–109, 1993. View at Publisher · View at Google Scholar · View at Scopus
  2. H. Malchow, F. M. Hilker, S. V. Petrovskii, and K. Brauer, “Oscillations and waves in a virally infected plankton system—part I: the lysogenic stage,” Ecological Complexity, vol. 1, no. 3, pp. 211–223, 2004. View at Publisher · View at Google Scholar · View at Scopus
  3. C. Dai, M. Zhao, and L. Chen, “Bifurcations and periodic solutions for an algae-fish semicontinuous system,” Abstract and Applied Analysis, vol. 2013, Article ID 619721, 11 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  4. R. Wang, Q. Liu, G. Sun, Z. Jin, and J. van de Koppel, “Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds,” Journal of the Royal Society Interface, vol. 6, no. 37, pp. 705–718, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. J. van de Koppel, M. Rietkerk, N. Dankers, and P. M. J. Herman, “Scale-dependent feedback and regular spatial patterns in young mussel beds,” The American Naturalist, vol. 165, no. 3, pp. E66–E77, 2005. View at Publisher · View at Google Scholar · View at Scopus
  6. Y. Wang, M. Zhao, C. Dai, and X. Pan, “Nonlinear dynamics of a nutrient-plankton model,” Abstract and Applied Analysis, vol. 2014, Article ID 451757, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  7. J. E. Pearson, “Complex patterns in a simple system,” Science, vol. 261, no. 5118, pp. 189–192, 1993. View at Publisher · View at Google Scholar · View at Scopus
  8. J. A. Sherratt, “Periodic travelling waves in cyclic predator-prey systems,” Ecology Letters, vol. 4, no. 1, pp. 30–37, 2001. View at Publisher · View at Google Scholar · View at Scopus
  9. J. A. Sherratt and M. J. Smith, “Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models,” Journal of the Royal Society Interface, vol. 5, no. 22, pp. 483–505, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. 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 MathSciNet · View at Scopus
  11. S. A. Levin, “The problem of pattern and scale in ecology,” Ecology, vol. 73, no. 6, pp. 1943–1967, 1992. View at Publisher · View at Google Scholar · View at Scopus
  12. W. M. Wang, Q. X. Liu, and J. Zhen, “Spatiotemporal complexity of a ratio-dependent predator-prey system,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 75, no. 5, pp. 051913–051921, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society of London B, vol. 231, pp. 37–72, 1952. View at Google Scholar
  14. 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
  15. J. D. Murray, “Mathematical Biology,” in Biomathematics, vol. 19, Springer, Berlin, Germany, 2nd edition, 1993. View at Google Scholar
  16. M. P. Hassell, H. N. Comins, and R. M. May, “Spatial structure and chaos in insect population dynamics,” Nature, vol. 353, no. 6341, pp. 255–258, 1991. View at Publisher · View at Google Scholar · View at Scopus
  17. J. Huisman, N. N. Pham Thi, D. M. Karl, and B. Sommeijer, “Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum,” Nature, vol. 439, no. 7074, pp. 322–325, 2006. View at Publisher · View at Google Scholar · View at Scopus
  18. J. K. Liu, Advanced Hydrobiology, Science Press, Beijing, China, 1999.
  19. K. Rinke, A. M. R. Huber, S. Kempke et al., “Lake-wide distributions of temperature, phytoplankton, zooplankton, and fish in the pelagic zone of a large lake,” Limnology and Oceanography, vol. 54, no. 4, pp. 1306–1322, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. H. Malchow, B. Radtke, M. Kallache, A. B. Medvinsky, D. A. Tikhonov, and S. V. Petrovskii, “Spatio-temporal pattern formation in coupled models of plankton dynamics and fish school motion,” Nonlinear Analysis: Real World Applications, vol. 1, no. 1, pp. 53–67, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. X. Lazzaro, R. W. Drenner, R. A. Stein, and J. D. Smith, “Planktivores and plankton dynamics: effects of fish biomass and planktivore type,” Canadian Journal of Fisheries and Aquatic Sciences, vol. 49, no. 7, pp. 1466–1473, 1992. View at Publisher · View at Google Scholar · View at Scopus
  22. M. Sieber, H. Malchow, and L. Schimansky-Geier, “Constructive effects of environmental noise in an excitable prey-predator plankton system with infected prey,” Ecological Complexity, vol. 4, no. 4, pp. 223–233, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. R. Ptacnik, S. Diehl, and S. Berger, “Performance of sinking and nonsinking phytoplankton taxa in a gradient of mixing depths,” Limnology and Oceanography, vol. 48, no. 5, pp. 1903–1912, 2003. View at Publisher · View at Google Scholar · View at Scopus
  24. G. C. Pitcher, D. R. Walker, and B. A. Mitchell-Innes, “Phytoplankton sinking rate dynamics in the southern Benguela upwelling system,” Marine Ecology Progress Series, vol. 55, pp. 261–269, 1989. View at Google Scholar
  25. P. K. Bienfang, “Phytoplankton sinking rates in oligotrophic waters off Hawaii, USA,” Marine Biology, vol. 61, no. 1, pp. 69–77, 1980. View at Publisher · View at Google Scholar · View at Scopus
  26. S. A. Condie and M. Bormans, “The influence of density stratification on particle settling, dispersion and population growth,” Journal of Theoretical Biology, vol. 187, no. 1, pp. 65–75, 1997. View at Publisher · View at Google Scholar · View at Scopus
  27. L. V. Lucas, J. E. Cloern, J. R. Koseff, S. G. Monismith, and J. K. Thompson, “Does the Sverdrup critical depth model explain bloom dynamics in estuaries?” Journal of Marine Research, vol. 56, no. 2, pp. 375–415, 1998. View at Publisher · View at Google Scholar · View at Scopus
  28. S. Diehl, “Phytoplankton, light, and nutrients in a gradient of mixing depths: theory,” Ecology, vol. 83, no. 2, pp. 386–398, 2002. View at Publisher · View at Google Scholar · View at Scopus
  29. H. L. Wang and J. F. Feng, Ecological Dynamics and Prediction of Red Tide, Tianjin University Press, Tianjin, China, 1st edition, 2006.
  30. M. L. Rosenzweig, “Paradox of enrichment: destabilization of exploitation ecosystems in ecological time,” Science, vol. 171, no. 3969, pp. 385–387, 1971. View at Publisher · View at Google Scholar · View at Scopus
  31. I. Hanski, “The functional response of predators: worries about scale,” Trends in Ecology and Evolution, vol. 6, no. 5, pp. 141–142, 1991. View at Publisher · View at Google Scholar · View at Scopus
  32. S. B. Hsu, T. W. Hwang, and Y. Kuang, “Rich dynamics of a ratio-dependent predator-prey models,” Bulletin of Mathematical Biology, vol. 61, pp. 489–506, 2011. View at Google Scholar
  33. 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. View at Publisher · View at Google Scholar · View at Scopus
  34. 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 · View at Scopus
  35. M. Haque, “Ratio-dependent predator-prey models of interacting populations,” Bulletin of Mathematical Biology, vol. 71, no. 2, pp. 430–452, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  36. J. D. Flores and E. G. Olivares, “Dynamics of a predator-prey model with Allee effect on prey and ratio-dependent functional response,” Ecological Complexity, vol. 18, pp. 59–65, 2014. View at Publisher · View at Google Scholar
  37. M. Sen, M. Banerjee, and A. Morozov, “Stage-structured ratio-dependent predator-prey models revisited: when should the maturation lag result in system destabilization?” Ecological Complexity, vol. 19, pp. 23–34, 2014. View at Publisher · View at Google Scholar
  38. Y. Song and X. Zou, “Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-Turing bifurcation point,” Computers & Mathematics with Applications, vol. 67, no. 10, pp. 1978–1997, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  39. S. P. Kuznetsov, E. Mosekilde, G. Dewel, and P. Borckmans, “Absolute and convective instabilities in a one-dimensional Brusselator flow model,” The Journal of Chemical Physics, vol. 106, no. 18, pp. 7609–7616, 1997. View at Publisher · View at Google Scholar · View at Scopus