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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 952840, 11 pages
http://dx.doi.org/10.1155/2014/952840
Research Article

Stability and Dynamical Analysis of a Biological System

1School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, 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 6 May 2014; Accepted 20 May 2014; Published 22 July 2014

Academic Editor: Imran Naeem

Copyright © 2014 Xinhong Pan et al. 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. P. Shen and Z. Peng, Microbiology, Higher Education Press, Beijing, China, 2003.
  2. E. Beltrami and T. O. Carroll, “Modeling the role of viral diseases in recurrent phytoplankton blooms,” Journal of Mathematical Biology, vol. 32, no. 8, pp. 857–863, 1994.
  3. A. M. Edwards and J. Brindley, “Zooplankton mortality and the dynamical behaviour of plankton population models,” Bulletin of Mathematical Biology, vol. 61, no. 2, pp. 303–339, 1999. View at Publisher · View at Google Scholar · View at Scopus
  4. A. Zingone, D. Sarno, and G. Forlani, “Seasonal dynamics in the abundance of Micromonas pusilla (Prasinophyceae) and its viruses in the Gulf of Naples (Mediterranean Sea),” Journal of Plankton Research, vol. 21, no. 11, pp. 2143–2159, 1999. View at Scopus
  5. J. E. Truscott and J. Brindley, “Ocean plankton populations as excitable media,” Bulletin of Mathematical Biology, vol. 56, no. 5, pp. 981–998, 1994. View at Scopus
  6. C. S. Hollling, “The components of predation as revealed by a study of small mammal predation of the European pine sawfly,” The Canadian Entomologist, vol. 91, no. 5, pp. 293–329, 1959.
  7. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Wang, J. Shi, and J. Wei, “Predator-prey system with strong Allee effect in prey,” Journal of Mathematical Biology, vol. 62, no. 3, pp. 291–331, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. P. J. Pal, T. Saha, M. Sen, and M. Banerjee, “A delayed predator-prey model with strong Allee effect in prey population growth,” Nonlinear Dynamics, vol. 68, no. 1-2, pp. 23–42, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Haque, “A detailed study of the Beddington-DeAngelis predator-prey model,” Mathematical Biosciences, vol. 234, no. 1, pp. 1–16, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Huang, G. Lu, and S. Ruan, “Existence of traveling wave solutions in a diffusive predator-prey model,” Journal of Mathematical Biology, vol. 46, no. 2, pp. 132–152, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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
  13. F. Yi, J. Wei, and J. Shi, “Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,” Journal of Differential Equations, vol. 246, no. 5, pp. 1944–1977, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. E. R. Abraham, “The generation of plankton patchiness by turbulent stirring,” Nature, vol. 391, no. 6667, pp. 577–580, 1998. View at Publisher · View at Google Scholar · View at Scopus
  15. R. Bhattacharyya and B. Mukhopadhyay, “Modeling fluctuations in a minimal plankton model: role of spatial heterogeneity and stochasticity,” Advances in Complex Systems, vol. 10, no. 2, pp. 197–216, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. Gascoigne and R. N. Lipcius, “Allee effects in marine systems,” Marine Ecology Progress Series, vol. 269, pp. 49–59, 2004. View at Scopus
  17. P. A. Stephens, W. J. Sutherland, and R. P. Freckleton, “What is the Allee effect?” Oikos, vol. 87, no. 1, pp. 185–190, 1999. View at Scopus
  18. A. M. Kramer, B. Dennis, A. M. Liebhold, and J. M. Drake, “The evidence for Allee effects,” Population Ecology, vol. 51, no. 3, pp. 341–354, 2009. View at Publisher · View at Google Scholar · View at Scopus
  19. W. Z. Lidicker Jr., “The Allee effect: its history and future importance,” The Open Ecology Journal, vol. 3, pp. 71–82, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. M. R. Owen and M. A. Lewis, “How predation can slow, stop or reverse a prey invasion,” Bulletin of Mathematical Biology, vol. 63, no. 4, pp. 655–684, 2001. View at Publisher · View at Google Scholar · View at Scopus
  21. F. Courchamp, L. Berec, and J. Gascoigne, “Allee effects in ecology and conservation,” Environmental Conservation, vol. 36, no. 1, pp. 80–85, 2008. View at Publisher · View at Google Scholar · View at Scopus
  22. W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, Ill, USA; AMS Press, New York, NY, USA, 1931.
  23. S. Ruan, “Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling,” Journal of Mathematical Biology, vol. 31, no. 6, pp. 633–654, 1993. View at Publisher · View at Google Scholar
  24. Y. Zhu, Y. Cai, S. Yan, and W. Wang, “Dynamical analysis of a delayed reaction-diffusion predator-prey system,” Abstract and Applied Analysis, vol. 2012, Article ID 323186, 23 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. F. Courchamp, T. Clutton-Brock, and B. Grenfell, “Inverse density dependence and the Allee effect,” Trends in Ecology and Evolution, vol. 14, no. 10, pp. 405–410, 1999. View at Publisher · View at Google Scholar · View at Scopus
  26. L. Berec, E. Angulo, and F. Courchamp, “Multiple Allee effects and population management,” Trends in Ecology and Evolution, vol. 22, no. 4, pp. 185–191, 2007. View at Publisher · View at Google Scholar · View at Scopus
  27. E. Angulo, G. W. Roemer, L. Berec, J. Gascoigne, and F. Courchamp, “Double Allee effects and extinction in the island fox,” Conservation Biology, vol. 21, no. 4, pp. 1082–1091, 2007. View at Publisher · View at Google Scholar · View at Scopus
  28. D. S. Boukal and L. Berec, “Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters,” Journal of Theoretical Biology, vol. 218, no. 3, pp. 375–394, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  29. E. González-Olivares, B. González-Yañez, J. M. Lorca, A. Rojas-Palma, and J. D. Flores, “Consequences of double Allee effect on the number of limit cycles in a predator-prey model,” Computers & Mathematics with Applications, vol. 62, no. 9, pp. 3449–3463, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. G. A. K. van Voorn, L. Hemerik, M. P. Boer, and B. W. Kooi, “Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect,” Mathematical Biosciences, vol. 209, no. 2, pp. 451–469, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. M.-H. Wang and M. Kot, “Speeds of invasion in a model with strong or weak Allee effects,” Mathematical Biosciences, vol. 171, no. 1, pp. 83–97, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. E. González-Olivares and A. Rojas-Palma, “Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey,” Bulletin of Mathematical Biology, vol. 73, no. 6, pp. 1378–1397, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. M. Liermann and R. Hilborn, “Depensation: evidence, models and implications,” Fish and Fisheries, vol. 2, no. 1, pp. 33–58, 2001. View at Publisher · View at Google Scholar · View at Scopus
  34. S.-R. Zhou, Y.-F. Liu, and G. Wang, “The stability of predator-prey systems subject to the Allee effects,” Theoretical Population Biology, vol. 67, no. 1, pp. 23–31, 2005. View at Publisher · View at Google Scholar · View at Scopus
  35. J. Zu and M. Mimura, “The impact of Allee effect on a predator-prey system with Holling type II functional response,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3542–3556, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, China, 1994. View at MathSciNet
  37. D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1981. View at MathSciNet
  38. J. Wang, J. Shi, and J. Wei, “Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 1276–1304, 2011. View at Publisher · View at Google Scholar · View at MathSciNet