Abstract and Applied Analysis

Stability and Dynamical Analysis of a Biological System

Xinhong Pan, Min Zhao, Yapei Wang, Hengguo Yu, Zengling Ma, and Qi Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This study considers the spatiotemporal dynamics of a reaction-diffusion phytoplankton-zooplankton system with a double Allee effect on prey under a homogeneous boundary condition. The qualitative properties are analyzed, including the local stability of all equilibria and the global asymptotic property of the unique positive equilibrium. We also discuss the Hopf bifurcation and the steady state bifurcation of the system. These results are expected to help understand the complexity of the Allee effect and the interaction between phytoplankton and zooplankton.

Article information

Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 952840, 11 pages.

First available in Project Euclid: 6 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Pan, Xinhong; Zhao, Min; Wang, Yapei; Yu, Hengguo; Ma, Zengling; Wang, Qi. Stability and Dynamical Analysis of a Biological System. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 952840, 11 pages. doi:10.1155/2014/952840. https://projecteuclid.org/euclid.aaa/1412605832

Export citation


  • P. Shen and Z. Peng, Microbiology, Higher Education Press, Beijing, China, 2003.
  • 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.
  • 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.
  • 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.
  • J. E. Truscott and J. Brindley, “Ocean plankton populations as excitable media,” Bulletin of Mathematical Biology, vol. 56, no. 5, pp. 981–998, 1994.
  • 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.
  • 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.
  • 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.
  • 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.
  • M. Haque, “A detailed study of the Beddington-DeAngelis predator-prey model,” Mathematical Biosciences, vol. 234, no. 1, pp. 1–16, 2011.
  • 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.
  • 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.
  • 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.
  • E. R. Abraham, “The generation of plankton patchiness by turbulent stirring,” Nature, vol. 391, no. 6667, pp. 577–580, 1998.
  • 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.
  • J. Gascoigne and R. N. Lipcius, “Allee effects in marine systems,” Marine Ecology Progress Series, vol. 269, pp. 49–59, 2004.
  • P. A. Stephens, W. J. Sutherland, and R. P. Freckleton, “What is the Allee effect?” Oikos, vol. 87, no. 1, pp. 185–190, 1999.
  • 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.
  • W. Z. Lidicker Jr., “The Allee effect: its history and future importance,” The Open Ecology Journal, vol. 3, pp. 71–82, 2010.
  • 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.
  • F. Courchamp, L. Berec, and J. Gascoigne, “Allee effects in ecology and conservation,” Environmental Conservation, vol. 36, no. 1, pp. 80–85, 2008.
  • W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, Ill, USA; AMS Press, New York, NY, USA, 1931.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • M. Liermann and R. Hilborn, “Depensation: evidence, models and implications,” Fish and Fisheries, vol. 2, no. 1, pp. 33–58, 2001.
  • 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.
  • 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.
  • Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, China, 1994.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1981.
  • 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. \endinput