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Nonlinear Dynamics of a Toxin‐Phytoplankton‐Zooplankton System with Self‐ and Cross‐Diffusion

Author

Listed:
  • Pengfei Wang
  • Min Zhao
  • Hengguo Yu
  • Chuanjun Dai
  • Nan Wang
  • Beibei Wang

Abstract

A nonlinear system describing the interaction between toxin‐producing phytoplankton and zooplankton was investigated analytically and numerically, where the system was represented by a couple of reaction‐diffusion equations. We analyzed the effect of self‐ and cross‐diffusion on the system. Some conditions for the local and global stability of the equilibrium were obtained based on the theoretical analysis. Furthermore, we found that the equilibrium lost its stability via Turing instability and patterns formation then occurred. In particular, the analysis indicated that cross‐diffusion can play an important role in pattern formation. Subsequently, we performed a series of numerical simulations to further study the dynamics of the system, which demonstrated the rich dynamics induced by diffusion in the system. In addition, the numerical simulations indicated that the direction of cross‐diffusion can influence the spatial distribution of the population and the population density. The numerical results agreed with the theoretical analysis. We hope that these results will prove useful in the study of toxic plankton systems.

Suggested Citation

  • Pengfei Wang & Min Zhao & Hengguo Yu & Chuanjun Dai & Nan Wang & Beibei Wang, 2016. "Nonlinear Dynamics of a Toxin‐Phytoplankton‐Zooplankton System with Self‐ and Cross‐Diffusion," Discrete Dynamics in Nature and Society, John Wiley & Sons, vol. 2016(1).
    • Handle: RePEc:wly:jnddns:v:2016:y:2016:i:1:n:4893451
    DOI: 10.1155/2016/4893451
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    References listed on IDEAS

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    1. Weiwei Zhang & Min Zhao, 2013. "Dynamical Complexity of a Spatial Phytoplankton‐Zooplankton Model with an Alternative Prey and Refuge Effect," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    2. Jorge L. Sarmiento & Tertia M. C. Hughes & Ronald J. Stouffer & Syukuro Manabe, 1998. "Simulated response of the ocean carbon cycle to anthropogenic climate warming," Nature, Nature, vol. 393(6682), pages 245-249, May.
    3. Chakraborty, Subhendu & Roy, Shovonlal & Chattopadhyay, J., 2008. "Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: A mathematical model," Ecological Modelling, Elsevier, vol. 213(2), pages 191-201.
    4. Jef Huisman & Nga N. Pham Thi & David M. Karl & Ben Sommeijer, 2006. "Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum," Nature, Nature, vol. 439(7074), pages 322-325, January.
    5. Yapei Wang & Min Zhao & Chuanjun Dai & Xinhong Pan, 2014. "Nonlinear Dynamics of a Nutrient‐Plankton Model," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    6. Yapei Wang & Min Zhao & Chuanjun Dai & Xinhong Pan, 2014. "Nonlinear Dynamics of a Nutrient-Plankton Model," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-10, January.
    7. Weiwei Zhang & Min Zhao, 2013. "Dynamical Complexity of a Spatial Phytoplankton-Zooplankton Model with an Alternative Prey and Refuge Effect," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-10, May.
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