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Dynamical analysis of a toxin-producing phytoplankton-zooplankton model with refuge

1. Jiangsu Key Laborary for NSLSCS, Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Nanjing 210023, China
2. Laboratory of Mathematical Parallel Systems (Lamps), Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

To study the impacts of toxin produced by phytoplankton and refuges provided for phytoplankton on phytoplankton-zooplankton interactions in lakes, we establish a simple phytoplankton-zooplankton system with Holling type Ⅱ response function. The existence and stability of positive equilibria are discussed. Bifurcation analyses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurcation, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the observed biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have a significant impact on the occurring and terminating of algal blooms in freshwater lakes.

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Keywords Phytoplankton; zooplankton; refuge; toxin; algal blooms; multiple equilibria; stability; Hopf bifurcation; Bogdanov-Takens bifurcation

Citation: Juan Li, Yongzhong Song, Hui Wan, Huaiping Zhu. Dynamical analysis of a toxin-producing phytoplankton-zooplankton model with refuge. Mathematical Biosciences and Engineering, 2017, 14(2): 529-557. doi: 10.3934/mbe.2017032

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