# American Institute of Mathematical Sciences

April  2017, 14(2): 529-557. doi: 10.3934/mbe.2017032

## 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

* Corresponding author: Huaiping Zhu

Received  December 18, 2015 Published  October 2016

Fund Project: This research was partially supported by NSFC grant (NO. 11271196) of China, China Scholarship Council (CSC), NSERC of Canada, and the NSF of the Jiangsu Higher Education Committee of China (No. 15KJD110004). The authors would like to thank the referees for their valuable comments and suggestions.

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.

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

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##### References:
Existence of the positive roots of $f(P)$ with $f(c)<0$
The existence and number of equilibrium of system (8) on the plane $(\theta, m)$ for different values of $d$, where the dotted line denotes $\theta=\beta_2-d$
A backward bifurcation occurs when $0<d<d_1$ and $m\in(0, m_*)$
The Hopf bifurcation diagram of system (8) with $m = 8$ and $\theta$ as a bifurcation parameter
The variation of phytoplankton and zooplankton with the increasing time and the phase plane diagram of system (3) with $m=8$ for different $\theta$ values, where the initial value is $(20,3)$
The Hopf bifurcation diagram of system (8) with $\theta=0.2$ and $m$ as a bifurcation parameter
The variation of phytoplankton and zooplankton with the increasing time and the phase plane diagram of system (3) with $\theta=0.2$ for different $m$ values, where the initial value is $(20,3)$
Bifurcation diagram of system (8) by choosing $m$ and $\theta$ as two parameters, where the values of $m$ and $\theta$ for $BT$ are $5.4248527$ and $0.20944969$ respectively. Note LP marked represents the limit point at which two positive equilibria collide into one positive equilibrium, and BP represents branch point at which positive equilibrium can disappears with the increase of the value of the parameter $\theta$ or$m$
The phase portraits of system (8) by perturbing $(K, m)$ in a small neighborhood of $(K^*, m^*)=(12.7759,0.6146)$
The phase portraits of system (8) by taking $(K, m, \theta)=(K^*, m^*, \theta^*)=(12.7759,0.6146, 0.2681)$ and perturbing $(K, m, \theta)$ in a small neighborhood of $(K^*, m^*, \theta^*)$. The partial enlarged details of $S_1$, $S_2$ and $S_3$ marked are shown by following fig. 11-13.
The local phase portraits of system (8) for $(\epsilon_1,\epsilon_2, \epsilon_3)=(0.01665, 0.0001, -0.0001)$ in Fig.10
The local phase portraits of system (8) for $(\epsilon_1,\epsilon_2, \epsilon_3)=(0.322,0,-0.0001)$ in Fig. 10
The local phase portraits of system (8) for $(\epsilon_1,\epsilon_2, \epsilon_3)=(0.0191099,0.000836,-0.0001)$ in Fig. 10
The biological interpretations of all parameters in system (3) with default values used for numerical studies
 Par. Description Value Unit Reference. $r$ Growth rate of phytoplankton 0.2 $h^{-1}$ 0.07-0.28 [16] $K$ Environmental carrying capacity 50 $l^{-1}$ 108 [15] $m$ Refuge capacity Par. $l^{-1}$ Defaulted $\beta_1$ Predation rate of zooplankton 1 $h^{-1}$ 0.6-1.4[16] $\beta_2$ Growth efficiency of zooplankton 0.15 $h^{-1}$ 0.2-0.5[16] $d$ Mortality rate of zooplankton 0.003 $h^{-1}$ 0.015-0.15[16] $a_1$ Half saturation constant 3 $l^{-1}$ Defaulted $a_2$ Half saturation constant 5.7 $l^{-1}$ 5.7[15] $\theta$ Toxin production rate Par. $h^{-1}$ Defaulted
 Par. Description Value Unit Reference. $r$ Growth rate of phytoplankton 0.2 $h^{-1}$ 0.07-0.28 [16] $K$ Environmental carrying capacity 50 $l^{-1}$ 108 [15] $m$ Refuge capacity Par. $l^{-1}$ Defaulted $\beta_1$ Predation rate of zooplankton 1 $h^{-1}$ 0.6-1.4[16] $\beta_2$ Growth efficiency of zooplankton 0.15 $h^{-1}$ 0.2-0.5[16] $d$ Mortality rate of zooplankton 0.003 $h^{-1}$ 0.015-0.15[16] $a_1$ Half saturation constant 3 $l^{-1}$ Defaulted $a_2$ Half saturation constant 5.7 $l^{-1}$ 5.7[15] $\theta$ Toxin production rate Par. $h^{-1}$ Defaulted
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