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:
[1]

H. DuanR. MaX. XuF. KongS. ZhangW. KongJ. Hao and L. Shang, Two-decade reconstruction of algal blooms in China-Lake Taihu, Environ. Sci. Technol., 43 (2009), 3522-3528.   Google Scholar

[2]

M. Wines, Spring Rain, Then Foul Algae in Ailing Lake Erie Report of The New York Times, 2013. Available from: http://www.nytimes.com/2013/03/15/science/earth/algae-blooms-threaten-lake-erie.html?&_r=0. Google Scholar

[3]

W. McLean and J. Macdonald, CLAS: Colby Liberal Arts Symposium, Lake Erie Algal Blooms 2014. Available from: http://digitalcommons.colby.edu/clas/2014/program/414/. Google Scholar

[4]

C. B. Lopez, E. B. Jewett, Q. Dortch, B. T. Walton and H. K. Hudnell, Scientific Assessment of Freshwater Harmful Algal Blooms Interagency Working Group on Harmful Algal Blooms, Hypoxia, and Human Health of the Joint Subcommittee on Ocean Science and Technology. Washington, DC. 2008. Available from: https://www.whitehouse.gov/sites/default/files/microsites/ostp/frshh2o0708.pdf. Google Scholar

[5]

E. Beltrami and T. O. Carroll, Modeling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol., 32 (1994), 857-863.  doi: 10.1007/BF00168802.  Google Scholar

[6]

J. Norberg and D. DeAngelis, Temperature effects on stocks and stability of a phytoplankton-zooplankton model and the dependence on light and nutrients, Ecol. Model., 95 (1997), 75-86.  doi: 10.1016/S0304-3800(96)00033-6.  Google Scholar

[7]

B. Mukhopadhyay and R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecol. Model., 198 (2006), 163-173.  doi: 10.1016/j.ecolmodel.2006.04.005.  Google Scholar

[8]

S. PalS. Chatterjee and J. Chattopadhyay, Role of toxin and nutrient for the occurrence and termination of plankton bloom results drawn from field observations and a mathematical model, Biosystems, 90 (2007), 87-100.  doi: 10.1016/j.biosystems.2006.07.003.  Google Scholar

[9]

T. Saha and M. Bandyopadhyay, Dynamical analysis of toxin producing phytoplankton-zooplankton interactions, Nonlinear Anal-Real., 10 (2009), 314-332.  doi: 10.1016/j.nonrwa.2007.09.001.  Google Scholar

[10]

Y. LvY. PeiS. Gao and C. Li, Harvesting of a phytoplankton-zooplankton model, Nonlinear Anal-Real., 11 (2010), 3608-3619.  doi: 10.1016/j.nonrwa.2010.01.007.  Google Scholar

[11]

M. BengfortU. FeudelF. M. Hilker and H. Malchow, Plankton blooms and patchiness generated by heterogeneous physical environments, Ecol. Complex., 20 (2014), 185-194.  doi: 10.1016/j.ecocom.2014.10.003.  Google Scholar

[12]

S. RanaS. SamantaS. BhattacharyaK. Al-KhaledA. Goswami and J. Chattopadhyay, The effect of nanoparticles on plankton dynamics: A mathematical model, Biosystems, 127 (2015), 28-41.  doi: 10.1016/j.biosystems.2014.11.003.  Google Scholar

[13]

T. G. HallamC. E. Clark and R. R. Lassiter, Effects of toxicants on populations: A qualitative approach Ⅰ. Equilibrium environmental exposure, Ecol. Model., 18 (1983), 291-304.  doi: 10.1016/0304-3800(83)90019-4.  Google Scholar

[14]

J. T. Turner and P. A. Tester, Toxic marine phytoplankton, zooplankton grazers, and pelagic food webs, Limnol. Oceanogr., 42 (1997), 1203-1214.  doi: 10.4319/lo.1997.42.5_part_2.1203.  Google Scholar

[15]

J. ChattopadhyayR. R. Sarkar and S. Mandal, Toxin producing plankton may act as a biological control for planktonic blooms-field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333-344.  doi: 10.1006/jtbi.2001.2510.  Google Scholar

[16]

J. ChattopadhyayR. R. Sarkar and A. El Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances, Math. Med. Biol., 19 (2002), 137-161.  doi: 10.1093/imammb/19.2.137.  Google Scholar

[17]

R. PalD. Basu and M. Banerjee, Modelling of phytoplankton allelopathy with Monod-Haldane-type functional response-A mathematical study, Biosystems, 95 (2009), 243-253.  doi: 10.1016/j.biosystems.2008.11.002.  Google Scholar

[18]

S. ChakrabortyS. ChatterjeeE. Venturino and J. Chattopadhyay, Recurring plankton bloom dynamics modeled via toxin-producing phytoplankton, J. Biol. Phys., 3 (2008), 271-290.  doi: 10.1007/s10867-008-9066-3.  Google Scholar

[19]

M. Banerjee and E. Venturino, A phytoplankton-toxic phytoplankton-zooplankton model, Ecol. Complex., 8 (2011), 239-248.  doi: 10.1016/j.ecocom.2011.04.001.  Google Scholar

[20]

M. M. MullinE. F. Stewart and F. J. Fuglister, Ingestion by planktonic grazers as a function of concentration of food1, Limnol. Oceanogr., 20 (1975), 259-262.  doi: 10.4319/lo.1975.20.2.0259.  Google Scholar

[21]

J. B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, B. Math. Biol., 57 (1995), 63-76.   Google Scholar

[22]

E. González-Olivares and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Model., 166 (2003), 135-146.   Google Scholar

[23]

L. ChenF. Chen and L. Chen, Qualitative analysis of a predator-rey model with Holling type Ⅱ functional response incorporating a constant prey refuge, Nonlinear anal-real., 11 (2010), 246-252.  doi: 10.1016/j.nonrwa.2008.10.056.  Google Scholar

[24]

T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear. Sci., 10 (2005), 681-691.  doi: 10.1016/j.cnsns.2003.08.006.  Google Scholar

[25]

Y. HuangF. Chen and L. Zhong, Stability analysis of a prey-predator model with Holling type Ⅲ response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672-683.  doi: 10.1016/j.amc.2006.04.030.  Google Scholar

[26]

D. E. Schindler and M. D. Scheuerell, Habitat coupling in lake ecosystems, Oikos, 98 (2002), 177-189.  doi: 10.1034/j.1600-0706.2002.980201.x.  Google Scholar

[27]

P. J. WilesL. A. van DurenC. HäseJ. Larsen and J. H. Simpson, Stratification and mixing in the Limfjorden in relation to mussel culture, J. Marine. Syst., 60 (2006), 129-143.  doi: 10.1016/j.jmarsys.2005.09.009.  Google Scholar

[28]

K. S. ChengS. B. Hsu and S. S. Lin, Some results on global stability of a predator-prey system, J. Math. Biol., 12 (1981), 115-126.  doi: 10.1007/BF00275207.  Google Scholar

[29]

L. P. Liou and K. S. Cheng, Global stability of a predator-prey system, J. Math. Biol., 26 (1988), 65-71.  doi: 10.1007/BF00280173.  Google Scholar

[30]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J, Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[31]

J. F. Talling, The annual cycle of stratification and phytoplankton growth in Lake Victoria (East Africa), Int. Revue Ges. Hydrobiol., 51 (1966), 545-621.  doi: 10.1002/iroh.19660510402.  Google Scholar

[32] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Academic press, 1969.   Google Scholar
[33]

G. Teschl, Ordinary Differential Equations and Dynamical Systems Am. Math. Soc., 2012. doi: 10.1090/gsm/140.  Google Scholar

[34]

H. K. Khalil and J. W. Grizzle, Nonlinear Systems Upper Saddle River: Prentice hall, 2000. Google Scholar

[35] P. Lawrence, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1991.  doi: 10.1007/978-1-4684-0392-3.  Google Scholar
[36] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Spring-Verlag, New York, 1995.  doi: 10.1007/978-1-4757-2421-9.  Google Scholar

show all references

References:
[1]

H. DuanR. MaX. XuF. KongS. ZhangW. KongJ. Hao and L. Shang, Two-decade reconstruction of algal blooms in China-Lake Taihu, Environ. Sci. Technol., 43 (2009), 3522-3528.   Google Scholar

[2]

M. Wines, Spring Rain, Then Foul Algae in Ailing Lake Erie Report of The New York Times, 2013. Available from: http://www.nytimes.com/2013/03/15/science/earth/algae-blooms-threaten-lake-erie.html?&_r=0. Google Scholar

[3]

W. McLean and J. Macdonald, CLAS: Colby Liberal Arts Symposium, Lake Erie Algal Blooms 2014. Available from: http://digitalcommons.colby.edu/clas/2014/program/414/. Google Scholar

[4]

C. B. Lopez, E. B. Jewett, Q. Dortch, B. T. Walton and H. K. Hudnell, Scientific Assessment of Freshwater Harmful Algal Blooms Interagency Working Group on Harmful Algal Blooms, Hypoxia, and Human Health of the Joint Subcommittee on Ocean Science and Technology. Washington, DC. 2008. Available from: https://www.whitehouse.gov/sites/default/files/microsites/ostp/frshh2o0708.pdf. Google Scholar

[5]

E. Beltrami and T. O. Carroll, Modeling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol., 32 (1994), 857-863.  doi: 10.1007/BF00168802.  Google Scholar

[6]

J. Norberg and D. DeAngelis, Temperature effects on stocks and stability of a phytoplankton-zooplankton model and the dependence on light and nutrients, Ecol. Model., 95 (1997), 75-86.  doi: 10.1016/S0304-3800(96)00033-6.  Google Scholar

[7]

B. Mukhopadhyay and R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecol. Model., 198 (2006), 163-173.  doi: 10.1016/j.ecolmodel.2006.04.005.  Google Scholar

[8]

S. PalS. Chatterjee and J. Chattopadhyay, Role of toxin and nutrient for the occurrence and termination of plankton bloom results drawn from field observations and a mathematical model, Biosystems, 90 (2007), 87-100.  doi: 10.1016/j.biosystems.2006.07.003.  Google Scholar

[9]

T. Saha and M. Bandyopadhyay, Dynamical analysis of toxin producing phytoplankton-zooplankton interactions, Nonlinear Anal-Real., 10 (2009), 314-332.  doi: 10.1016/j.nonrwa.2007.09.001.  Google Scholar

[10]

Y. LvY. PeiS. Gao and C. Li, Harvesting of a phytoplankton-zooplankton model, Nonlinear Anal-Real., 11 (2010), 3608-3619.  doi: 10.1016/j.nonrwa.2010.01.007.  Google Scholar

[11]

M. BengfortU. FeudelF. M. Hilker and H. Malchow, Plankton blooms and patchiness generated by heterogeneous physical environments, Ecol. Complex., 20 (2014), 185-194.  doi: 10.1016/j.ecocom.2014.10.003.  Google Scholar

[12]

S. RanaS. SamantaS. BhattacharyaK. Al-KhaledA. Goswami and J. Chattopadhyay, The effect of nanoparticles on plankton dynamics: A mathematical model, Biosystems, 127 (2015), 28-41.  doi: 10.1016/j.biosystems.2014.11.003.  Google Scholar

[13]

T. G. HallamC. E. Clark and R. R. Lassiter, Effects of toxicants on populations: A qualitative approach Ⅰ. Equilibrium environmental exposure, Ecol. Model., 18 (1983), 291-304.  doi: 10.1016/0304-3800(83)90019-4.  Google Scholar

[14]

J. T. Turner and P. A. Tester, Toxic marine phytoplankton, zooplankton grazers, and pelagic food webs, Limnol. Oceanogr., 42 (1997), 1203-1214.  doi: 10.4319/lo.1997.42.5_part_2.1203.  Google Scholar

[15]

J. ChattopadhyayR. R. Sarkar and S. Mandal, Toxin producing plankton may act as a biological control for planktonic blooms-field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333-344.  doi: 10.1006/jtbi.2001.2510.  Google Scholar

[16]

J. ChattopadhyayR. R. Sarkar and A. El Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances, Math. Med. Biol., 19 (2002), 137-161.  doi: 10.1093/imammb/19.2.137.  Google Scholar

[17]

R. PalD. Basu and M. Banerjee, Modelling of phytoplankton allelopathy with Monod-Haldane-type functional response-A mathematical study, Biosystems, 95 (2009), 243-253.  doi: 10.1016/j.biosystems.2008.11.002.  Google Scholar

[18]

S. ChakrabortyS. ChatterjeeE. Venturino and J. Chattopadhyay, Recurring plankton bloom dynamics modeled via toxin-producing phytoplankton, J. Biol. Phys., 3 (2008), 271-290.  doi: 10.1007/s10867-008-9066-3.  Google Scholar

[19]

M. Banerjee and E. Venturino, A phytoplankton-toxic phytoplankton-zooplankton model, Ecol. Complex., 8 (2011), 239-248.  doi: 10.1016/j.ecocom.2011.04.001.  Google Scholar

[20]

M. M. MullinE. F. Stewart and F. J. Fuglister, Ingestion by planktonic grazers as a function of concentration of food1, Limnol. Oceanogr., 20 (1975), 259-262.  doi: 10.4319/lo.1975.20.2.0259.  Google Scholar

[21]

J. B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, B. Math. Biol., 57 (1995), 63-76.   Google Scholar

[22]

E. González-Olivares and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Model., 166 (2003), 135-146.   Google Scholar

[23]

L. ChenF. Chen and L. Chen, Qualitative analysis of a predator-rey model with Holling type Ⅱ functional response incorporating a constant prey refuge, Nonlinear anal-real., 11 (2010), 246-252.  doi: 10.1016/j.nonrwa.2008.10.056.  Google Scholar

[24]

T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear. Sci., 10 (2005), 681-691.  doi: 10.1016/j.cnsns.2003.08.006.  Google Scholar

[25]

Y. HuangF. Chen and L. Zhong, Stability analysis of a prey-predator model with Holling type Ⅲ response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672-683.  doi: 10.1016/j.amc.2006.04.030.  Google Scholar

[26]

D. E. Schindler and M. D. Scheuerell, Habitat coupling in lake ecosystems, Oikos, 98 (2002), 177-189.  doi: 10.1034/j.1600-0706.2002.980201.x.  Google Scholar

[27]

P. J. WilesL. A. van DurenC. HäseJ. Larsen and J. H. Simpson, Stratification and mixing in the Limfjorden in relation to mussel culture, J. Marine. Syst., 60 (2006), 129-143.  doi: 10.1016/j.jmarsys.2005.09.009.  Google Scholar

[28]

K. S. ChengS. B. Hsu and S. S. Lin, Some results on global stability of a predator-prey system, J. Math. Biol., 12 (1981), 115-126.  doi: 10.1007/BF00275207.  Google Scholar

[29]

L. P. Liou and K. S. Cheng, Global stability of a predator-prey system, J. Math. Biol., 26 (1988), 65-71.  doi: 10.1007/BF00280173.  Google Scholar

[30]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J, Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[31]

J. F. Talling, The annual cycle of stratification and phytoplankton growth in Lake Victoria (East Africa), Int. Revue Ges. Hydrobiol., 51 (1966), 545-621.  doi: 10.1002/iroh.19660510402.  Google Scholar

[32] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Academic press, 1969.   Google Scholar
[33]

G. Teschl, Ordinary Differential Equations and Dynamical Systems Am. Math. Soc., 2012. doi: 10.1090/gsm/140.  Google Scholar

[34]

H. K. Khalil and J. W. Grizzle, Nonlinear Systems Upper Saddle River: Prentice hall, 2000. Google Scholar

[35] P. Lawrence, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1991.  doi: 10.1007/978-1-4684-0392-3.  Google Scholar
[36] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Spring-Verlag, New York, 1995.  doi: 10.1007/978-1-4757-2421-9.  Google Scholar
Figure 1.  Existence of the positive roots of $f(P)$ with $f(c)<0$
Figure 2.  The existence and number of equilibrium of system (8) on the plane $(\theta, m)$ for different values of $d$, where the dotted line denotes $\theta=\beta_2-d$
Figure 3.  A backward bifurcation occurs when $0<d<d_1$ and $m\in(0, m_*)$
Figure 4.  The Hopf bifurcation diagram of system (8) with $m = 8$ and $\theta$ as a bifurcation parameter
Figure 5.  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 $(20,3)$
Figure 6.  The Hopf bifurcation diagram of system (8) with $\theta=0.2$ and $m$ as a bifurcation parameter
Figure 7.  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 $(20,3)$
Figure 8.  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$
Figure 9.  The phase portraits of system (8) by perturbing $(K, m)$ in a small neighborhood of $(K^*, m^*)=(12.7759,0.6146)$
Figure 10.  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.
Figure 11.  The local phase portraits of system (8) for $(\epsilon_1,\epsilon_2, \epsilon_3)=(0.01665, 0.0001, -0.0001)$ in Fig.10
Figure 12.  The local phase portraits of system (8) for $(\epsilon_1,\epsilon_2, \epsilon_3)=(0.322,0,-0.0001)$ in Fig. 10
Figure 13.  The local phase portraits of system (8) for $(\epsilon_1,\epsilon_2, \epsilon_3)=(0.0191099,0.000836,-0.0001)$ in Fig. 10
Table 1.  The biological interpretations of all parameters in system (3) with default values used for numerical studies
Par.DescriptionValueUnitReference.
$r$Growth rate of phytoplankton0.2$h^{-1}$0.07-0.28 [16]
$K$Environmental carrying capacity50$l^{-1}$108 [15]
$m$Refuge capacityPar.$l^{-1}$Defaulted
$\beta_1$Predation rate of zooplankton1 $h^{-1}$0.6-1.4[16]
$\beta_2$Growth efficiency of zooplankton0.15$h^{-1}$0.2-0.5[16]
$d$Mortality rate of zooplankton0.003$h^{-1}$0.015-0.15[16]
$a_1$Half saturation constant3$l^{-1}$Defaulted
$a_2$Half saturation constant5.7 $l^{-1}$5.7[15]
$\theta$Toxin production ratePar.$h^{-1}$Defaulted
Par.DescriptionValueUnitReference.
$r$Growth rate of phytoplankton0.2$h^{-1}$0.07-0.28 [16]
$K$Environmental carrying capacity50$l^{-1}$108 [15]
$m$Refuge capacityPar.$l^{-1}$Defaulted
$\beta_1$Predation rate of zooplankton1 $h^{-1}$0.6-1.4[16]
$\beta_2$Growth efficiency of zooplankton0.15$h^{-1}$0.2-0.5[16]
$d$Mortality rate of zooplankton0.003$h^{-1}$0.015-0.15[16]
$a_1$Half saturation constant3$l^{-1}$Defaulted
$a_2$Half saturation constant5.7 $l^{-1}$5.7[15]
$\theta$Toxin production ratePar.$h^{-1}$Defaulted
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