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November  2020, 13(11): 3073-3081. doi: 10.3934/dcdss.2020135

Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B5A3, Canada

* Corresponding author

Received  January 2019 Revised  February 2019 Published  November 2019

In this paper, we analyze a nutrient-phytoplankton model with toxic effects governed by a Holling-type Ⅲ functional. We show the model can undergo two saddle-node bifurcations and a Hopf bifurcation. This results in very interesting dynamics: the model can have at most three positive equilibria and can exhibit relaxation oscillations. Our results provide some insights on understanding the occurrence and control of phytoplankton blooms.

Citation: Juping Ji, Lin Wang. Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3073-3081. doi: 10.3934/dcdss.2020135
References:
[1]

S. ChakrabortyS. ChatterjeeE. Venturino and J. Chattopadhyay, Recurring plankton bloom dynamics modeled via toxin producing phytoplankton, J. Biol. Phys., 33 (2007), 271-290.  doi: 10.1007/s10867-008-9066-3.  Google Scholar

[2]

S. ChakrabortyP. K. TiwariA. K. Misra and J. Chattopadhyay, Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton, Math. Biosci., 264 (2015), 94-100.  doi: 10.1016/j.mbs.2015.03.010.  Google Scholar

[3]

J. ChattopadhyayR. R. Sarkarw and S. Mandalw, 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

[4]

G. M. Hallegraeff, A review of harmful algal blooms and their apparent global increase, Phycologia, 32 (1993), 79-99.  doi: 10.2216/i0031-8884-32-2-79.1.  Google Scholar

[5]

S.-B. Hsu and J. P. Shi, Relaxation oscillations profile of limit cycle in predator-prey system, Discret. Contin. Dyn. Syst.-B, 11 (2009), 893-911.  doi: 10.3934/dcdsb.2009.11.893.  Google Scholar

[6]

W. S. LiuD. M. Xiao and Y. F. Yi, Relaxation oscillations in a class of predator-prey systems, J. Differential Equations, 188 (2003), 306-331.  doi: 10.1016/S0022-0396(02)00076-1.  Google Scholar

[7]

T. G. NielsenT. Kirboe and P. K. Bjrnsen, Effects of a Chrysochromulina polylepis subsurface bloom on the planktonic community, Mar. Ecol. Prog. Ser., 62 (1990), 21-35.  doi: 10.3354/meps062021.  Google Scholar

[8]

S. PalS. Chatterjee and J. Chattopadhyay, Role of toxin and nutrient for the occurence and termination of plankton bloom-results drawn from field observations and a mathematical model, Biosystems, 90 (2007), 87-100.   Google Scholar

[9]

S. Roy and J. Chattopadhyay, Toxin-alleopathy among phytoplankton species prevents competition exclusion, J. Biol. Syst., 15 (2007), 73-93.   Google Scholar

[10]

H. Y. ShuX. HuL. Wang and J. Watmough, Delay induced stability switch, multi-type bistability and chaos in an intraguild predation model, J. Math. Biol., 71 (2015), 1269-1298.  doi: 10.1007/s00285-015-0857-4.  Google Scholar

[11] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry, and Engineering,, Second edition, Westview Press, Boulder, CO, 2015.   Google Scholar

show all references

References:
[1]

S. ChakrabortyS. ChatterjeeE. Venturino and J. Chattopadhyay, Recurring plankton bloom dynamics modeled via toxin producing phytoplankton, J. Biol. Phys., 33 (2007), 271-290.  doi: 10.1007/s10867-008-9066-3.  Google Scholar

[2]

S. ChakrabortyP. K. TiwariA. K. Misra and J. Chattopadhyay, Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton, Math. Biosci., 264 (2015), 94-100.  doi: 10.1016/j.mbs.2015.03.010.  Google Scholar

[3]

J. ChattopadhyayR. R. Sarkarw and S. Mandalw, 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

[4]

G. M. Hallegraeff, A review of harmful algal blooms and their apparent global increase, Phycologia, 32 (1993), 79-99.  doi: 10.2216/i0031-8884-32-2-79.1.  Google Scholar

[5]

S.-B. Hsu and J. P. Shi, Relaxation oscillations profile of limit cycle in predator-prey system, Discret. Contin. Dyn. Syst.-B, 11 (2009), 893-911.  doi: 10.3934/dcdsb.2009.11.893.  Google Scholar

[6]

W. S. LiuD. M. Xiao and Y. F. Yi, Relaxation oscillations in a class of predator-prey systems, J. Differential Equations, 188 (2003), 306-331.  doi: 10.1016/S0022-0396(02)00076-1.  Google Scholar

[7]

T. G. NielsenT. Kirboe and P. K. Bjrnsen, Effects of a Chrysochromulina polylepis subsurface bloom on the planktonic community, Mar. Ecol. Prog. Ser., 62 (1990), 21-35.  doi: 10.3354/meps062021.  Google Scholar

[8]

S. PalS. Chatterjee and J. Chattopadhyay, Role of toxin and nutrient for the occurence and termination of plankton bloom-results drawn from field observations and a mathematical model, Biosystems, 90 (2007), 87-100.   Google Scholar

[9]

S. Roy and J. Chattopadhyay, Toxin-alleopathy among phytoplankton species prevents competition exclusion, J. Biol. Syst., 15 (2007), 73-93.   Google Scholar

[10]

H. Y. ShuX. HuL. Wang and J. Watmough, Delay induced stability switch, multi-type bistability and chaos in an intraguild predation model, J. Math. Biol., 71 (2015), 1269-1298.  doi: 10.1007/s00285-015-0857-4.  Google Scholar

[11] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry, and Engineering,, Second edition, Westview Press, Boulder, CO, 2015.   Google Scholar
Figure 1.  The saddle-node bifurcation curves in ($ \theta,b) $ space
Figure 2.  Bifurcation diagram of Model (1). Parameter values used here are: $ a = 0.5 $, $ \varepsilon = 1 $, $ \delta = 0.05 $, $ \mu = 0.08 $, $ \xi = 0.18 $ and $ \theta = 2 $
Figure 3.  A phase portrait of Model (1). Parameter values are: $ a = 0.5 $, $ \varepsilon = 1 $, $ \delta = 0.05 $, $ \mu = 0.08 $, $ \theta = 2 $, $ \xi = 0.18 $ and $ b = 2.3 $
Figure 4.  A numerical solution of Model (1): relaxation oscillations are observed. Parameter values are: $ a = 0.5 $, $ \varepsilon = 1 $, $ \delta = 0.05 $, $ \mu = 0.08 $, $ \xi = 0.18 $, $ \theta = 2 $.and $ b = 2.01 $
Figure 5.  Bifurcation diagram of Model (1) using $ \theta $ as the bifurcation parameter. Other parameter values are: $ a = 0.5 $, $ \varepsilon = 1 $, $ \delta = 0.05 $, $ \mu = 0.08 $, $ \xi = 0.18 $ and $ b = 2.5 $
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