Figure 1.
The saddle-node bifurcation curves in ($ \theta,b) $ space
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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 |
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.
Keywords:
Nutrient-phytoplankton model,
toxic effect,
stability,
saddle-node bifurcation,
hopf bifurcation.
Mathematics Subject Classification:
37C10, 37G10, 92B05.
Citation:
Juping Ji, Lin Wang. Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects. Discrete and Continuous Dynamical Systems - S,
2020, 13
(11)
: 3073-3081.
doi: 10.3934/dcdss.2020135
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References:
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S. Chakraborty, S. Chatterjee, E. 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. |
| [2] |
S. Chakraborty, P. K. Tiwari, A. 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. |
| [3] |
J. Chattopadhyay, R. 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. |
| [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. |
| [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. |
| [6] |
W. S. Liu, D. 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. |
| [7] |
T. G. Nielsen, T. 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. |
| [8] |
S. Pal, S. 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.
|
| [9] |
S. Roy and J. Chattopadhyay,
Toxin-alleopathy among phytoplankton species prevents competition exclusion, J. Biol. Syst., 15 (2007), 73-93.
|
| [10] |
H. Y. Shu, X. Hu, L. 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. |
| [11] |
S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry, and Engineering,, Second edition, Westview Press, Boulder, CO, 2015.
|
show all references
References:
| [1] |
S. Chakraborty, S. Chatterjee, E. 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. |
| [2] |
S. Chakraborty, P. K. Tiwari, A. 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. |
| [3] |
J. Chattopadhyay, R. 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. |
| [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. |
| [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. |
| [6] |
W. S. Liu, D. 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. |
| [7] |
T. G. Nielsen, T. 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. |
| [8] |
S. Pal, S. 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.
|
| [9] |
S. Roy and J. Chattopadhyay,
Toxin-alleopathy among phytoplankton species prevents competition exclusion, J. Biol. Syst., 15 (2007), 73-93.
|
| [10] |
H. Y. Shu, X. Hu, L. 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. |
| [11] |
S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry, and Engineering,, Second edition, Westview Press, Boulder, CO, 2015.
|
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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