Stability and Hopf Bifurcation Analysis of a Phytoplankton-Zooplankton Model Under Temperature Factor and Stage-Structure Population of Phytoplankton
Main Article Content
Abstract
In the paper, we built a predator-prey model to simulate and study the dynamics of zooplankton and phytoplankton populations under the temperature impact, in which the stage structure is considered in the zooplankton population. Our model is an ordinary differential system of three nonlinear equations with some parameters as temperature-dependent functions and uses the generalized Holling response function. The non-negative and boundedness of the model solutions have been proven. The behaviors of our system are shown by the local stability conditions of the equilibria, especially the co-existence case. The stage transformation of zooplankton was studied through the Hopf bifurcation results of varying the temperature. The analysis and simulation results indicate that the ideal temperature for the co-existence is about 12-21 degrees Celsius. The zooplankton's transformation decreases when the temperature increases, leading to an imbalance in the system. Besides that, we also provided simulation figures to illustrate the found theoretical results.
Keywords
phytoplankton-zooplankton system, predator-prey model, stage-structure population, water temperature, stability analysis, Hopf bifurcation
Article Details
References
[1] B. C. Rall, O. Vucic-pestic, R. B. Ehnes,
M. Emmerson and U. Brose, Temperature,
predator-prey interaction strength and population
stability, Global Change Biology, 16, 2010, 2145-
2157.
https://doi.org/10.1111/j.1365-2486.2009.02124.x
[2] J. L. Lessard and D. B. Hayes, Effects of elevated
water temperature on fish and macroinvertebrate
communities below small dams, River Research
and Applications, 19, 2003, 721-732.
https://doi.org/10.1002/rra.713
[3] J. C. Morrill, R. C. Bales and M. H. Conklin,
Estimating stream temperature from air
temperature: implications for future water quality,
Journal of Environmental Engineering, 131, 2005,
139-146.
https://doi.org/10.1061/(ASCE)0733-
9372(2005)131:1(139)
[4] L. S. Peck, K. E. Webb and D. M. Bailey, Extreme
sensitivity of biological function to temperature in
antarctic marine species, Functional Ecology, 18,
2004, 625-630.
https://doi.org/10.1111/j.0269-8463.2004.00903.x
[5] P. A. Staehr and K. A. J. Sand-Jensen, Seasonal
changes in temperature and nutrient control of
photosynthesis, respiration and growth of natural
phytoplankton communities, Freshwater Biology,
51, 2. 2006, 249-262.
https://doi.org/10.1111/j.1365-2427.2005.01490.x
[6] A. Toseland, S. J. Daines, J. R. Clark, A. Kirkham,
J. Strauss, C. Uhlig, T. M. Lenton, K. Valentin,
G. A. Pearson, V. Moulton and T. Mock, The
impact of temperature on marine phytoplankton
resource allocation and metabolism, Nature Climate
Change, 3, 2013.
https://doi.org/10.1038/nclimate1989
[7] K. E. Havens, R .M Ointo-Coelho, M. Beklioglu,
K. S. Christoffersen, E.Jeppesen, T. Lauridsen,
A. Mazumder, G. Methot, B .P. Alloul,
U. N. Tavsanoglu, S. Erdogan and J. Vijverberg,
Temperature effects on body size of freshwater
crustacean zooplankton from greenland to the
tropics, Hydrobiologia 743, 2015, 27-35.
https://doi.org/10.1007/s10750-014-2000-8
[8] J. G. Choi, T. C. Lippmann and E. L. Harvey,
Analytical population dynamics underlying harmful
algal blooms triggered by prey avoidance,
Ecological Modelling, 481, 2023, 110366.
https://doi.org/10.1016/j.ecolmodel.2023.110366
[9] A. Gera, R. Gayathri, P. Ezhilarasan, V. R. Rao and
M.V.R Murthy, Coupled physical-biogeochemical
simulations of upwelling, ecological response to
fresh water, Ecological Modelling, 476, 2023,
110246.
https://doi.org/10.1016/j.ecolmodel.2022.110246
[10] T. Chu, H. V. Moeller and K. M. Archibald,
Competition between phytoplankton and
mixotrophs leads to metabolic character
displacement, Ecological Modelling, 481, 2023,
110331.
https://doi.org/10.1016/j.ecolmodel.2023.110331
[11] A. Mandal, P. K. Tiwari and S. Pal,
A nonautonomous model for the effects of refuge
and additional food on the dynamics of
phytoplankton-zooplankton system, Ecological
Complexity, 46, 2021, 100927.
https://doi.org/10.1016/j.ecocom.2021.100927
[12] K. Agnihotri and H. Kaur, Optimal control of
harvesting effort in a phytoplankton-zooplankton
model with infected zooplankton under the
influence of toxicity, Mathematics and Computers
in Simulation, 190, 2021, 946-964.
https://doi.org/10.1016/j.matcom.2021.06.022
[13] S. N. Raw and S. R. Sahu, Strong stability with
impact of maturation delay and diffusion on a toxin
producing phytoplankton-zooplankton model,
Mathematics and Computers in Simulation, 210,
2023, 547-570.
https://doi.org/10.1016/j.matcom.2023.03.023
[14] Q. Zhao, S. Liu and X. Niu, Effect of water
temperature on the dynamic behavior of
phytoplankton - zooplankton model, Applied
Mathematics and Computation, 378, 2020, 125211.
https://doi.org/10.1016/j.amc.2020.125211
[15] J. M. Jackson and P. H. Lenz, Predator-prey
interactions in the plankton: larval fish feeding on
evasive copepods, Sci Rep 6, 33585, 2016.
https://doi.org/10.1038/srep33585
[16] I. Mclaren, Effects of temperature on growth of
zooplankton, and the adaptive value of vertical
migration, Journal of the Fisheries Research Board
of Canada, 20, 2011, 685-727.
https://doi.org/10.1139/f63-046
[17] W. Uszko, S. Diehl, G. Englund and
P. Amarasekare, Effects of warming on predator-prey interactions - a resource-based approach and a
theoretical synthesis, Ecology Letters, 20, 2017,
513-523.
https://doi.org/10.1111/ele.12755
M. Emmerson and U. Brose, Temperature,
predator-prey interaction strength and population
stability, Global Change Biology, 16, 2010, 2145-
2157.
https://doi.org/10.1111/j.1365-2486.2009.02124.x
[2] J. L. Lessard and D. B. Hayes, Effects of elevated
water temperature on fish and macroinvertebrate
communities below small dams, River Research
and Applications, 19, 2003, 721-732.
https://doi.org/10.1002/rra.713
[3] J. C. Morrill, R. C. Bales and M. H. Conklin,
Estimating stream temperature from air
temperature: implications for future water quality,
Journal of Environmental Engineering, 131, 2005,
139-146.
https://doi.org/10.1061/(ASCE)0733-
9372(2005)131:1(139)
[4] L. S. Peck, K. E. Webb and D. M. Bailey, Extreme
sensitivity of biological function to temperature in
antarctic marine species, Functional Ecology, 18,
2004, 625-630.
https://doi.org/10.1111/j.0269-8463.2004.00903.x
[5] P. A. Staehr and K. A. J. Sand-Jensen, Seasonal
changes in temperature and nutrient control of
photosynthesis, respiration and growth of natural
phytoplankton communities, Freshwater Biology,
51, 2. 2006, 249-262.
https://doi.org/10.1111/j.1365-2427.2005.01490.x
[6] A. Toseland, S. J. Daines, J. R. Clark, A. Kirkham,
J. Strauss, C. Uhlig, T. M. Lenton, K. Valentin,
G. A. Pearson, V. Moulton and T. Mock, The
impact of temperature on marine phytoplankton
resource allocation and metabolism, Nature Climate
Change, 3, 2013.
https://doi.org/10.1038/nclimate1989
[7] K. E. Havens, R .M Ointo-Coelho, M. Beklioglu,
K. S. Christoffersen, E.Jeppesen, T. Lauridsen,
A. Mazumder, G. Methot, B .P. Alloul,
U. N. Tavsanoglu, S. Erdogan and J. Vijverberg,
Temperature effects on body size of freshwater
crustacean zooplankton from greenland to the
tropics, Hydrobiologia 743, 2015, 27-35.
https://doi.org/10.1007/s10750-014-2000-8
[8] J. G. Choi, T. C. Lippmann and E. L. Harvey,
Analytical population dynamics underlying harmful
algal blooms triggered by prey avoidance,
Ecological Modelling, 481, 2023, 110366.
https://doi.org/10.1016/j.ecolmodel.2023.110366
[9] A. Gera, R. Gayathri, P. Ezhilarasan, V. R. Rao and
M.V.R Murthy, Coupled physical-biogeochemical
simulations of upwelling, ecological response to
fresh water, Ecological Modelling, 476, 2023,
110246.
https://doi.org/10.1016/j.ecolmodel.2022.110246
[10] T. Chu, H. V. Moeller and K. M. Archibald,
Competition between phytoplankton and
mixotrophs leads to metabolic character
displacement, Ecological Modelling, 481, 2023,
110331.
https://doi.org/10.1016/j.ecolmodel.2023.110331
[11] A. Mandal, P. K. Tiwari and S. Pal,
A nonautonomous model for the effects of refuge
and additional food on the dynamics of
phytoplankton-zooplankton system, Ecological
Complexity, 46, 2021, 100927.
https://doi.org/10.1016/j.ecocom.2021.100927
[12] K. Agnihotri and H. Kaur, Optimal control of
harvesting effort in a phytoplankton-zooplankton
model with infected zooplankton under the
influence of toxicity, Mathematics and Computers
in Simulation, 190, 2021, 946-964.
https://doi.org/10.1016/j.matcom.2021.06.022
[13] S. N. Raw and S. R. Sahu, Strong stability with
impact of maturation delay and diffusion on a toxin
producing phytoplankton-zooplankton model,
Mathematics and Computers in Simulation, 210,
2023, 547-570.
https://doi.org/10.1016/j.matcom.2023.03.023
[14] Q. Zhao, S. Liu and X. Niu, Effect of water
temperature on the dynamic behavior of
phytoplankton - zooplankton model, Applied
Mathematics and Computation, 378, 2020, 125211.
https://doi.org/10.1016/j.amc.2020.125211
[15] J. M. Jackson and P. H. Lenz, Predator-prey
interactions in the plankton: larval fish feeding on
evasive copepods, Sci Rep 6, 33585, 2016.
https://doi.org/10.1038/srep33585
[16] I. Mclaren, Effects of temperature on growth of
zooplankton, and the adaptive value of vertical
migration, Journal of the Fisheries Research Board
of Canada, 20, 2011, 685-727.
https://doi.org/10.1139/f63-046
[17] W. Uszko, S. Diehl, G. Englund and
P. Amarasekare, Effects of warming on predator-prey interactions - a resource-based approach and a
theoretical synthesis, Ecology Letters, 20, 2017,
513-523.
https://doi.org/10.1111/ele.12755