American Institute of Mathematical Sciences

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October  2016, 21(8): 2703-2728. doi: 10.3934/dcdsb.2016069

Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy

Zhipeng Qiu 1, and  Huaiping Zhu 2,

1. 

Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094
2. 

Laboratory of Mathematical Parallel Systems (Lamps), Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

Received  September 2015 Revised  February 2016 Published  September 2016

Understanding the plankton dynamics can help us take effective measures to settle the critical issue on how to keep plankton ecosystem balance. In this paper, a nutrient-phytoplankton-zooplankton (NPZ) model is formulated to understand the mechanism of plankton dynamics. To account for the harmful effect of the phytoplankton allelopathy, a prototype for a non-monotone response function is used to model zooplankton grazing, and nonlinear phytoplankton mortality is also included in the NPZ model. Using the model, we will focus on understanding how the phytoplankton allelopathy and nonlinear phytoplankton mortality affect the plankton population dynamics. We first examine the existence of multiple equilibria and provide a detailed classification for the equilibria, then stability and local bifurcation analysis are also studied. Sufficient conditions for Hopf bifurcation and zero-Hopf bifurcation are given respectively. Numerical simulations are finally conducted to confirm and extend the analytic results. Both theoretical and numerical findings imply that the phytoplankton allelopathy and nonlinear phytoplankton mortality may lead to a rich variety of complex dynamics of the nutrient-plankton system. The results of this study suggest that the effects of the phytoplankton allelopathy and nonlinear phytoplankton mortality should receive more attention to understand the plankton dynamics.
Keywords: bifurcation, Nutrient-phytoplankton-zooplankton model, nonlinear phytoplankton mortality, allelopathy, complex dynamics..
Mathematics Subject Classification: Primary: 92D25, 92D40; Secondary: 34A4.
Citation: Zhipeng Qiu, Huaiping Zhu. Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2703-2728. doi: 10.3934/dcdsb.2016069
References:
[1]

S. Busenberg, S. K. Kumar, P. Austin and G. Wake, The dynamics of a model of a plankton-nutrient interaction,, Bull. Math. Biol., 52 (1990), 677.   Google Scholar

[2]

J. Chattopadhyay, R. R. Sarker and S. Mandal, Toxin producing plankton may act as a biological control for plankton blooms-filed study and mathemtical modelling,, J. Theor. Biol., 215 (2002), 333.   Google Scholar

[3]

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[5]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models,, Bull. Math. Biol., 61 (1999), 303.  doi: 10.1006/bulm.1998.0082.  Google Scholar

[6]

P. G. Falkowski, The role of phytoplankton photosynthesis in global biogeochemical cycles,, Photosyntheis Research, 39 (1994), 235.  doi: 10.1007/BF00014586.  Google Scholar

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P. J. S. Franks, NPZ models of plankton dynamics: Their construction, coupling to physics, and application,, J. Oceanography, 58 (2002), 379.   Google Scholar

[8]

H. I. Freedman and Y. T. Xu, Models of competition in the chemostat with instantaneous and delays nutrient recycling,, J. Math. Biol., 31 (1993), 513.  doi: 10.1007/BF00173890.  Google Scholar

[9]

J. P. Grover, D. L. Roelke and B. W. Brooks, Modeling of planton community dynamics characterized by algal toxicity and allelopathy: A focus on historical Promnesium parvum blooms in a Texas reservior,, Ecological Modeling, 227 (2012), 147.   Google Scholar

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B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation,, Cambridge University, (1981).   Google Scholar

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S. B. Hsu and P. Waltman, A survey of mathematical models fo compettion with an inhibitor,, Math. Biosci., 187 (2004), 53.  doi: 10.1016/j.mbs.2003.07.004.  Google Scholar

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A. Huppert, B. Blasius and L. Stone, A model of phytoplankton blooms,, The American Naturalist, 159 (2002), 156.  doi: 10.1086/324789.  Google Scholar

[14]

A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms,, J. Theoret. Biol., 236 (2005), 276.  doi: 10.1016/j.jtbi.2005.03.012.  Google Scholar

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S. E. Jorgenson, A eutrophication model for a lake,, Ecol. Model., 2 (1976), 147.  doi: 10.1016/0304-3800(76)90030-2.  Google Scholar

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Yu. A. Kuznetsov, Elements of Apllied Bifurcation Theory,, $2^{nd}$ edition, (1998).   Google Scholar

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C. Lalli and T. Parsons, Biological Oceanography: An Introduction,, Butterworth-Heinemann, (1993).   Google Scholar

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J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method,, Academic Press, (1961).   Google Scholar

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W. M. Liu, Crition of Hopf bifurcations without using eigenvalues,, J. Math. Anal. Appl., 182 (1994), 250.  doi: 10.1006/jmaa.1994.1079.  Google Scholar

[20]

A. Mukhopadhyay, J. Chattopadhyay and P. K. Tapaswi, A delay differential equations of plankton allelopathy,, Math. Biosci., 149 (1998), 167.  doi: 10.1016/S0025-5564(98)00005-4.  Google Scholar

[21]

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

[22]

R. Pal, D. Basu and M. Banerjee, Modeling of phytoplankton allelpathy with Monod-Haldane-type functional response-a mathematical study,, BioSystems, 95 (2009), 243.   Google Scholar

[23]

J. Jiang, Z. Qiu, J. Wu and H. Zhu, Threshold conditions for West Nile virus outbreaks,, Bull. Math. Biol., 71 (2009), 627.  doi: 10.1007/s11538-008-9374-6.  Google Scholar

[24]

G. A. Riley, H. Stommel and D. P. Burrpus, Qualitative ecology of the plankton of the Western North Atlantic,, Bull. Bingh. Ocean. Coll., 12 (1949), 1.   Google Scholar

[25]

S. Roy, The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy,, Theoret. Populat. Biol., 75 (2009), 68.  doi: 10.1016/j.tpb.2008.11.003.  Google Scholar

[26]

S. G. Ruan and X. Z. He, Global stability in Chemostat-type competition models with nutrient recycling,, SIAM J. Appl. Math., 58 (1998), 170.  doi: 10.1137/S0036139996299248.  Google Scholar

[27]

J. B. Shukla, A. K. Misra and P. Chandra, Modeling and analysis of the algal bloom in a lake caused by discharge of nutrients,, Applied Math. Comput., 196 (2008), 782.  doi: 10.1016/j.amc.2007.07.010.  Google Scholar

[28]

H. L. Smith and P. Waltman, Pertubation of a globally stable steady state,, Proc. Amer. Math. Soc., 127 (1999), 447.  doi: 10.1090/S0002-9939-99-04768-1.  Google Scholar

[29]

J. Sole, E. Garcia-Ladona, P. Ruardij and M. Estrada, Modelling allelopathy among marine algae,, Ecol. Model., 183 (2005), 373.  doi: 10.1016/j.ecolmodel.2004.08.021.  Google Scholar

[30]

J. H. Steele and E. W. Henderson, A simple plankton model,, The American Naturalist, 117 (1981), 676.  doi: 10.1086/283752.  Google Scholar

[31]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[32]

H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus,, Math. Biosci., 227 (2010), 20.  doi: 10.1016/j.mbs.2010.05.006.  Google Scholar

[33]

W. Wang and X. Zhao, An epidemic model in a patchy environment,, Math. Biosci., 190 (2004), 97.  doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[34]

J. S. Wroblewski, J. L. Sarmiento and G. R. Fliel, An ocean basin scale model of plankton dynamics in the North Atlantic. Solutions for the climatological oceanographic condition in May,, Global Biogeochem. Cycles., 2 (1988), 199.  doi: 10.1029/GB002i003p00199.  Google Scholar

[35]

T. Yoshizawa, Stability Theory by Liapunov's Second Method,, The mathematical Society of Japan, (1966).   Google Scholar

show all references

References:
[1]

S. Busenberg, S. K. Kumar, P. Austin and G. Wake, The dynamics of a model of a plankton-nutrient interaction,, Bull. Math. Biol., 52 (1990), 677.   Google Scholar

[2]

J. Chattopadhyay, R. R. Sarker and S. Mandal, Toxin producing plankton may act as a biological control for plankton blooms-filed study and mathemtical modelling,, J. Theor. Biol., 215 (2002), 333.   Google Scholar

[3]

R. Cropp and J. Norbury, Simple predator-prey interactions control dynamics in a plankton food web model,, Ecol. Model., 220 (2009), 1552.  doi: 10.1016/j.ecolmodel.2009.04.003.  Google Scholar

[4]

A. W. Edwards and J. Brindley, Oscillatory behavior in a three-component plankton population model,, Dynam. Stabili. Syst., 11 (1996), 347.   Google Scholar

[5]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models,, Bull. Math. Biol., 61 (1999), 303.  doi: 10.1006/bulm.1998.0082.  Google Scholar

[6]

P. G. Falkowski, The role of phytoplankton photosynthesis in global biogeochemical cycles,, Photosyntheis Research, 39 (1994), 235.  doi: 10.1007/BF00014586.  Google Scholar

[7]

P. J. S. Franks, NPZ models of plankton dynamics: Their construction, coupling to physics, and application,, J. Oceanography, 58 (2002), 379.   Google Scholar

[8]

H. I. Freedman and Y. T. Xu, Models of competition in the chemostat with instantaneous and delays nutrient recycling,, J. Math. Biol., 31 (1993), 513.  doi: 10.1007/BF00173890.  Google Scholar

[9]

J. P. Grover, D. L. Roelke and B. W. Brooks, Modeling of planton community dynamics characterized by algal toxicity and allelopathy: A focus on historical Promnesium parvum blooms in a Texas reservior,, Ecological Modeling, 227 (2012), 147.   Google Scholar

[10]

J. Hale, Ordinary Differential Equations,, Krieger, (1980).   Google Scholar

[11]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation,, Cambridge University, (1981).   Google Scholar

[12]

S. B. Hsu and P. Waltman, A survey of mathematical models fo compettion with an inhibitor,, Math. Biosci., 187 (2004), 53.  doi: 10.1016/j.mbs.2003.07.004.  Google Scholar

[13]

A. Huppert, B. Blasius and L. Stone, A model of phytoplankton blooms,, The American Naturalist, 159 (2002), 156.  doi: 10.1086/324789.  Google Scholar

[14]

A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms,, J. Theoret. Biol., 236 (2005), 276.  doi: 10.1016/j.jtbi.2005.03.012.  Google Scholar

[15]

S. E. Jorgenson, A eutrophication model for a lake,, Ecol. Model., 2 (1976), 147.  doi: 10.1016/0304-3800(76)90030-2.  Google Scholar

[16]

Yu. A. Kuznetsov, Elements of Apllied Bifurcation Theory,, $2^{nd}$ edition, (1998).   Google Scholar

[17]

C. Lalli and T. Parsons, Biological Oceanography: An Introduction,, Butterworth-Heinemann, (1993).   Google Scholar

[18]

J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method,, Academic Press, (1961).   Google Scholar

[19]

W. M. Liu, Crition of Hopf bifurcations without using eigenvalues,, J. Math. Anal. Appl., 182 (1994), 250.  doi: 10.1006/jmaa.1994.1079.  Google Scholar

[20]

A. Mukhopadhyay, J. Chattopadhyay and P. K. Tapaswi, A delay differential equations of plankton allelopathy,, Math. Biosci., 149 (1998), 167.  doi: 10.1016/S0025-5564(98)00005-4.  Google Scholar

[21]

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

[22]

R. Pal, D. Basu and M. Banerjee, Modeling of phytoplankton allelpathy with Monod-Haldane-type functional response-a mathematical study,, BioSystems, 95 (2009), 243.   Google Scholar

[23]

J. Jiang, Z. Qiu, J. Wu and H. Zhu, Threshold conditions for West Nile virus outbreaks,, Bull. Math. Biol., 71 (2009), 627.  doi: 10.1007/s11538-008-9374-6.  Google Scholar

[24]

G. A. Riley, H. Stommel and D. P. Burrpus, Qualitative ecology of the plankton of the Western North Atlantic,, Bull. Bingh. Ocean. Coll., 12 (1949), 1.   Google Scholar

[25]

S. Roy, The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy,, Theoret. Populat. Biol., 75 (2009), 68.  doi: 10.1016/j.tpb.2008.11.003.  Google Scholar

[26]

S. G. Ruan and X. Z. He, Global stability in Chemostat-type competition models with nutrient recycling,, SIAM J. Appl. Math., 58 (1998), 170.  doi: 10.1137/S0036139996299248.  Google Scholar

[27]

J. B. Shukla, A. K. Misra and P. Chandra, Modeling and analysis of the algal bloom in a lake caused by discharge of nutrients,, Applied Math. Comput., 196 (2008), 782.  doi: 10.1016/j.amc.2007.07.010.  Google Scholar

[28]

H. L. Smith and P. Waltman, Pertubation of a globally stable steady state,, Proc. Amer. Math. Soc., 127 (1999), 447.  doi: 10.1090/S0002-9939-99-04768-1.  Google Scholar

[29]

J. Sole, E. Garcia-Ladona, P. Ruardij and M. Estrada, Modelling allelopathy among marine algae,, Ecol. Model., 183 (2005), 373.  doi: 10.1016/j.ecolmodel.2004.08.021.  Google Scholar

[30]

J. H. Steele and E. W. Henderson, A simple plankton model,, The American Naturalist, 117 (1981), 676.  doi: 10.1086/283752.  Google Scholar

[31]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[32]

H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus,, Math. Biosci., 227 (2010), 20.  doi: 10.1016/j.mbs.2010.05.006.  Google Scholar

[33]

W. Wang and X. Zhao, An epidemic model in a patchy environment,, Math. Biosci., 190 (2004), 97.  doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[34]

J. S. Wroblewski, J. L. Sarmiento and G. R. Fliel, An ocean basin scale model of plankton dynamics in the North Atlantic. Solutions for the climatological oceanographic condition in May,, Global Biogeochem. Cycles., 2 (1988), 199.  doi: 10.1029/GB002i003p00199.  Google Scholar

[35]

T. Yoshizawa, Stability Theory by Liapunov's Second Method,, The mathematical Society of Japan, (1966).   Google Scholar

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