# American Institute of Mathematical Sciences

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

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

 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.
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] 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

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
 [1] 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 [2] Sze-Bi Hsu, Chiu-Ju Lin. Dynamics of two phytoplankton species competing for light and nutrient with internal storage. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1259-1285. doi: 10.3934/dcdss.2014.7.1259 [3] Sze-Bi Hsu, Feng-Bin Wang. On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1479-1501. doi: 10.3934/cpaa.2011.10.1479 [4] Mohammad A. Tabatabai, Wayne M. Eby, Sejong Bae, Karan P. Singh. A flexible multivariable model for Phytoplankton growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 913-923. doi: 10.3934/mbe.2013.10.913 [5] Jean-Jacques Kengwoung-Keumo. Dynamics of two phytoplankton populations under predation. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1319-1336. doi: 10.3934/mbe.2014.11.1319 [6] Khalid Boushaba. A multi layer method applied to a model of phytoplankton. Networks & Heterogeneous Media, 2007, 2 (1) : 37-54. doi: 10.3934/nhm.2007.2.37 [7] Christopher K.R.T. Jones, Bevin Maultsby. A dynamical approach to phytoplankton blooms. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 859-878. doi: 10.3934/dcds.2017035 [8] Jianquan Li, Yicang Zhou, Jianhong Wu, Zhien Ma. Complex dynamics of a simple epidemic model with a nonlinear incidence. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 161-173. doi: 10.3934/dcdsb.2007.8.161 [9] Hua Nie, Feng-Bin Wang. Competition for one nutrient with recycling and allelopathy in an unstirred chemostat. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2129-2155. doi: 10.3934/dcdsb.2015.20.2129 [10] Jean-Jacques Kengwoung-Keumo. Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation. Mathematical Biosciences & Engineering, 2016, 13 (4) : 787-812. doi: 10.3934/mbe.2016018 [11] Linfeng Mei, Wei Dong, Changhe Guo. Concentration phenomenon in a nonlocal equation modeling phytoplankton growth. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 587-597. doi: 10.3934/dcdsb.2015.20.587 [12] Chiu-Ju Lin. Competition of two phytoplankton species for light with wavelength. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 523-536. doi: 10.3934/dcdsb.2016.21.523 [13] Ming Chen, Meng Fan, Xing Yuan, Huaiping Zhu. Effect of seasonal changing temperature on the growth of phytoplankton. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1091-1117. doi: 10.3934/mbe.2017057 [14] Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112 [15] Linfeng Mei, Sze-Bi Hsu, Feng-Bin Wang. Growth of single phytoplankton species with internal storage in a water column. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 607-620. doi: 10.3934/dcdsb.2016.21.607 [16] Danfeng Pang, Hua Nie, Jianhua Wu. Single phytoplankton species growth with light and crowding effect in a water column. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 41-74. doi: 10.3934/dcds.2019003 [17] Chuangxia Huang, Hua Zhang, Lihong Huang. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3337-3349. doi: 10.3934/cpaa.2019150 [18] Meihong Qiao, Anping Liu, Qing Tang. The dynamics of an HBV epidemic model on complex heterogeneous networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1393-1404. doi: 10.3934/dcdsb.2015.20.1393 [19] Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221 [20] Hermann J. Eberl, Messoud A. Efendiev, Dariusz Wrzosek, Anna Zhigun. Analysis of a degenerate biofilm model with a nutrient taxis term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 99-119. doi: 10.3934/dcds.2014.34.99

2018 Impact Factor: 1.008