# 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 and 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), 677-696. [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), 333-344. [3] R. Cropp and J. Norbury, Simple predator-prey interactions control dynamics in a plankton food web model, Ecol. Model., 220 (2009), 1552-1565. doi: 10.1016/j.ecolmodel.2009.04.003. [4] A. W. Edwards and J. Brindley, Oscillatory behavior in a three-component plankton population model, Dynam. Stabili. Syst., 11 (1996), 347-370. [5] A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models, Bull. Math. Biol., 61 (1999), 303-339. doi: 10.1006/bulm.1998.0082. [6] P. G. Falkowski, The role of phytoplankton photosynthesis in global biogeochemical cycles, Photosyntheis Research, 39 (1994), 235-258. doi: 10.1007/BF00014586. [7] P. J. S. Franks, NPZ models of plankton dynamics: Their construction, coupling to physics, and application, J. Oceanography, 58 (2002), 379-387. [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), 513-527. doi: 10.1007/BF00173890. [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), 147-161. [10] J. Hale, Ordinary Differential Equations, Krieger, Malabar, 1980. [11] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University, Cambridge, 1981. [12] S. B. Hsu and P. Waltman, A survey of mathematical models fo compettion with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004. [13] A. Huppert, B. Blasius and L. Stone, A model of phytoplankton blooms, The American Naturalist, 159 (2002), 156-171. doi: 10.1086/324789. [14] A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theoret. Biol., 236 (2005), 276-290. doi: 10.1016/j.jtbi.2005.03.012. [15] S. E. Jorgenson, A eutrophication model for a lake, Ecol. Model., 2 (1976), 147-165. doi: 10.1016/0304-3800(76)90030-2. [16] Yu. A. Kuznetsov, Elements of Apllied Bifurcation Theory, $2^{nd}$ edition, Springer-Verlag, New York, 1998. [17] C. Lalli and T. Parsons, Biological Oceanography: An Introduction, Butterworth-Heinemann, 1993. [18] J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, Academic Press, New York, 1961. [19] W. M. Liu, Crition of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256. doi: 10.1006/jmaa.1994.1079. [20] A. Mukhopadhyay, J. Chattopadhyay and P. K. Tapaswi, A delay differential equations of plankton allelopathy, Math. Biosci., 149 (1998), 167-189. doi: 10.1016/S0025-5564(98)00005-4. [21] B. Mukhopadhyay and R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecological Modelling, 198 (2006), 163-173. doi: 10.1016/j.ecolmodel.2006.04.005. [22] R. Pal, D. Basu and M. Banerjee, Modeling of phytoplankton allelpathy with Monod-Haldane-type functional response-a mathematical study, BioSystems, 95 (2009), 243-253. [23] J. Jiang, Z. Qiu, J. Wu and H. Zhu, Threshold conditions for West Nile virus outbreaks, Bull. Math. Biol., 71 (2009), 627-647. doi: 10.1007/s11538-008-9374-6. [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), 1-169. [25] S. Roy, The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy, Theoret. Populat. Biol., 75 (2009), 68-75. doi: 10.1016/j.tpb.2008.11.003. [26] S. G. Ruan and X. Z. He, Global stability in Chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math., 58 (1998), 170-192. doi: 10.1137/S0036139996299248. [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), 782-790. doi: 10.1016/j.amc.2007.07.010. [28] H. L. Smith and P. Waltman, Pertubation of a globally stable steady state, Proc. Amer. Math. Soc., 127 (1999), 447-453. doi: 10.1090/S0002-9939-99-04768-1. [29] J. Sole, E. Garcia-Ladona, P. Ruardij and M. Estrada, Modelling allelopathy among marine algae, Ecol. Model., 183 (2005), 373-384. doi: 10.1016/j.ecolmodel.2004.08.021. [30] J. H. Steele and E. W. Henderson, A simple plankton model, The American Naturalist, 117 (1981), 676-691. doi: 10.1086/283752. [31] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [32] H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Math. Biosci., 227 (2010), 20-28. doi: 10.1016/j.mbs.2010.05.006. [33] W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [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), 199-218. doi: 10.1029/GB002i003p00199. [35] T. Yoshizawa, Stability Theory by Liapunov's Second Method, The mathematical Society of Japan, Tokyo, 1966.

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##### 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), 677-696. [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), 333-344. [3] R. Cropp and J. Norbury, Simple predator-prey interactions control dynamics in a plankton food web model, Ecol. Model., 220 (2009), 1552-1565. doi: 10.1016/j.ecolmodel.2009.04.003. [4] A. W. Edwards and J. Brindley, Oscillatory behavior in a three-component plankton population model, Dynam. Stabili. Syst., 11 (1996), 347-370. [5] A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton population models, Bull. Math. Biol., 61 (1999), 303-339. doi: 10.1006/bulm.1998.0082. [6] P. G. Falkowski, The role of phytoplankton photosynthesis in global biogeochemical cycles, Photosyntheis Research, 39 (1994), 235-258. doi: 10.1007/BF00014586. [7] P. J. S. Franks, NPZ models of plankton dynamics: Their construction, coupling to physics, and application, J. Oceanography, 58 (2002), 379-387. [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), 513-527. doi: 10.1007/BF00173890. [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), 147-161. [10] J. Hale, Ordinary Differential Equations, Krieger, Malabar, 1980. [11] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University, Cambridge, 1981. [12] S. B. Hsu and P. Waltman, A survey of mathematical models fo compettion with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004. [13] A. Huppert, B. Blasius and L. Stone, A model of phytoplankton blooms, The American Naturalist, 159 (2002), 156-171. doi: 10.1086/324789. [14] A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theoret. Biol., 236 (2005), 276-290. doi: 10.1016/j.jtbi.2005.03.012. [15] S. E. Jorgenson, A eutrophication model for a lake, Ecol. Model., 2 (1976), 147-165. doi: 10.1016/0304-3800(76)90030-2. [16] Yu. A. Kuznetsov, Elements of Apllied Bifurcation Theory, $2^{nd}$ edition, Springer-Verlag, New York, 1998. [17] C. Lalli and T. Parsons, Biological Oceanography: An Introduction, Butterworth-Heinemann, 1993. [18] J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, Academic Press, New York, 1961. [19] W. M. Liu, Crition of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256. doi: 10.1006/jmaa.1994.1079. [20] A. Mukhopadhyay, J. Chattopadhyay and P. K. Tapaswi, A delay differential equations of plankton allelopathy, Math. Biosci., 149 (1998), 167-189. doi: 10.1016/S0025-5564(98)00005-4. [21] B. Mukhopadhyay and R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecological Modelling, 198 (2006), 163-173. doi: 10.1016/j.ecolmodel.2006.04.005. [22] R. Pal, D. Basu and M. Banerjee, Modeling of phytoplankton allelpathy with Monod-Haldane-type functional response-a mathematical study, BioSystems, 95 (2009), 243-253. [23] J. Jiang, Z. Qiu, J. Wu and H. Zhu, Threshold conditions for West Nile virus outbreaks, Bull. Math. Biol., 71 (2009), 627-647. doi: 10.1007/s11538-008-9374-6. [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), 1-169. [25] S. Roy, The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy, Theoret. Populat. Biol., 75 (2009), 68-75. doi: 10.1016/j.tpb.2008.11.003. [26] S. G. Ruan and X. Z. He, Global stability in Chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math., 58 (1998), 170-192. doi: 10.1137/S0036139996299248. [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), 782-790. doi: 10.1016/j.amc.2007.07.010. [28] H. L. Smith and P. Waltman, Pertubation of a globally stable steady state, Proc. Amer. Math. Soc., 127 (1999), 447-453. doi: 10.1090/S0002-9939-99-04768-1. [29] J. Sole, E. Garcia-Ladona, P. Ruardij and M. Estrada, Modelling allelopathy among marine algae, Ecol. Model., 183 (2005), 373-384. doi: 10.1016/j.ecolmodel.2004.08.021. [30] J. H. Steele and E. W. Henderson, A simple plankton model, The American Naturalist, 117 (1981), 676-691. doi: 10.1086/283752. [31] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [32] H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Math. Biosci., 227 (2010), 20-28. doi: 10.1016/j.mbs.2010.05.006. [33] W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [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), 199-218. doi: 10.1029/GB002i003p00199. [35] T. Yoshizawa, Stability Theory by Liapunov's Second Method, The mathematical Society of Japan, Tokyo, 1966.
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