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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 |
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. |
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), 677-696. |
[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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