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Stability and Hopf bifurcation analysis of a nutrient-phytoplankton model with delay effect. (English) Zbl 1470.37114

Summary: A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

MSC:

37N25 Dynamical systems in biology
92D25 Population dynamics (general)
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
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[1] Riley, G. A.; Stommel, H.; Bumpus, D. F., Quantitative ecology of the plankton of the western North Atlantic, Bulletin of the Bingham Oceanographic Collection, 12, 1-169 (1949)
[2] Patten, B. C., Mathematical models of plankton production, Internationale Revue der gesamten Hydrobiologie und Hydrographie, 53, 3, 357-408 (1968) · doi:10.1002/iroh.19680530302
[3] Platt, T.; Denman, K. L.; Jassby, A. D.; Goldberg, E. D.; McCase, I. N.; O’Brien, J.; Steele, J. H., Modelling the productivity of phytoplankton, The Sea, Vol. 6: Marine Modelling, 857-890 (1977), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[4] Zhang, W.; Zhao, M., Dynamical complexity of a spatial phytoplankton-zooplankton model with an alternative prey and refuge effect, Journal of Applied Mathematics, 2013 (2013) · Zbl 1266.92070 · doi:10.1155/2013/608073
[5] Evans, G. T.; Parslow, J. S., A model of annual plankton cycles, Biological Oceanography, 3, 3, 327-427 (1985)
[6] Steele, J. H.; Henderson, E. W., Simulation of vertical structure in a plankton ecosystem, Scottish Fisheries Research Report, 5 (1976)
[7] Taylor, A. H., Characteristic properties of models for the vertical distribution of phytoplankton under stratification, Ecological Modelling, 40, 3-4, 175-199 (1988)
[8] Busenberg, S.; Kumar, S. K.; Austin, P.; Wake, G., The dynamics of a model of a plankton-nutrient interaction, Bulletin of Mathematical Biology, 52, 5, 677-696 (1990) · Zbl 0704.92020 · doi:10.1016/S0092-8240(05)80373-8
[9] Ruan, S. G., Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, Journal of Mathematical Biology, 31, 6, 633-654 (1993) · Zbl 0779.92021 · doi:10.1007/BF00161202
[10] Taylor, A. H.; Harris, J. R. W.; Aiken, J.; Nihoul, J. C., The interaction of physical and biological process in a model of the vertical distribution of phytoplankton under stratification, Marine Interfaces Ecohydrodynamics. Marine Interfaces Ecohydrodynamics, Elsevier Oceanography Series, 42, 313-330 (1986), Elsevier
[11] Medvinsky, A. B.; Petrovskii, S. V.; Tikhonova, I. A.; Malchow, H.; Li, B.-L., Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44, 3, 311-370 (2002) · Zbl 1001.92050 · doi:10.1137/S0036144502404442
[12] Pardo, O., Global stability for a phytoplankton-nutrient system, Journal of Biological Systems, 8, 2, 195-209 (2000)
[13] Li, Y.; Li, C., Stability and Hopf bifurcation analysis on a delayed Leslie-Gower predator-prey system incorporating a prey refuge, Applied Mathematics and Computation, 219, 9, 4576-4589 (2013) · Zbl 1418.34090 · doi:10.1016/j.amc.2012.10.069
[14] Zhang, C.-H.; Yan, X.-P.; Cui, G.-H., Hopf bifurcations in a predator-prey system with a discrete delay and a distributed delay, Nonlinear Analysis: Real World Applications, 11, 5, 4141-4153 (2010) · Zbl 1206.34104 · doi:10.1016/j.nonrwa.2010.05.001
[15] Lian, F.; Xu, Y., Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay, Applied Mathematics and Computation, 215, 4, 1484-1495 (2009) · Zbl 1187.34116 · doi:10.1016/j.amc.2009.07.003
[16] Chen, Y.; Zhang, F., Dynamics of a delayed predator-prey model with predator migration, Applied Mathematical Modelling, 37, 3, 1400-1412 (2013) · Zbl 1351.34082 · doi:10.1016/j.apm.2012.04.012
[17] Jana, S.; Chakraborty, M.; Chakraborty, K.; Kar, T. K., Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge, Mathematics and Computers in Simulation, 85, 57-77 (2012) · Zbl 1258.34161 · doi:10.1016/j.matcom.2012.10.003
[18] Zuo, W.; Wei, J., Stability and Hopf bifurcation in a diffusive predatory-prey system with delay effect, Nonlinear Analysis: Real World Applications, 12, 4, 1998-2011 (2011) · Zbl 1221.35053 · doi:10.1016/j.nonrwa.2010.12.016
[19] Hu, G.-P.; Li, W.-T., Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects, Nonlinear Analysis: Real World Applications, 11, 2, 819-826 (2010) · Zbl 1181.37119 · doi:10.1016/j.nonrwa.2009.01.027
[20] Zhu, Y.; Cai, Y.; Yan, S.; Wang, W., Dynamical analysis of a delayed reaction-diffusion predator-prey system, Abstract and Applied Analysis, 2012 (2012) · Zbl 1256.35183 · doi:10.1155/2012/323186
[21] Lian, X.; Yan, S.; Wang, H., Pattern formation in predator-prey model with delay and cross diffusion, Abstract and Applied Analysis, 2013 (2013) · Zbl 1297.35030 · doi:10.1155/2013/147232
[22] Zhao, M.; Wang, X.; Yu, H.; Zhu, J., Dynamics of an ecological model with impulsive control strategy and distributed time delay, Mathematics and Computers in Simulation, 82, 8, 1432-1444 (2012) · Zbl 1251.92049 · doi:10.1016/j.matcom.2011.08.009
[23] Yu, H.; Zhao, M.; Agarwal, R. P., Stability and dynamics analysis of time delayed eutrophication ecological model based upon the Zeya reservoir, Mathematics and Computers in Simulation, 97, 53-67 (2014) · Zbl 1461.92132 · doi:10.1016/j.matcom.2013.06.008
[24] Zhao, M., Hopf bifurcation analysis for a semiratio-dependent predator-prey system with two delays, Abstract and Applied Analysis, 2013 (2013) · Zbl 1297.34094 · doi:10.1155/2013/495072
[25] Yan, S.; Lian, X.; Wang, W.; Upadhyay, R. K., Spatiotemporal dynamics in a delayed diffusive predator model, Applied Mathematics and Computation, 224, 524-534 (2013) · Zbl 1334.92382 · doi:10.1016/j.amc.2013.08.045
[26] Zhang, W.; Liu, H.; Xu, C., Local bifurcations for a delay differential model of plankton allelopathy, International Journal of Nonlinear Science, 15, 4, 340-349 (2013) · Zbl 1394.92116
[27] Lu, Z.; Liu, X., Analysis of a predator-prey model with modified Holling-Tanner functional response and time delay, Nonlinear Analysis: Real World Applications, 9, 2, 641-650 (2008) · Zbl 1142.34053 · doi:10.1016/j.nonrwa.2006.12.016
[28] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation. Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41 (1981), Cambridge, Mass, USA: Cambridge University Press, Cambridge, Mass, USA · Zbl 0474.34002
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