Stability and selective harvesting of a phytoplankton-zooplankton system. (English) Zbl 1442.92198

Summary: Considering that some zooplankton can be harvested for food in some bodies of water, a phytoplankton-zooplankton model with continuous harvesting of zooplankton only is proposed and investigated. By using environmental carrying capacity as a parameter, possible dynamic behaviors, such as stability, global stability, Hopf bifurcation, and transcritical bifurcations, are analyzed. The optimal harvesting policy is disposed by imposing a tax per unit biomass of zooplankton. The problem of determining the optimal harvest policy is solved by using Pontryagin’s maximum principle subject to the state equations and the control constraints, and the impact of tax is also discussed. Finally, some numerical simulations are performed to justify analytical findings.


92D40 Ecology
37N25 Dynamical systems in biology
Full Text: DOI


[1] Chattopadhayay, J.; Sarkar, R. R.; Mandal, S., Toxin-producing plankton may act as a biological control for planktonic blooms—Field study and mathematical modelling, Journal of Theoretical Biology, 215, 3, 333-344 (2002) · doi:10.1006/jtbi.2001.2510
[2] Chattopadhyay, J.; Sarkar, R. R.; El Abdllaoui, A., A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA Journal of Mathematics Applied in Medicine and Biology, 19, 2, 137-161 (2002) · Zbl 1013.92046 · doi:10.1093/imammb/19.2.137
[3] He, X.; Ruan, S., Global stability in chemostat-type plankton models with delayed nutrient recycling, Journal of Mathematical Biology, 37, 3, 253-271 (1998) · Zbl 0935.92034 · doi:10.1007/s002850050128
[4] Das, K.; Ray, S., Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecological Modelling, 215, 1-3, 69-76 (2008) · doi:10.1016/j.ecolmodel.2008.02.019
[5] Roy, S., The coevolution of two phytoplankton species on a single resource: allelopathy as a pseudo-mixotrophy, Theoretical Population Biology, 75, 1, 68-75 (2009) · Zbl 1210.92058 · doi:10.1016/j.tpb.2008.11.003
[6] Gakkhar, S.; Singh, A., A delay model for viral infection in toxin producing phytoplankton and zooplankton system, Communications in Nonlinear Science and Numerical Simulation, 15, 11, 3607-3620 (2010) · Zbl 1222.37098 · doi:10.1016/j.cnsns.2010.01.010
[7] Saha, T.; Bandyopadhyay, M., Dynamical analysis of toxin producing Phytoplankton-Zooplankton interactions, Nonlinear Analysis: Real World Applications, 10, 1, 314-332 (2009) · Zbl 1154.37384 · doi:10.1016/j.nonrwa.2007.09.001
[8] Zhao, J.; Wei, J., Stability and bifurcation in a two harmful phytoplankton-zooplankton system, Chaos, Solitons and Fractals, 39, 3, 1395-1409 (2009) · Zbl 1197.37131 · doi:10.1016/j.chaos.2007.05.019
[9] Kar, T. K., Management of a fishery based on continuous fishing effort, Nonlinear Analysis: Real World Applications, 5, 4, 629-644 (2004) · Zbl 1102.91349 · doi:10.1016/j.nonrwa.2004.01.003
[10] Jang, S.; Baglama, J.; Rick, J., Nutrient-phytoplankton-zooplankton models with a toxin, Mathematical and Computer Modelling, 43, 1-2, 105-118 (2006) · Zbl 1086.92053 · doi:10.1016/j.mcm.2005.09.030
[11] Clark, C. W., Bioeconomic Moddelling and Fisheries Management (1985), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[12] Clark, C. W., Mathematical Bio-Economics: The Optimal Management of Renewable Resources (1990), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0712.90018
[13] Kar, T. K., Conservation of a fishery through optimal taxation: a dynamic reaction model, Communications in Nonlinear Science and Numerical Simulation, 10, 2, 121-131 (2005) · Zbl 1058.34060 · doi:10.1016/S1007-5704(03)00101-1
[14] Ganguly, S.; Chaudhuri, K. S., Regulation of a single-species fishery by taxation, Ecological Modelling, 82, 1, 51-60 (1995) · doi:10.1016/0304-3800(94)00079-W
[15] Chaudhuri, K.; Johnson, T., Bioeconomic dynamics of a fishery modeled as an S-system, Mathematical Biosciences, 99, 2, 231-249 (1990) · Zbl 0699.92024 · doi:10.1016/0025-5564(90)90006-K
[16] Pradhan, T.; Chaudhuri, K. S., Bioeconomic modelling of a single species fishery with Gompertz law of growth, Journal of Biological Systems, 6, 4, 393-409 (1998)
[17] Hale, J. K., Theory of Functional Differential Equations (1976), New York, NY, USA: Springer, New York, NY, USA
[18] Anderson, D. M., Turning back the harmful red tide, Nature, 338, 513-514 (1997)
[19] Edmonson, W. T., Daphia in lake Washington, Limnology and Oceanography, 27, 272-293 (1982)
[20] Shapiro, J.; Lamarra, V.; Lynch, M., Water Quality Management Through Biological Control (1975), University of Florida
[21] Arrow, K. J.; Kurz, M., Public Investment, The Rate of Return and Optimal Fiscal Policy (1970), Baltimore, Md, USA: John Hopkins, Baltimore, Md, USA
[22] Pontryagin, L. S.; Boltyanski, V. G.; Gamkrelidze, R. V.; Mishchenco, E. F., The Mathematical Theory of Optimal Processes (1987), New York, NY, USA: John Wiley & Sons, New York, NY, USA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.