一类状态脉冲反馈控制下的营养盐–浮游植物动力系统的相关动力学分析
Related Dynamic Analysis of a Nutrient-Phytoplankton Dynamic System under State Impulsive Feedback Control
DOI: 10.12677/AAM.2018.72018, PDF,   
作者: 段 涵：温州大学，浙江 温州
关键词: 浮游植物；脉冲；周期解；稳定性；Phytoplankton； Impulsive； Periodic Solution； Stability
摘要: 基于状态脉冲反馈控制理论与非线性动力系统理论，本文在动态建模过程中引入Michaelis-Menten功能反应函数来刻画浮游植物与营养盐的相互作用机制，构建了一类状态脉冲反馈控制下的营养盐–浮游植物动力系统，对所建动力系统进行一些定性分析，建立了该动力系统阶-1周期解的存在性、唯一性和轨道渐近稳定的理论判据，这些研究工作为全面研究状态脉冲反馈控制理论在水体富营养化防治中的应用提供理论支撑。
Abstract: On the basis of state impulsive feedback control theory and nonlinear dynamic system theory, the Michaelis-Menten functional response function is introduced to describe the interaction mechanism of phytoplankton and nutrients in the process of dynamic modeling; a nutrient-phytoplankton dynamic system under state impulsive feedback control has been structured. Some qualitative analysis of the dynamic system has been investigated to establish some theoretical criterions for the existence, uniqueness and asymptotic stability of the order-1 periodic solution. The research work can provide a theoretical support for the comprehensive study of the application of state impulsive feedback control theory to the prevention and control of eutrophication in water body.
文章引用：段涵. 一类状态脉冲反馈控制下的营养盐–浮游植物动力系统的相关动力学分析[J]. 应用数学进展, 2018, 7(2): 145-151. https://doi.org/10.12677/AAM.2018.72018

参考文献

[1] Reigada, R., Hillary, R.M., Bees, M.A., Sancho, J.M. and Sagues, F. (2003) Plankton Blooms Induced by Turbulent Flows. Proceedings of the Royal Society B: Biological Sciences, 270, 875-880.
[Google Scholar] [CrossRef] [PubMed]
[2] Zhang, W.W. and Zhao, M. (2013) Dynamical Complexity of a Spatial Phytoplankton-Zooplankton Model with an Alternative Prey and Refuge Effect. Journal of Applied Mathematics, 2013, 707-724.
[3] Abbas, S., Banerjee, M. and Hungerbuhler, N. (2010) Existence, Uniqueness and Stability Analysis of Allelopathic Stimulatory Phytoplankton Model. Journal of Mathematical Analysis and Applications, 367, 249-259.
[Google Scholar] [CrossRef] 
[4] El Saadi, N. and Bah, A. (2011) Numerical Treatment of a Nonlocal Model for Phytoplankton Aggregation. Applied Mathematics and Computation, 218, 8279-8287.
[Google Scholar] [CrossRef] 
[5] Fan, A., Han, P. and Wang, K. (2013) Global Dynamics of a Nu-trient-Plankton System in the Water Ecosystem. Applied Mathematics and Computation, 219, 8269-8276.
[6] Dai, C. and Zhao, M. (2013) Bifurcation and Periodic Solutions for an Algae-Fish Semi-Continuous System. Abstract and Ap-plied Analysis, 2013, 1-11.
[7] Dai, C. and Zhao, M. (2014) Nonlinear Analysis in a Nutrient-Algae-Zooplankton System with Sinking of Algae. Abstract and Applied Analysis, 2014, 1-11.
[Google Scholar] [CrossRef] 
[8] Luo, J.H. (2013) Phytoplankton-Zooplankton Dynamics in Periodic Environments Taking into Account Eutrophication. Mathematical Biosciences, 245, 126-136.
[Google Scholar] [CrossRef] [PubMed]
[9] Jang, S.R. (2000) Dynamics of Variable-Yield Nutri-ent-Phytoplankton-Zooplankton Models with Nutrient Recycling and Self-Shading. Journal of Mathematical Biology, 40, 229-250.
[Google Scholar] [CrossRef] [PubMed]
[10] Alvarez-Vázquez, L.J., Fernández, F.J. and R. Muñoz-Sola, R. (2009) Mathematical Analysis of a Three-Dimensional Eutrophication Model. Journal of Mathematical Analysis and Applications, 349, 135-155.
[Google Scholar] [CrossRef] 
[11] Chakraborty, K., Jana, S. and Kar, T.K. (2012) Global Dynamics and Bifurcation in a Stage Structured Prey-Predator Fishery Model with Harvesting. Applied Mathematics and Computation, 218, 9271-9290.
[Google Scholar] [CrossRef] 
[12] Wang, Y., Zhao, M., Dai, C. and Pan, X. (2014) Nonlinear Dy-namics of a Nutrient-Plankton Model. Abstract and Applied Analysis, 2014, Article ID: 451757.
[Google Scholar] [CrossRef] 
[13] Gao, M., Shi, H. and Li, Z. (2009) Chaos in a Seasonally and Periodi-cally Forced Phytoplankton-Zooplankton System. Nonlinear Analysis Real World Applications, 10, 1643-1650.
[Google Scholar] [CrossRef] 
[14] Zhao, J. and Wei, J. (2009) Stability and Bifurcation in a Two Harmful Phytoplankton-Zooplankton System. Chaos Solitons and Fractals, 39, 1395-1409.
[Google Scholar] [CrossRef] 
[15] Yu, H., Zhao, M. and Wang, Q. (2013) Analysis of Mathematics and Sustainability in an Impulsive Eutrophication Controlling System. Abstract and Applied Analysis, 2013, Article ID: 726172.
[Google Scholar] [CrossRef] 
[16] Banerjee and Venturino, E. (2011) A Phytoplankton-Toxic Phytoplankton-Zooplankton Model. Ecological Complexity, 8, 239-248.
[Google Scholar] [CrossRef] 
[17] Deng, Y.L., Zhao, M., Yu, H.G. and Wang, Y. (2015) Dy-namical Analysis of a Nitrogen-Phosphorus-Phytoplankton Model. Discrete Dynamics in Nature and Society, 2015, Article ID: 823026.
[Google Scholar] [CrossRef] 
[18] Wang, P.F. and Zhao, M. (2016) Nonlinear Dy-namics of a Toxin-Phytoplankton-Zooplankton System with Self- and Cross-Diffusion. Discrete Dynamics in Nature and Society, 2016, Article ID: 4893451.
[Google Scholar] [CrossRef] 
[19] Smith, R.J. and Wolkowicz, G.S.K. (2003) Growth and Competition in the Nutrient Driven Self-Cycling Fermentation Process. Canadian Applied Mathematics Quarterly, 10, 171-177.
[20] Zhao, Z., Luo, C., Pang, L. and Chen, Y. (2016) Nonlinear Modeling of the Interaction between Phyto-plankton and Zooplankton with the Impulsive Feedback Control. Chaos Solitons and Fractals, 87, 255-261.
[Google Scholar] [CrossRef] 
[21] Pang, G.P. and Chen, L.S. (2014) Periodic Solution of the Sys-tem with Impulsive State Feedback Control. Nonlinear Dynamics, 78, 743-753.
[Google Scholar] [CrossRef] 
[22] Zhao, Z., Pang, L.Y. and Song, X.Y. (2017) Optimal Control of Phytoplankton-Fish Model with the Impulsive Feedback Control. Nonlinear Dynamics, 88, 2003-2011.
[Google Scholar] [CrossRef] 
[23] Dai, C.J., Zhao, M. and Chen, L.S. (2012) Dynamic Complexity of an Ivlev-Type Prey-Predator System with Impulsive State Feedback Control. Journal of Applied Mathematics, 2012, 341-360.
[Google Scholar] [CrossRef] 
[24] Nie, L.F., Teng, Z.D., Hu, L. and Peng, J.G. (2009) Existence and Stability of Periodic Solution of a Predator-Prey Model with State-Dependent Impulsive Effects. Mathematics and Computers in Simulation, 79, 2122-2134.
[Google Scholar] [CrossRef] 
[25] Jiang, G. and Lu, Q. (2007) Impulsive State Feedback Control of a Predator-Prey Model. Journal of Computational and Applied Mathematics, 200, 193-207.
[Google Scholar] [CrossRef] 
[26] Jiang, G. and Lu, Q. (2006) The Dynamics of a Prey-Predator Model with Impulsive State Feedback Control. Discrete and Continuous Dynamical Systems, 6, 1301-1320.
[Google Scholar] [CrossRef] 
[27] Liu, X. and Chen, L. (2003) Complex Dynamics of Holling Type II Lotka-Volterra Predator-Prey System with Impulsive Perturbations on the Predator. Chaos Solitons and Fractals, 2, 311-320.
[Google Scholar] [CrossRef] 
[28] Liu, B., Zhang, Y. and Chen, L. (2005) Dynamic Com-plexities in a Lotka-Volterra Predator-Prey Model Concerning Impulsive Control Strategy. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2, 517-531.
[Google Scholar] [CrossRef] 
[29] Samoilenko, A.M. and Perestyuk, N.A. (1995) Impulsive Dif-ferential Equations. World Scientific, Singapore.
[Google Scholar] [CrossRef] 
[30] Zavalishchin, S.T. and Sesekin, A.N. (1997) Dynamic Impulse Systems Theory and Applications. Mathematics and Its Applications. Kluwer, Dordrecht, 394.
[31] Lakshmikantham, V. and Liu, X. (1989) On Quasi-Stability for Impulsive Differential Systems. Nonlinear Analysis, 13, 819-828.
[Google Scholar] [CrossRef] 
[32] Li, Z.X., Chen, L.S. and Liu, Z.J. (2012) Periodic Solution of a Chemostat Model with Variable Yield and Impulsive State Feedback Control. Applied Mathematical Modelling, 36, 1255-1266.
[33] Zhao, Z., Zhang, J.Y., Pang, L.Y. and Chen, Y. (2015) Nonlinear Modelling of Ethanol Inhibition with the State Feedback Control. Journal of Applied Mathematics and Computing, 48, 205-219.
[34] Dai, C.J., Zhao, M. and Yu, H.G. (2016) Dynamics Induced by Delay in a Nutrient-Phytoplankton Model with Diffusion. Ecological Complexity, 26, 29-36.
[Google Scholar] [CrossRef] 
[35] Liang, Z.Q. and Zeng, X.P. (2017) Periodic Solution of a Leslie Predator-Prey System with Ratio-Dependent and State Impulsive Feedback Control. Nonlinear Dynamics, 89, 2941-2955.
[Google Scholar] [CrossRef] 

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