一类具有双时滞效应的溶解氧-浮游生物模型的动力学分析
Dynamic Analysis of a Dissolved Oxygen-Plankton Model with Two Time Delays
DOI: 10.12677/AAM.2017.67099, PDF, HTML, XML, 下载: 1,593  浏览: 1,891  国家自然科学基金支持
作者: 郭庆:温州大学,浙江 温州
关键词: 溶解氧浮游生物时滞Hopf分支稳定性Dissolved Oxygen Plankton Two Time Delays Hopf Bifurcation Stability
摘要: 基于海洋生态系统内在动态变化特性和海洋生物种群之间的动态作用机制,本论文构建了一类具有双时滞效应的溶解氧-浮游生物动态模型,并对其相关动力学性质进行理论分析,解析出模型内平衡点具有局部渐近稳定性和发生Hopf分支的阈值条件,并详细探析了Hopf分支的相关动力学性质。本研究成果有利于从动力学的角度揭示溶解氧与海洋浮游生物之间的相互作用机制,有助于深入理解时滞效应如何影响海洋生态系统的动态运行趋势,为进一步研究海洋浮游生物之间相互制约、相互协调的生长动态机制提供一定的理论支撑。
Abstract: Based on the internal dynamic characteristics of marine ecosystem and the dynamic mechanism of marine biological population, this paper constructed a dissolved oxygen-plankton dynamic model involving two time delays, and made theoretical analysis of corresponding dynamics. We obtained the locally asymptotic stability of positive equilibrium and the threshold conditions of occurring Hopf bifurcation were gained. Therefore, we analyzed the dynamics of Hopf bifurcation in detail. This result provides a great help for the interaction between dissolved oxygen and marine plankton in dynamics, and is helpful to deeply understand how the delay affects the dynamic trend of the marine ecosystem, and furthermore provide certain theoretical support for the study of dynamic growth mechanism of mutual restriction and mutual coordination in marine plankton.
文章引用:郭庆. 一类具有双时滞效应的溶解氧-浮游生物模型的动力学分析[J]. 应用数学进展, 2017, 6(7): 816-830. https://doi.org/10.12677/AAM.2017.67099

参考文献

[1] Rana, S., Samanta, S., Bhattacharya, S., et al. (2015) The Effect of Nanoparticles on Plankton Dynamics: A Mathematical Model. BioSystems, 127, 28-41.
https://doi.org/10.1016/j.biosystems.2014.11.003
[2] Dai, C., Zhao, M. and Yu, H. (2016) Dynamics Induced by Delay in a Nutrient-Phytoplankton Model with Diffusion. Ecological Complexity, 26, 29-36.
https://doi.org/10.1016/j.ecocom.2016.03.001
[3] Edwards, A.M. and Brindley, J. (1999) Zooplankton Mortality and the Dynamical Behaviour of Plankton Population Models. Bulletin of Mathematical Biology, 61, 303-339.
https://doi.org/10.1006/bulm.1998.0082
[4] Chattopadhyay, J. and Pal, S. (2002) Viral Infection on Phytoplankton-Zooplankton System—A Mathematical Model. Ecological Modelling, 151, 15-28.
https://doi.org/10.1016/S0304-3800(01)00415-X
[5] Mei, L. and Zhang, X. (2012) Existence and Nonexistence of Positive Steady States in Multi-Species Phytoplankton Dynamics. Journal of Differential Equations, 253, 2025-2063.
https://doi.org/10.1016/j.jde.2012.06.011
[6] Ruan, S. (1995) The Effect of Delays on Stability and Persistence in Plankton Models. Nonlinear Analysis, 24, 575-585.
https://doi.org/10.1016/0362-546X(95)93092-I
[7] Zhao, J., Tian, J.P. and Wei, J. (2016) Minimal Model of Plankton Systems Revisited with Spatial Diffusion and Maturation Delay. Bulletin of Mathematical Biology, 78, 381.
https://doi.org/10.1007/s11538-016-0147-3
[8] Chatterjee, A. and Pal, S. (2016) Plankton Nutrient Interaction Model with Effect of Toxin in Presence of Modified Traditional Holling Type II Functional Response. Systems Science & Control Engineering, 4, 20-30.
https://doi.org/10.1080/21642583.2015.1136801
[9] Misra, A.K. (2011) Modeling the Depletion of Dissolved Oxygen due to Algal Bloom in a Lake by Taking Holling Type-III Interaction. Applied Mathematics & Computation, 217, 8367-8376.
https://doi.org/10.1016/j.amc.2011.03.034
[10] Smith, D.W and Piedrahita, R.H. (1988) The Relation Between Phytoplankton and Dissolved Oxygen in Fish Ponds. Aquaculture, 68, 249-265.
https://doi.org/10.1016/0044-8486(88)90357-2
[11] Sekerci, Y. and Petrovskii, S. (2015) Mathematical Modelling of Plankton-Oxygen Dynamics under the Climate Change. Bulletin of Mathematical Biology, 77, 2325-2353.
https://doi.org/10.1007/s11538-015-0126-0
[12] Sekerci, Y. and Petrovskii, S. (2015) Mathematical Modelling of Spatiotemporal Dynamics of Oxygen in a Plankton System. Mathematical Modelling of Natural Phenomena, 10, 96-114.
https://doi.org/10.1051/mmnp/201510207
[13] Murillo, A.A., Delong, E.F. and Ulloa, O. (2014) Enhanced Metabolic Versatility of Planktonic Sulfur-Oxidizing γ-Proteobacteria in an Oxygen-Deficient Coastal Ecosystem. Frontiers in Marine Science, 1, 18.
https://doi.org/10.3389/fmars.2014.00018
[14] Kharel, S., Kumar, S. and Singh, C. (2013) Modelling Effect of the Depleting Dissolved Oxygen on the Existence of Interacting Planktonic Population. Elixir. Applied Mathematics, 55, 12739-12742.
[15] Naik, V.K. and Manjapp, S. (2010) Prediction of Dissolved Oxygen through Mathematical Modeling. International Journal of Environmental Research, 4, 153-160.
[16] Misra, A.K. (2010) Modeling the Dep-letion of Dissolved Oxygen in a Lake Due to Submerged Macrophytes. Nonlinear Analysis: Modelling and Control, 15, 185-198.
[17] Shukla, J.B., Misra, A.K. and Chandra, P. (2008) Mathematical Modeling and Analysis of the Depletion of Dissolved Oxygen in Eutrophied Water Bodies Affected by Organic Pollutants. Nonlinear Analysis Real World Applications, 9, 1851-1865.
[18] Dhar, J. and Baghel, R.S. (2016) Role of Dissolved Oxygen on the Plankton Dynamics in Spatio-Temporal Domain. Modeling Earth Systems & Environment, 2, 6.
https://doi.org/10.1007/s40808-015-0061-y
[19] Sharma, A., Sharma, A.K. and Agnihotri, K. (2014) The Dynamic of Plankton-Nutrient Interaction with Delay. Applied Mathematics & Computation, 231, 503-515.
[20] Lian, F.Y. and Xu, Y.T. (2009) Hopf Bifurcation Analysis of a Predator-Prey System with Holling Type IV Functional Response and Time Delay. Applied Mathematics & Computation, 215, 1484-1495.
[21] Zhao, H.Y., Huang, X.X. and Zhang, X.B. (2015) Hopf Bifurcation and Harvesting Control of a Bioeconomic Plankton Model with Delay and Diffusion Terms. Physica A, 421, 300-315.
[22] Das, K. and Ray, S. (2008) Effect of Delay on Nutrient Cycling in Phytop-lankton-Zooplankton Interactions in Estuarine System. Ecological Modelling, 215, 69-76.
[23] Rehim, M., Zhang, Z.Z. and Muhammadhaji, A. (2016) Mathematical Analysis of a Nutrient-Plankton System with Delay. SpringerPlus, 5, 1-22.
https://doi.org/10.1186/s40064-016-2435-7
[24] Chen, S.S., Shi, J.P. and Wei, J.J. (2013) Time Delay-Induced Instabilities and Hopf Bifurcations in General Reaction-Diffusion Systems. Journal of Nonlinear Science, 23, 1-38.
https://doi.org/10.1007/s00332-012-9138-1
[25] Song, Y.L. and Wei, J.J. (2005) Local Hopf Bifurcation and Global Periodic Solutions in a Delayed Predator-Prey System. Journal of Mathematical Analysis & Applications, 301, 1-21.
[26] Shi, R.X. and Yu, J. (2017) Hopf Bifurcation Analysis of Two Zooplankton-Phytoplankton Model with Two Delays. Chaos Solitons & Fractals, 100, 62-73.
[27] Liu, C., Wang, L.P., Zhang, Q.L. and Yan, Y. (2017) Dynamical Analysis in a Bioeconomic Phytoplankton Zooplankton System with Double Time Delays and Environmental Stochasticity. Physica A Statistical Mechanics & Its Applications, 82, 682-698.
[28] Liu, J. (2015) Hopf Bifurcation Analysis for an SIRS Epidemic Model with Logistic Growth and Delays. Journal of Applied Mathematics & Computing, 50, 1-20.
[29] Hassard, B.D., Kazarinoff, N.D. and Wan, Y. (1981) Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge.
[30] Song, Y.L. and Wei, J.J. (2004) Bifurcation Analysis for Chen’s System with Delayed Feedback and Its Application to Control of Chaos. Chaos Solitons Fractals, 22, 75-91.