Search Advanced

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 6728-6752. doi: 10.3934/mbe.2019336
Previous Article Next Article
Research article Special Issues

Mathematical modeling and analysis of harmful algal blooms in flowing habitats

  • 1.
    Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan
  • 2.
    Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan; and Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung branch, Keelung 204, Taiwan
  • 3.
    Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada
  • Received: 30 January 2019 Accepted: 08 June 2019 Published: 26 July 2019

  • In this paper, we survey recent developments of mathematical modeling and analysis of the dynamics of harmful algae in riverine reservoirs. To make the models more realistic, a hydraulic storage zone is incorporated into a flow reactor model and new mathematical challenges arise from the loss of compactness of the solution maps. The key point in the study of the evolution dynamics is to prove the existence of global attractors for the model systems and the principal eigenvalues for the associated linearized systems without compactness.

    Citation: Sze-Bi Hsu, Feng-Bin Wang, Xiao-Qiang Zhao. Mathematical modeling and analysis of harmful algal blooms in flowing habitats[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6728-6752. doi: 10.3934/mbe.2019336

    Related Papers:

  • In this paper, we survey recent developments of mathematical modeling and analysis of the dynamics of harmful algae in riverine reservoirs. To make the models more realistic, a hydraulic storage zone is incorporated into a flow reactor model and new mathematical challenges arise from the loss of compactness of the solution maps. The key point in the study of the evolution dynamics is to prove the existence of global attractors for the model systems and the principal eigenvalues for the associated linearized systems without compactness.


    加载中


    [1] D. L. Roelke, J. P. Grover, B. W. Brooks, et al., A decade of fishkilling Prymnesium parvum blooms in Texas: roles of inflow and salinity, J. Plankton Res., 33 (2011), 243–253.
    [2] G. M. Southard, L. T. Fries and A. Barkoh, Prymnesium parvum: the Texas experience, J. Am. Water Resources Assoc., 46 (2010), 14–23.
    [3] T. L. James and A. De La Cruz, Prymnesium parvum Carter (Chrysophyceae) as a suspect of mass mortalities of fish and shellfish communities in western Texas, Texas J. Sci., 41 (1989), 429–430.
    [4] D. L. Roelke, A. Barkoh, B. W. Brooks, et al., A chronicle of a killer alga in the west: ecology, assessment, and management of Prymnesium parvum blooms, Hydrobiologia, 764 (2016), 29–50.
    [5] V. M. Lundgrena, D. L. Roelke, J. P. Grover, et al., Interplay between ambient surface water mixing and manipulatedhydraulic flushing: Implications for harmful algal bloom mitigation, Ecol. Eng., 60 (2013), 289–298.
    [6] C. G. R. Maier, M. D. Burch and M. Bormans, Flow management strategies to control blooms of the cyanobacterium, Anabaena circinalis, in the river Murray at Morgan, South Australia, Regul. Rivers Res. Mgmt., 17 (2001), 637–650.
    [7] S. M. Mitrovic, L. Hardwick, R. Oliver, et al., Use of flow management to control saxitoxin pro-ducing cyanobacterial blooms in the Lower Darling River, Australia, J. Plankton Res., 33 (2011), 229–241.
    [8] D. L. Roelke, G. M. Gable and T. W. Valenti, Hydraulic flushing as a Prymnesium parvum bloom terminating mechanism in a subtropical lake, Harmful Algae, 9 (2010), 323–332.
    [9] J. P. Grover, S. B. Hsu and F. B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone, Math. Biosci., 222 (2009), 42–52.
    [10] J. P. Grover, K. W. Crane, J. W. Baker, et al., Spatial variation of harmful algae and their toxins in flowing-water habitats: a theoretical exploration, J. Plankton Res., 33 (2011), 211–227.
    [11] S. B. Hsu, F. B. Wang and X. Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, J. Dynam. Differ. Eq., 23 (2011), 817–842.
    [12] S. B. Hsu, F. B. Wang and X. Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Diff. Eqns., 255 (2013), 265–297.
    [13] F. B. Wang, S. B. Hsu and X. Q. Zhao, A reaction-diffusion-advection model of harmful algae growth with toxin degradation, J. Diff. Eqns., 259 (2015), 3178–3201.
    [14] C. M. Kung and B. Baltzis, The growth of pure and simple microbial competitors in a moving distributed medium, Math. Biosci., 111 (1992), 295–313.
    [15] F. B. Wang, A system of partial differential equations modeling the competition for two comple-mentary resources in flowing habitats, J. Diff. Eqns., 249 (2010), 2866–2888.
    [16] H. L. Smith and X. Q. Zhao, Dynamics of a periodically pulsed bio-reactor model, J. Diff. Eqns.,155 (1999), 368–404.
    [17] F. B. Wang and C. C. Huang, A reaction-advection-diffusion system modeling the competition for two complementary resources with seasonality in a flowing habitat, J. Math. Anal. Appl., 428 (2015), 145–164.
    [18] C. S. Reynolds, Potamoplankton: paradigms, paradoxes and prognoses, in Algae and the Aquatic Environment, F. E. Round, ed., Biopress, Bristol, UK, 1990.
    [19] X. Q. Zhao, Dynamical systems in population biology, second edition, Springer, New York, 2017.
    [20] W. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652–1673.
    [21] S. B. Hsu, J. López-Gómez, L. Mei, et al., A pivotal eigenvalue problem in river ecology, J. Diff. Eqns., 259 (2015), 2280–2316.
    [22] X. Liang, L. Zhang and X. Q. Zhao, The principal eigenvalue for degenerate periodic reaction-diffusion systems, SIAM J. Math. Anal., 49 (2017), 3603–3636.
    [23] R. D. Nussbaum, Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theo-rem, in Fixed Point Theory (E. Fadell and G. Fournier, eds.), 309–331, Lecture Notes in Mathe-matics 886, Springer, New York/Berlin, 1981.
    [24] M. Ballyk, D. Le, D. A. Jones, et al., Effects of random motility on microbial growth and compe-tition in a flow reactor, SIAM J. Appl. Math., 59 (1998), 573–596.
    [25] H. L. Smith, Monotone Dynamical Systems:An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.
    [26] R. Aris, Mathematical Modeling, a Chemical Engineer's Perspective, Academic Press, New York, 1999.
    [27] M. Ballyk, D. A. Jones and H. L. Smith, The Freter Model of Biofilm Formation: a review, a book chapter in "Structured Population Models in Biology and Epidemiology", eds P.Magal, S. Ruan, Lecture Notes in Mathematics, Mathematical Biosciences subseries, Springer, 2008.
    [28] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1988.
    [29] P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251–275.
    [30] X. Yu and X. Q. Zhao, A periodic reaction-advection-diffusion model for a stream population, J. Diff. Eqns., 258 (2015), 3037–3062.
    [31] W. M. Hirsch, H. L. Smith and X. Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dynam. Differ. Eq., 13 (2001), 107–131.
    [32] M. Murata and T. Yasumoto, The structure elucidation and biological activities of high molec-ular weight algal toxins: maitotoxins, prymnesins and zooxanthellatoxins, Nat Prod. Rep., 17 (2000),293–314.
    [33] R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion sys-tems, Trans. Amer. Math. Soc., 321 (1990), 1–44.
    [34] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population struc-ture and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188–211.
    [35] F. B. Wang, S. B. Hsu and W. Wang, Dynamics of harmful algae with seasonal temperature varia-tions in the cove-main lake, Discrete Cont. Dyn. S., 21 (2016), 313–335.
    [36] K. E. Bencala and R. A. Walters, Simulation of solute transport in a mountain pool-and-riffle stream: a transient storage model, Water Resour. Res., 19 (1983), 718–724.
    [37] Y. Jin, F. M. Hilker, P. M. Steffler, et al., Seasonal invasion dynamics in a spatially heterogeneous river with fluctuating flows, B. Math. Biol., 76 (2014), 1522–1565.
    [38] Y. Jin and M. A. Lewis, Seasonal influences on population spread and persistence in streams: critical domain size, SIAM J. Appl. Math., 71 (2011), 1241–1262.
    [39] F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749–772.
    [40] H. W. McKenzie, Y. Jin, J. Jacobsen, et al., R0 Analysis of a spationtemporal model for a stream population, SIAM J. Appl. Dyn. Syst., 11 (2012), 567–596.
    [41] E. Pachepsky, F. Lutscher, R. M. Nisbet, et al., Persistence, spread and the drift paradox, Theor. Popul. Biol., 67 (2005), 61–73.
    [42] D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219–1237.
    [43] Q. Huang, Y. Jin and M. A. Lewis, R0 analysis of a benthic-drift model for a stream population, SIAM J. Appl. Dyn. Syst., 15 (2016), 287–321. Erratum: 16(1), (2017), 770.
    [44] Y. Jin and F. B. Wang, Dynamics of a benthic-drift model for two competitive species, J. Math. Anal. Appl., 462 (2018), 840–860.
    [45] F. Lutscher, M. A. Lewis and E. McCauley, Effects of Heterogeneity on Spread and Persistence in Rivers, B. Math. Biol., 68 (2006), 2129–2160.
    [46] K. Y. Lam, Y. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dynam., 9 (2015), 188–212.
    [47] K. Y. Lam, Y. Lou and F. Lutscher, The emergence of range limits in advective environments, SIAM J. Appl. Math., 76 (2016), 641–662.
    [48] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319–1342.
    [49] Y. Lou, X. Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. de Mathématiques Pures et Appliquées, 121 (2019), 47–82.
    [50] F. Lutscher, E. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in sys-tems with unidirectional flow, Theor. Popul. Biol., 71 (2007), 267–277.
    [51] X. Q. Zhao and P. Zhou, On a Lotka-Volterra competition model: the effects of advection and spatial variation, Calc. Var. Partial Dif., 55 (2016), 55–73.
    [52] P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356–380.
    [53] J. P. Grover, D. L. Roelke and B. W. Brooks, Modeling of plankton community dynamics charac-terized by algal toxicity and allelopathy: A focus on historical Prymnesium parvum blooms in a Texas reservoir, Ecol. Model., 227 (2012), 147–161.
    [54] S. B. Hsu, F. B. Wang and X. Q. Zhao, A reaction-diffusion model of harmful algae and zooplank-ton in an ecosystem, J. Math. Anal. Appl., 451 (2017), 659–677.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(4033) PDF downloads(729) Cited by(4)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog