
Mathematical Biosciences and Engineering, 2019, 16(6): 67286752. doi: 10.3934/mbe.2019336.
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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: , Accepted: , Published:
Special Issues: Recent Advances in Mathematical Population Dynamics
Keywords: harmful algae; zooplankton; principal eigenvalue; basic reproduction ratio; global attractor; threshold dynamics; persistence and extinction
Citation: SzeBi Hsu, FengBin Wang, XiaoQiang Zhao. Mathematical modeling and analysis of harmful algal blooms in flowing habitats. Mathematical Biosciences and Engineering, 2019, 16(6): 67286752. doi: 10.3934/mbe.2019336
References:
 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 producing 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 flowingwater 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 bioreactor 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 reactiondiffusionadvection 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 complementary resources in flowing habitats, J. Diff. Eqns., 249 (2010), 2866–2888.
 16. H. L. Smith and X. Q. Zhao, Dynamics of a periodically pulsed bioreactor model, J. Diff. Eqns.,155 (1999), 368–404.
 17. F. B. Wang and C. C. Huang, A reactionadvectiondiffusion 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 reactiondiffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652–1673.
 21. S. B. Hsu, J. LópezGómez, 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 reactiondiffusion systems, SIAM J. Math. Anal., 49 (2017), 3603–3636.
 23. R. D. Nussbaum, Eigenvectors of nonlinear positive operator and the linear KreinRutman theorem, in Fixed Point Theory (E. Fadell and G. Fournier, eds.), 309–331, Lecture Notes in Mathematics 886, Springer, New York/Berlin, 1981.
 24. M. Ballyk, D. Le, D. A. Jones, et al., Effects of random motility on microbial growth and competition 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, SpringerVerlag, 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 reactionadvectiondiffusion 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 molecular 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 reactiondiffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1–44.
 34. H. R. Thieme, Spectral bound and reproduction number for infinitedimensional population structure 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 variations in the covemain lake, Discrete Cont. Dyn. S., 21 (2016), 313–335.
 36. K. E. Bencala and R. A. Walters, Simulation of solute transport in a mountain poolandriffle 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., R_{0} 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, R_{0} analysis of a benthicdrift 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 benthicdrift 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 LotkaVolterra competitiondiffusionadvection 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 systems with unidirectional flow, Theor. Popul. Biol., 71 (2007), 267–277.
 51. X. Q. Zhao and P. Zhou, On a LotkaVolterra 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 LotkaVolterra competitiondiffusionadvection system, J. Funct. Anal., 275 (2018), 356–380.
 53. J. P. Grover, D. L. Roelke and B. W. Brooks, Modeling of plankton community dynamics characterized 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 reactiondiffusion model of harmful algae and zooplankton in an ecosystem, J. Math. Anal. Appl., 451 (2017), 659–677.
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