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
[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. |