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Algae–Bacteria Interactions with Nutrients and Light: A Reaction–Diffusion–Advection Model

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Abstract

In a poorly mixed water column, a reaction–diffusion–advection system is proposed to model algae–bacteria interactions containing nutrients and light. The basic ecological reproductive indices for the invasion of algae and bacteria into aquatic ecosystems are rigorously derived. All possibilities for the survival or extinction of algae and bacteria are obtained by analyzing nonnegative steady-state solutions. We further explore the influence of spatial factors and abiotic factors on algal or bacterial dynamics. Our results show that bacteria effectively reduce the biomass of algae and prevent them from moving to the water surface and ultimately reduce the probability of harmful algal blooms.

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References

  • Chang, X.Y., Shi, J.P., Wang, H.: Spatial modeling and dynamics of organic matter biodegradation in the absence or presence of bacterivorous grazing. Math. Biosci. 331, 108501 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Codeço, C.T., Grover, J.P.: Competition along a spatial gradient of resource supply: a microbial experimental model. Am. Nat. 157(3), 300–315 (2001)

    Article  Google Scholar 

  • Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8(2), 321–340 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  • Crane, K.W., Grover, J.P.: Coexistence of mixotrophs, autotrophs, and heterotrophs in planktonic microbial communities. J. Theor. Biol. 262(3), 517–527 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Davies, C.M., Wang, H.: Incorporating carbon dioxide into a stoichiometric producer-grazer model. J. Math. Biol. 83, 49 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Du, Y.H., Hsu, S.B.: On a nonlocal reaction–diffusion problem arising from the modeling of phytoplankton growth. SIAM J. Math. Anal. 42(3), 1305–1333 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards, K.F.: Mixotrophy in nanoflagellates across environmental gradients in the ocean. Proc. Natl. Acad. Sci. USA 116(13), 6211–6220 (2019)

    Article  Google Scholar 

  • Grover, J.P.: Sink or swim? Vertical movement and nutrient storage in phytoplankton. J. Theor. Biol. 432(7), 38–48 (2017)

    Article  MATH  Google Scholar 

  • Hale, J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence (1988)

    MATH  Google Scholar 

  • Heggerud, C.M., Wang, H., Lewis, M.A.: Transient dynamics of a stoichiometric cyanobacteria model via multiple-scale analysis. SIAM J. Appl. Math. 80(3), 1223–1246 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Hsu, S.B., Lou, Y.: Single phytoplankton species growth with light and advection in a water column. SIAM J. Appl. Math. 70(8), 2942–2974 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Hsu, S.B., Wang, F.B., Zhao, X.Q.: A reaction–diffusion model of harmful algae and zooplankton in an ecosystem. J. Math. Anal. Appl. 451(2), 659–677 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Huisman, J., Weissing, F.J.: Light-limited growth and competition for light in well-mixed aquatic environments: an elementary model. Ecology 75(2), 507–520 (1994)

    Article  Google Scholar 

  • Huisman, J., Arrayás, M., Ebert, U., Sommeijer, B.: How do sinking phytoplankton species manage to persist? Am. Nat. 159(3), 245–254 (2002)

    Article  Google Scholar 

  • Huisman, J., Pham Thi, N.N., Karl, D.M., Sommeijer, B.: Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum. Nature 439, 322–325 (2006)

    Article  Google Scholar 

  • Jäger, C.G., Diehl, S.: Resource competition across habitat boundaries: asymmetric interactions between benthic and pelagic producers. Ecol. Monogr. 84(2), 287–302 (2014)

    Article  Google Scholar 

  • Jäger, C.G., Diehl, S., Emans, M.: Physical determinants of phytoplankton production, algal stoichiometry, and vertical nutrient fluxes. Am. Nat. 175(4), 91–104 (2010)

    Article  Google Scholar 

  • Klausmeier, C.A., Litchman, E.: Algal games: the vertical distribution of phytoplankton in poorly mixed water columns. Limnol. Oceanogr. 46(8), 1998–2007 (2001)

    Article  Google Scholar 

  • Kong, J.D., Salceanu, P., Wang, H.: A stoichiometric organic matter decomposition model in a chemostat culture. J. Math. Biol. 76(3), 609–644 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Loladze, I., Kuang, Y., Elser, J.J.: Stoichiometry in producer-grazer systems: linking energy flow with element cycling. Bull. Math. Biol. 62, 1137–1162 (2000)

    Article  MATH  Google Scholar 

  • Magal, P., Zhao, X.Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37(1), 251–275 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Medina-Sánchez, J.M., Villar-Argaiz, M., Carrillo, P.: Neither with nor without you: a complex algal control on bacterioplankton in a high mountain lake. Limnol. Oceanogr. 49(5), 1722–1733 (2004)

    Article  Google Scholar 

  • Mei, L.F., Wang, F.B.: Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column. Discrete Contin. Dyn. Syst. Ser. B 26(4), 2115–2132 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Mischaikow, K., Smith, H., Thieme, H.R.: Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions. Trans. Am. Math. Soc. 347(5), 1669–1685 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Nie, H., Hsu, S.B., Wu, J.H.: Coexistence solutions of a competition model with two species in a water column. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2691–2714 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Nie, H., Hsu, S.B., Grover, J.P.: Algal competition in a water column with excessive dioxide in the atmosphere. J. Math. Biol. 72(7), 1845–1892 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Nie, H., Wang, B., Wu, J.H.: Invasion analysis on a predator-prey system in open advective environments. J. Math. Biol. 81(6), 1429–1463 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)

    MATH  Google Scholar 

  • Ryabov, A.B., Rudolf, L., Blasius, B.: Vertical distribution and composition of phytoplankton under the influence of an upper mixed layer. J. Theor. Biol. 263(1), 120–133 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Shi, J.P., Wang, X.F.: On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246(7), 2788–2812 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, H.L., Zhao, X.Q.: Robust persistence for semidynamical systems. Nonlinear Anal. 47(9), 6169–6179 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Thieme, H.R.: Convergence results and a Poincaré–Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30(7), 755–763 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Vasconcelos, F.R., Diehl, S., Rodríguez, P., Hedström, P., Karlsson, J., Byström, P.: Asymmetrical competition between aquatic primary producers in a warmer and browner world. Ecology 97(10), 2580–2592 (2016)

    Article  Google Scholar 

  • Wang, H., Smith, H.L., Kuang, Y., Elser, J.J.: Dynamics of stoichiometric bacteria-algae interactions in the epilimnion. SIAM J. Appl. Math. 68(2), 503–522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, H., Kuang, Y., Loladze, I.: Dynamics of a mechanistically derived stoichiometric producer-grazer model. J. Biol. Dyn. 2(3), 286–296 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, H., Sterner, R.W., Elser, J.J.: On the “strict homeostasis’’ assumption in ecological stoichiometry. Ecol. Model. 243, 81–88 (2012)

    Article  Google Scholar 

  • Wang, H., Lu, Z., Raghavan, A.: Weak dynamical threshold for the “strict homeostasis’’ assumption in ecological stoichiometry. Ecol. Model. 384, 233–240 (2018)

    Article  Google Scholar 

  • Wang, Y., Shi, J.P., Wang, J.F.: Persistence and extinction of population in reaction–diffusion–advection model with strong Allee effect growth. J. Math. Biol. 78(7), 2093–2140 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, H., Garcia, P.V., Ahmed, S., Heggerud, C.M.: Mathematical comparison and empirical review of the Monod and Droop forms for resource-based population dynamics. Ecol. Model. 466, 109887 (2022)

    Article  Google Scholar 

  • Wüest, A., Lorke, A.: Small-scale hydrodynamics in lakes. Annu. Rev. Fluid Mech. 35(1), 373–412 (2003)

    Article  MATH  Google Scholar 

  • Yan, Y.W., Zhang, J.M., Wang, H.: Dynamics of stoichiometric autotroph–mixotroph–bacteria interactions in the epilimnion. Bull. Math. Biol. 84, 5 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Yoshiyama, K., Mellard, J.P., Litchman, E., Klausmeier, C.A.: Phytoplankton competition for nutrients and light in a stratified water column. Am. Nat. 174(2), 190–203 (2009)

    Article  MATH  Google Scholar 

  • Zhang, J.M., Kong, J.D., Shi, J.P., Wang, H.: Phytoplankton competition for nutrients and light in a stratified lake: a mathematical model connecting epilimnion and hypolimnion. J. Nonlinear Sci. 31, 35 (2021a)

  • Zhang, J.M., Shi, J.P., Chang, X.Y.: A model of algal growth depending on nutrients and inorganic carbon in a poorly mixed water column. J. Math. Biol. 83, 15 (2021b)

  • Zhao, X.Q.: Dynamical Systems in Population Biology. Springer, New York (2003)

    Book  MATH  Google Scholar 

Download references

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Author notes

  1. Yawen Yan and Jimin Zhang have contributed equally to this work and can be regarded as co-first authors.

Authors and Affiliations

  1. School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, Heilongjiang, People’s Republic of China

    Yawen Yan & Jimin Zhang

  2. College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, Heilongjiang, People’s Republic of China

    Yawen Yan

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Hao Wang

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  1. Yawen Yan
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Correspondence to Hao Wang.

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Communicated by Philip K. Maini.

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Yawen Yan and Jimin Zhang are supported by NSFC-11971088 and NSFHLJ-LH2019A022; Hao Wang is supported by NSERC Discovery Grant RGPIN-2020-03911 and NSERC Accelerator Grant RGPAS-2020-00090.

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Yan, Y., Zhang, J. & Wang, H. Algae–Bacteria Interactions with Nutrients and Light: A Reaction–Diffusion–Advection Model. J Nonlinear Sci 32, 56 (2022). https://doi.org/10.1007/s00332-022-09815-8

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