Advertisement

The Effect of Toxin and Human Impact on Marine Ecosystem

Chapter
  • 761 Downloads
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 41)

Abstract

We formed a plankton-nutrient interaction model which consists of phytoplankton, herbivorous zooplankton, dissolved limiting nutrient with general nutrient uptake functions, instantaneous nutrient recycling, and harvesting on plankton population. Afterward, we modified and expanded the primary model by considering the effect of sunlight, additional nutrients, harmful chemicals, and carbon dioxide into account. Phytoplankton obtain carbohydrate supply from the carbon dioxide in the air, and the overall nutrient uptake rate increases in the presence of sunlight. So, the effect of sunlight and additional food is discussed. Hypothetically, carbon dioxide accelerates the growth of phytoplankton but we considered some limiting factors which abate the process. Some assumptions were made to construct the system equations. We assumed that phytoplankton releases toxic substances to defend themselves from the predation of zooplankton. The entire system was studied analytically, and the threshold values for the existence and stability of various steady states were discussed. Further, we discussed whether pollution emissions can variate the dynamics of the primary system, bring in recurrence bloom therein toxic phytoplankton can be applied to a great extent to sustain system stability.

Notes

Acknowledgement

The research is partially supported by the University Grants Commission, New Delhi [grant number MRP-MAJ-MATH-2013-609].

References

  1. 1.
    M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol, 9 (1973) 264–272.Google Scholar
  2. 2.
    K.DEIMLING, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 2008.Google Scholar
  3. 3.
    Y.Du, S.B. Hsu, Concentration phenomena in a nonlocal quasilinear problem modelling phytoplankton I: existence, SIAM J. Math. Anal. 40 (2008) 1419–1440.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Y.Du, S.B. Hsu, Concentration phenomena in a nonlocal quasilinear problem modelling phytoplankton II: limiting profile, SIAM J. Math. Anal. 40 (2008) 1441–1470.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Y.Du, S.B. Hsu, On a nonlocal reaction-diffusion problem arising from the modelling of the phytoplankton growth: SIAM J. Math. Anal. 42 (2010) 1305–1333.Google Scholar
  6. 6.
    Evans, G.T. and Parslow, J.S. : A model of annual plankton cycles. Biol. Oceanogr. 3,(1985), 327–427.Google Scholar
  7. 7.
    Dancer EN(1984), On positive solutions of some pairs of differential equation, Trans Amer Math Soc 284 729–743.MathSciNetCrossRefGoogle Scholar
  8. 8.
    U. Ebert, M. Arrayas, N. Temme, B. Sommeijer, J. Huisman, Critical condition for phytoplankton blooms, Bull. Math. Biol. 63 (2001) 1095–1124.CrossRefGoogle Scholar
  9. 9.
    H.L. Smith, P.E. Waltman, The Theory of the Chemostat, Cambridge University Press. (2008).Google Scholar
  10. 10.
    Courant R, Hilbert D(1953) Methods of Mathematical Physics, Vol. I Willey Interscience, New York.Google Scholar
  11. 11.
    X.-Q. ZHAO, Dynamical Systems in Population Biology, Springer, New York, 2003.CrossRefGoogle Scholar
  12. 12.
    S. Chakraborty, S. Roy, J. Chattopadhyay, Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: a mathematical model , E col. Model, 213, (2008), 191–201.CrossRefGoogle Scholar
  13. 13.
    Brewer PG, Goldman JC(1976), Alkalinity changes generated by phytoplankton growth, Limnol Oceanogr 21 108–117.CrossRefGoogle Scholar
  14. 14.
    Busenberg, S., Kishore, K.S., Austin, P. and Wake, G. : The dynamics of a model of a plankton - nutrient interaction. J. Math. Biol. 52, (1990), 677–696.CrossRefGoogle Scholar
  15. 15.
    D. Tilman, Resource Competition and community structure. Princeton University Press New Jersey, 1982.Google Scholar
  16. 16.
    Greer AT, Cowen RK, Guiland CM, McManus MA, Sevadjian JC, Timmerman AHV (2013), Relationships between phytoplankton thin layers and the fine-scale vertical distributions of two trophic levels of zooplankton. Journal of Plankton Research 35, 939–956.CrossRefGoogle Scholar
  17. 17.
    Badger MR, Price GD, Long BM, Woodger FJ(2006) The environmental plasticity and ecological genomics of cyanobacterial CO 2 concentrating mechanisms, J. Exp Bot 57 249–265.CrossRefGoogle Scholar
  18. 18.
    Ruan, S. : Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling. J. Math. Biol. 31, (1993), 633–654.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pardo, O. : Global stability for a phytoplankton-nutrient system. J. Biological Systems 8, (2000), 195–209.Google Scholar
  20. 20.
    Edwards, A.M. and Brindley, J. : Zooplankton mortality and the dynamical behaviour of plankton population models. Bull. Math. Biol., 61, (1999), 303–339 .CrossRefGoogle Scholar
  21. 21.
    Cushing DH (1975), Marine ecology and fisheries, Cambridge University Press,. London.Google Scholar
  22. 22.
    Edwards, A.M. and Yool, A. : The role of higher predation in plankton population models. J. Plankton Res., 22, (2000), 1085–1112.CrossRefGoogle Scholar
  23. 23.
    Ruan, S. : Oscillations in Plankton Models with Nutrient Recycling J. Theor. Biol. 208, (2001), 15–26.CrossRefGoogle Scholar
  24. 24.
    Bairagi, N., Pal, S., Chatterjee, S., Chattopadhyay, J.: Nutrient, non-toxic phytoplankton, toxic phytoplankton and zooplankton interaction in the open marine system. In: Hosking, R.J., Venturino, E.(Eds), Aspects of Mathematical Modelling. Mathematics and Biosciences in Interaction. Birkhauser Verlag Basel, Switzerland,(2008), 41–63.Google Scholar
  25. 25.
    Oscar Angulo, J.C. Lopez-Marcos, M.A. Lopez-Marcos, Numerical Analysis of a Size-Structured Population Model with a Dynamical Resource, Biomath 3 (2014), 1403241, http://dx.doi.org/10.11145/j.biomath.2014.03.241.
  26. 26.
    Mitra, A and Flynn, K.J. : Promotion of harmful algal blooms by zooplankton predatory activity. Biol. Lett., 2 (2),(2006),194–197.CrossRefGoogle Scholar
  27. 27.
    Morozov, A. and Arashkevich, E. : Patterns of Zooplankton Functional Response in Communities with Vertical Heterogeneity : a Model Study. Math Model. Nat. Phenom. 3 (3), (2008), 131–149.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Poggiale, J.-C., Gauduchon, M., Auger, P. : Enrichment Paradox Induced by Spatial Heterogeneity in a Phytoplankton-Zooplankton System. Math Model. Nat. Phenom. 3, (2008), 87–102.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rene Alt, Jean-Luc Lamotte, Stochastic Arithmetic as a Tool to Study the Stability of Biological Models, Biomath 2 (2013), 1312291, http://dx.doi.org/10.11145/j.biomath.2013.12.291.MathSciNetzbMATHGoogle Scholar
  30. 30.
    Condon RH, Duarte CM, Pitt KA, Robinson KL, Lucas CH, Sutherland KR, Mianzan HW, Bogeberg M, Purcell JE, Decker MB, and others (2013) Recurrent jellyfish blooms are a consequence of global oscillations. Proc Natl Acad Sci U S A 110: 1000–1005CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KalyaniKalyaniIndia

Personalised recommendations