Skip to main content

Advertisement

SpringerLink
Log in

A nonautonomous model for the effect of environmental toxins on plankton dynamics

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The increasing input of environmental toxins in aquatic systems raises concerns regarding the environmental exposure and impact of toxins on natural aquatic environments. Phytoplankton and zooplankton appear to be among the most sensitive aquatic organisms to environmental toxins. Moreover, toxin-producing phytoplankton plays an important role in regulating the real aquatic ecosystems. In this paper, the combined effects of these factors on the dynamics of phytoplankton–zooplankton interactions are investigated. The phytoplankton grows logistically, but their growth rate is suppressed due to the presence of environmental toxins. The zooplankton is assumed to be generalist and follows logistic growth in the absence of phytoplankton. Also, it is considered that toxicants in the environment are increased constantly due to different natural and human behaviors. Global sensitivity analysis helps to identify the most significant parameters that reduce the environmental toxins in the system. Among these, the input rate of environmental toxins, contact rate between phytoplankton and environmental toxins, and environmental toxins-induced growth suppression of phytoplankton have destabilizing effect on the dynamics of system, while the depletion rate of environmental toxins has stabilizing effect. Therefore, it is imperative to modulate the depletion rate of environmental toxins to prevent the crash of the aquatic food web system. Further, we incorporate seasonal variations in the model, letting the parameters become functions of time. Sufficient conditions for the existence and stability of positive periodic solutions are obtained. We also derive conditions for existence, uniqueness and stability of a positive almost periodic solution. Large values of time-dependent toxin release by phytoplankton and input rate of environmental toxins induce periodic solutions of the nonautonomous system while the corresponding autonomous system exhibits a stable focus. Interestingly, our nonautonomous system exhibits bursting patterns for two slow rationally related excitation frequencies. Finally, we convert our deterministic autonomous model into stochastic model by introducing additive noise term. We find that the stability of the system gets disturbed in the presence of environmental fluctuation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Riley, R.A., Stommel, H., Burrpus, D.P.: Qualitative ecology of the plankton of the Western North Atlantic. Bull. Bingham Oceanogr. Collect. Yale Univ. 12, 1–169 (1949)

    Google Scholar 

  2. Chen, M., et al.: The dynamics of temperature and light on the growth of phytoplankton. J. Theor. Biol. 385, 8–19 (2015)

    MATH  Google Scholar 

  3. Sekerci, Y., Petrovskii, S.: Mathematical modelling of plankton–oxygen dynamics under the climate change. Bull. Math. Biol. 77, 2325–2353 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Moss, B.R.: Ecology of fresh waters: man and medium, past to future, 3rd edn. Wiley-Blackwell (2009)

  5. Edwards, A.M., Brindley, J.: Oscillatory behavior in a three component plankton population model. Dyn. Stab. Syst. 11, 347–370 (1996)

    MATH  Google Scholar 

  6. Behrenfeld, M.J., Falkowski, P.G.: A consumer’s guide to phytoplankton primary productivity models. Limnol. Oceanogr. 42, 1479–1491 (1997)

    Google Scholar 

  7. Hoppe, G., et al.: Bacterial growth and primary production along a north-south transect of the Atlantic Ocean. Nature 416, 168–171 (2002)

    Google Scholar 

  8. Huppert, A., et al.: A model for seasonal phytoplankton blooms. J. Theor. Biol. 236, 276–290 (2005)

    MathSciNet  Google Scholar 

  9. Yang, X., et al.: Stability and bifurcation in a stoichiometric producer-grazer model with knife edge. SIAM J. Appl. Dyn. Syst. 15, 2051–2077 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Chakraborty, S., Roy, S., Chattopadhyay, J.: Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: a mathematical model. Ecol. Model. 213, 191–201 (2008)

    Google Scholar 

  11. Das, K., Ray, S.: Effect of delay on nutrient cycling in phytoplankton–zooplankton interactions in estuarine system. Ecol. Model. 215, 69–76 (2008)

    Google Scholar 

  12. Jang, S.J., Baglama, J., Rick, J.: Nutrient–phytoplankton–zooplankton models with a toxin. Math. Comput. Model. 43(1/2), 105–118 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Panja, P., Mondal, S.K.: Stability analysis of coexistence of three species prey–predator model. Nonlinear Dyn. 81(1–2), 373–382 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Mukhopadhyay, B., Bhattacharyya, R.: Modelling phytoplankton allelopathy in a nutrient–plankton model with spatial heterogeneity. Ecol. Model. 198, 163–173 (2006)

    Google Scholar 

  15. Roy, S.: The coevolution of two phytoplankton species on a single resource: allelopathy as a pseudo-mixotrophy. Theor. Popul. Biol. 75, 68–75 (2009)

    MATH  Google Scholar 

  16. Silva, M.D., Jang, S.R.-J.: Dynamical behavior of systems of two phytoplankton and one zooplankton populations with toxin producing phytoplankton. Math. Methods Appl. Sci. 40, 4295–4309 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Cronberg, G.: Changes in the phytoplankton of Lake Trummen included by restoration. Hydrobiologia 86, 185–193 (1982)

    Google Scholar 

  18. Leah, R.T., Moss, B., Forrest, D.E.: The role of predation in causing major changes in the limnology. Int. Revue ges. Hydrobiol. 65(2), 223–247 (1990)

    Google Scholar 

  19. Scheffer, M.: Alternative stable states in eutrophic shallow fresh water systems: a minimal model. Hydrobiol. Bull. 23, 73–85 (1989)

    Google Scholar 

  20. Moss, B.: Engineering and biological approaches to the restoration from eutrophication of shallow lakes in which aquatic plant communities are important components. Hydrobiol. 200(201), 367–377 (1990)

    Google Scholar 

  21. Boney, A.D.: Phytoplankton. Edward Arnold Ltd., London (1976)

    Google Scholar 

  22. Odum, E.P.: Fundamentals of Ecology. W. B. Saunders Company, Philadelphia (1971)

    Google Scholar 

  23. Kirk, K., Gilbert, J.: Variations in herbivore response to chemical defences: zooplankton foraging on toxic cyanobacteria. Ecology 73, 2208–2213 (1992)

    Google Scholar 

  24. Kozlowsky-Suzuki, B., et al.: Feeding, reproduction and toxin accumulation by the copepods Acartia bifilosa and Eurytemora affinis in the presence of the toxic cyanobacterium Nodularia spumigena. Mar. Ecol. Prog. Ser. 249, 237–249 (2003)

    Google Scholar 

  25. Liu, C., et al.: Dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations. Appl. Math. Comput. 316, 115–137 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Chattopadhyay, J.: Effect of toxic substances on a two-species competitive system. Ecol. Model. 84, 287–289 (1996)

    Google Scholar 

  27. Chattopadhayay, J., Sarkar, R.R., Mandal, S.: Toxin producing plankton may act as a biological control for planktonic blooms-field study and mathematical modeling. J. Theor. Biol. 215(3), 333–344 (2002)

    Google Scholar 

  28. Sarkar, R.R., Chattopadhayay, J.: Occurrence of planktonic blooms under environmental fluctuations and its possible control mechanism—mathematical models and experimental observations. J. Theor. Biol. 224, 501–516 (2003)

    Google Scholar 

  29. Pal, R., Basu, D., Banerjee, M.: Modelling of phytoplankton allelopathy with Monod–Haldane-type functional response—a mathematical study. Biosystems 95, 243–253 (2009)

    Google Scholar 

  30. Schmidt, L.E., Hansen, P.J.: Allelopathy in the prymnesiophyte Chyrsochromulina polylepis: effect of cell concentration, growth phase and pH. Mar. Ecol. Prog. Ser. 216, 67–81 (2001)

    Google Scholar 

  31. Windust, A.J., Wright, J.L.C., McLachlan, J.L.: The effects of the diarrhetic shellfish poisoning toxins okadaic acid and dinophysistoxin-1, on the growth of microalgae. Mar. Biol. 126, 19–25 (1996)

    Google Scholar 

  32. Hansen, F.C.: Trophic interaction between zooplankton and Phaeocystis cf. globosa. Helgol. Meeresunters. 49, 283–293 (1995)

    Google Scholar 

  33. Nielsen, T.G., Kiorboe, T., Bjornsen, P.K.: Effects of a Chrysochromulina polylepis sub surface bloom on the plankton community. Mar. Ecol. Prog. Ser. 62, 21–35 (1990)

    Google Scholar 

  34. Buskey, E.J., Hyatt, C.J.: Effect of the Texas (USA) brown tide alga on planktonic grazers. Mar. Ecol. Prog. Ser. 126, 285–292 (1995)

    Google Scholar 

  35. Banerjee, M., Venturino, E.: A phytoplankton–toxic phytoplankton–zooplankton model. Ecol. Complex. 8, 239–248 (2011)

    Google Scholar 

  36. Moratou-Apostolopoulou, M., Ignatiades, L.: Pollution effects on the phytoplankton–zooplankton relationships in an inshore environment. Hydrobiologia 75(2), 259–266 (1980)

    Google Scholar 

  37. Tchounwou, P.B., et al.: Heavy metals toxicity and the environment. Exs 101, 133–164 (2012)

    Google Scholar 

  38. Huang, Y.J., et al.: The chronic effects of oil pollution on marine phytoplankton in a subtropical bay. Chin. Environ. Monit. Assess. 176(1), 517–530 (2011)

    Google Scholar 

  39. U.S. Environmental Protection Agency Great Lkes National Program Office Significant Activities Report. http://www.epa.gov/glnpo/aoc/waukegan.html

  40. Labille, J., Brant, J.: Stability of nanoparticles in water. Nanomedicine 5(6), 985–998 (2010)

    Google Scholar 

  41. Miao, A.J., et al.: The algal toxicity of silver engineered nanoparticles and detoxification by exopolymeric substances. Environ. Pollut. 157, 3034–3041 (2009)

    Google Scholar 

  42. Miao, A., et al.: Zinc oxide-engineered nanoparticles: dissolution and toxicity to marine phytoplankton. Environ. Toxicol. Chem. 29(12), 2814–2822 (2010)

    Google Scholar 

  43. Miller, R.J., et al.: \(\text{ TiO }_2\) nanoparticles are phototoxic to marine phytoplankton. PLoS ONE 7(1), e30321 (2012)

    Google Scholar 

  44. Rana, S., et al.: The effect of nanoparticles on plankton dynamics: a mathematical model. BioSystems 127, 28–41 (2015)

    Google Scholar 

  45. Panja, P., Mondal, S.K., Jana, D.K.: Effects of toxicants on Phytoplankton–Zooplankton–Fish dynamics and harvesting. Chaos Solit. Fract. 104, 389–399 (2017)

    MathSciNet  MATH  Google Scholar 

  46. Chakraborty, S., Chattopadhyay, J.: Nutrient–phytoplankton–zooplankton dynamics in the presence of additional food source—a mathematical study. J. Biol. Syst. 16(04), 547–564 (2008)

    MATH  Google Scholar 

  47. Roy, S., Chattopadhyay, J.: Disease-selective predation may lead to prey extinction. Math. Methods Appl. Sci. 28, 1257–1267 (2005)

    MathSciNet  MATH  Google Scholar 

  48. Das, K.P., Roy, S., Chattopadhyay, J.: Effect of disease-selective predation on prey infected by contact and external sources. BioSystems 95, 188–199 (2009)

    Google Scholar 

  49. Timms, R.M., Moss, B.: Prevention of growth of potentially dense phytoplankton populations by zooplankton grazing, in the presence of zooplanktivorous fish, in a shallow wetland ecosystem. Limnol. Oceanogr. 29, 472–486 (1984)

    Google Scholar 

  50. Murdoch, W.W., Bence, J.: General predators and unstable prey populations -. In: Kerfoot, W.C., Sih, A. (eds.) Predation: Direct and Indrect Impacts on Aquatic Communities, pp. 17–30. University Press of New England, Hanover (1987)

    Google Scholar 

  51. Yu, X., Yuan, S., Zhang, T.: Survival and ergodicity of a stochastic phytoplankton–zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment. Appl. Math. Comput. 347, 249–264 (2019)

    MathSciNet  MATH  Google Scholar 

  52. Jang, S.R.-J., Baglama, J., Rick, J.: Plankton–toxin interaction with a variable input nutrient. J. Biol. Dyn. 2(1), 14–30 (2008)

    MathSciNet  MATH  Google Scholar 

  53. Chakraborty, S., Chatterjee, S., Venturino, E., Chattopadhyay, J.: Recurring plankton bloom dynamics modeled via toxin-Producing phytoplankton. J. Biol. Phys. 33(4), 271–290 (2007)

    Google Scholar 

  54. Rueter, J.R., Chisholm, S.W., Morel, F.: Effects of copper toxicity on silicon acid uptake and growth in Thalassiosira pseudonana. J. Phycol. 17, 270–278 (1981)

    Google Scholar 

  55. Vardi, A., et al.: Dinoflagellate-cyanobacterium communication may determine the composition of phytoplankton assemblage in a mesotrophic lake. Curr. Biol. 12(20), 1767–1772 (2002)

    Google Scholar 

  56. Scheffer, M.: Fish and nutrients interplay determines algal biomass: a minimal model. Oikos 62, 271–282 (1991)

    Google Scholar 

  57. Beretta, E., Kuang, Y.: Modelling and analysis of a marine bacteriophase infection. Math. Biosci. 149, 57–67 (1998)

    MathSciNet  MATH  Google Scholar 

  58. Abate, A., Tiwari, A., Sastry, S.: Box invariance in biologically-inspired dynamical systems. Automatica 45, 1601–1610 (2009)

    MathSciNet  MATH  Google Scholar 

  59. Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc., New York (1989)

    MATH  Google Scholar 

  60. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations, 4th edn. Wiley, New York (1989)

    MATH  Google Scholar 

  61. Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Application of Hopf-Bifurcation. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  62. Graneli, E., Johansson, N.: Effects of the toxic haptophyte Prymnesium parvum on the survival and feeding of a ciliate: the influence of different nutrient conditions. Mar. Ecol. Prog. Ser. 254, 49–56 (2003)

    Google Scholar 

  63. Johansson, N., Graneli, E.: Influence of different nutrient conditions on cell density, chemical composition and toxicity of Prymnesium parvum (Haptophyta) in semi-continuous cultures. J. Exp. Mar. Biol. Ecol. 239, 243–258 (1999)

    Google Scholar 

  64. Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, Boston (1992)

    MATH  Google Scholar 

  65. Yoshizawa, T.: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, vol. 14. Springer, New York (2012)

    MATH  Google Scholar 

  66. Blower, S.M., Dowlatabadi, M.: Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int. Stat. Rev. 62, 229–243 (1994)

    MATH  Google Scholar 

  67. Marino, S., et al.: A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254(1), 178–196 (2008)

    MathSciNet  MATH  Google Scholar 

  68. Izhikevich, E.M., et al.: Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci. 26, 161–167 (2003)

    Google Scholar 

  69. Schuster, S., Knoke, B., Marhl, M.: Differential regulation of proteins by bursting calcium oscillations—a theoretical study. BioSystems 81, 49–63 (2005)

    Google Scholar 

  70. Li, X.H., Bi, Q.S.: Single-Hopf bursting in periodic perturbed Belousov—Zhabotinsky reaction with two time scales. Chin. Phys. Lett. 30, 010503–6 (2013)

    Google Scholar 

  71. Kingni, S.T., et al.: Bursting oscillations in a 3d system with asymmetrically distributed equilibria: mechanism, electronic implementation and fractional derivation effect. Chaos Solit. Fract. 71, 29–40 (2015)

    MathSciNet  MATH  Google Scholar 

  72. Ibarz, B., Casado, J.M., Sanjun, M.A.F.: Map-based models in neuronal dynamics. Phys. Rep. 501, 1–74 (2011)

    Google Scholar 

  73. Rinzel, J.: Bursting oscillation in an excitable membrane model. In: Sleeman, B.D., Jarvis, R.J. (eds.) Ordinary and Partial Differential Equations, pp. 304–316. Springer, Berlin (1985)

    Google Scholar 

  74. Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10(06), 1171–1266 (2000)

    MathSciNet  MATH  Google Scholar 

  75. Han, X., et al.: Frequency-truncation fast-slow analysis for parametrically and externally excited systems with two slow incommensurate excitation frequencies. Commun. Nonlinear Sci. Numer. Simulat. 72, 16–25 (2019)

    MathSciNet  Google Scholar 

  76. Wang, C., Tang, J., Ma, J.: Minireview on signal exchange between nonlinear circuits and neurons via field coupling. Eur. Phys. J. Spec. Top. 228, 1907–1924 (2019)

    Google Scholar 

  77. Han, X., et al.: Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations. Nonlinear Dyn. 88, 2693–2703 (2017)

    Google Scholar 

  78. Han, X., et al.: Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies. Phy. Rev. E 92, 012911 (2015)

    MathSciNet  Google Scholar 

  79. Han, X., Bi, Q., Kurths, J.: Route to bursting via pulse-shaped explosion. Phy. Rev. E 98, 010201(R) (2018)

    Google Scholar 

  80. Yu, X., Yuan, S., Zhang, T.: The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms. Nonlinear Dyn. 91(3), 1653–1668 (2018)

    MATH  Google Scholar 

  81. Lan, G., Wei, C., Zhang, S.: Long time behaviors of single-species population models with psychological effect and impulsive toxicant in polluted environments. Phys. A 521, 528–542 (2019)

    MathSciNet  Google Scholar 

  82. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM 43, 525–546 (2001)

    MathSciNet  MATH  Google Scholar 

  83. Navarro, E., et al.: Toxicity of silver nanoparticles to Chlamydomonas reinhardtii. Environ. Sci. Technol. 42(23), 8959–8964 (2008)

    Google Scholar 

  84. Chhater, A., et al.: Bacterial consortia for crude oil spill remediation. Water Sci. Technol. 34, 187–193 (1996)

    Google Scholar 

  85. Duval, B.D.: UV from Sunlight Excites Nanoparticles to Kill Phytoplankton in Lab Setting (2012). http://earthsky.org/human-world/uv-from-sunlight-excites-nanoparticles-to-kill-phytoplankton-in-lab-setting/

Download references

Acknowledgements

The authors express their gratitude to the associate editor and the reviewers whose comments and suggestions have helped the improvements of this paper.

Funding

The research work of Arindam Mandal is supported by University Grants Commission in the form of Senior Research Fellowship (Ref. No: 19/06/2016(i)EU-V). Pankaj Kumar Tiwari is thankful to University Grants Commissions, New Delhi, India, for providing financial support in form of D.S. Kothari postdoctoral fellowship (No. F.4-2/2006 (BSR)/MA/17-18/0021). Research of Samares Pal is partially supported by DST-FIST programme of University of Kalyani (474 (Sanc.)/ST/P/S & T/16 G-22/2018).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kalyani, Kalyani, 741235, India

    Arindam Mandal, Pankaj Kumar Tiwari & Samares Pal

  2. Department of Mathematics, Bankura University, Bankura, West Bengal, 722155, India

    Sudip Samanta

  3. Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, via Carlo Alberto 10, 10123, Turin, Italy

    Ezio Venturino

Authors
  1. Arindam Mandal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pankaj Kumar Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sudip Samanta
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ezio Venturino
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Samares Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samares Pal.

Ethics declarations

Conflicts of interest

The authors declare that there is no conflict of interests regarding the publication of this article.

Ethical standard

The authors state that this research complies with ethical standards. This research does not involve either human participants or animals.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mandal, A., Tiwari, P.K., Samanta, S. et al. A nonautonomous model for the effect of environmental toxins on plankton dynamics. Nonlinear Dyn 99, 3373–3405 (2020). https://doi.org/10.1007/s11071-020-05480-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05480-2

Keywords

Search

Navigation