SpringerLink
Log in

Bifurcation analysis of an algal blooms dynamical model in trophic interaction

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we revisit the algal blooms model of plankton interactions initially proposed by Das and Sarkar (DCDIS-A, 14(3):401–414, 2007), where the oscillatory mode in the interaction between phytoplankton and zooplankton is observed. We provide a detailed analysis of the dependence of the equilibria and their stability on various parameters in the model. The bifurcation behaviors around equilibrium (e.g., Hopf bifurcation, Bogdanov–Takens bifurcation) are further found. Meanwhile, numerical simulations verify and illustrate the effectiveness of our theoretical results.

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

Similar content being viewed by others

References

  1. Belshiasheela, I.R., Ghosh, M.: Impact of overfishing of large predatory fish on algal blooms: a mathematical study. Nonlinear Stud. 27(2), 405–413 (2020)

    MathSciNet  Google Scholar 

  2. Cai, L., Chen, G., **ao, D.: Multiparametric bifurcations of an epidemiological model with strong Allee effect. J. Math. Biol. 67(2), 185–215 (2013)

    Article  MathSciNet  Google Scholar 

  3. Chow, S.N., Li, C., Wang, D.: Normal forms and bifurcations of planar vector fields. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  4. Das, K., Sarkar, A.K.: Effect of algal blooms due to trophic interaction: a qualitative study. Dyn. Cont. Discret. Impulsive Syst. Ser. A 14(3), 401–414 (2007)

    MathSciNet  Google Scholar 

  5. Dumortier, F., Roussarie, R., Sotomayor, J.: Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3. Ergod. Theory Dyn. Syst. 7(3), 375–413 (1987)

    Article  MathSciNet  Google Scholar 

  6. Dai, Y., Zhao, Y., Sang, B.: Four limit cycles in a predator-prey system of Leslie type with generalized Holling type III functional response. Nonlinear Anal. Real World Appl. 50, 218–239 (2019)

    Article  MathSciNet  Google Scholar 

  7. Franks, P.J.S.: Recent advances in modelling of harmful algal blooms. In: Glibert, P., Berdalet, E., Burford, M., Pitcher, G., Zhou, M. (eds.) Global ecology and oceanography of harmful algal blooms, pp. 359–377. Springer, Cham (2018)

    Chapter  Google Scholar 

  8. Gazi, N.H., Das, K.: Structural stability analysis of an algal bloom mathematical model in tropic interaction. Nonlinear Anal. Real World Appl. 11(4), 2191–2206 (2010)

    Article  MathSciNet  Google Scholar 

  9. Grattan, L.M., Holobaugh, S., Morris, J.G.: Harmful algal blooms and public health. Harmful Algae 57, 2–8 (2016)

    Article  Google Scholar 

  10. Huang, J., Gong, Y., Ruan, S.: Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discret. Contin. Dyn. Syst. Ser. B 18(8), 2101–2121 (2013)

    MathSciNet  Google Scholar 

  11. Henderson, A., Kose, E., Lewis, A., Swanson, E.R.: Mathematical modeling of algal blooms due to swine CAFOs in Eastern North Carolina. Discret. Cont. Dyn. Syst. 15(3), 555–572 (2022)

    Article  MathSciNet  Google Scholar 

  12. Lamontage, Y., Coutu, C., Rousseau, C.: Bifurcation analysis of a predator-prey system with generalized Holling type III functional response. J. Dyn. Differ. Equ. 20(3), 535–571 (2008)

    Article  Google Scholar 

  13. Li, C., Li, J., Ma, Z.: Codimension 3 B-T bifurcation in an epidemic model with nonlinear incidence. Discret. Contin. Dyn. Syst. Ser. B 20(4), 1107–1116 (2015)

    Article  MathSciNet  Google Scholar 

  14. Misra, A.K., Tiwari, P.K., Chandra, P.: Modeling the control of algal bloom in a lake by applying some external efforts with time delay. Differ. Equ. Dyn. Syst. 29(3), 539–568 (2021)

    Article  MathSciNet  Google Scholar 

  15. Miller, M., Joshi, H.R.: Modeling harmful algal blooms in the western basin of lake erie and an economic solution. Neural Parallel Sci. Comput. 25, 403–416 (2017)

    MathSciNet  Google Scholar 

  16. Perko, L.: Differential equations and dynamical systems, 3rd edn. Springer, New York (2001)

    Book  Google Scholar 

  17. Song, D., Fan, M., Chen, M., Wang, H.: Dynamics of a periodic stoichiometric model with application in predicting and controlling algal bloom in Bohai Sea off China. Math. Biosci. Eng. 16(1), 119–138 (2019)

    Article  MathSciNet  Google Scholar 

  18. Sarkar, R.R., Pal, J., Das, K.P., Chattopadhyay, J.: Control of harmful algal blooms through input of competitive phytoplankton and the effect of environmental variability. J Calcutta Math. Soc. 4, 1–8 (2008)

    MathSciNet  Google Scholar 

  19. Thakur, N.K., Tiwari, S.K., Upadhyay, R.K.: Harmful algal blooms in fresh and marine water systems: the role of toxin producing phytoplankton. Int. J. Biomath. 9(3), 1650043 (2016)

    Article  MathSciNet  Google Scholar 

  20. Timm, U., Totaro, S., Okubo, A.: Self-and mutual shading effect on competing algal distribution. Nonlinear Anal. Theory Methods Appl. 17(6), 559–576 (1991)

    Article  MathSciNet  Google Scholar 

  21. **ao, D., Zhang, F.: Multiple bifurcation of a predator-prey system. Discret. Contin. Dyn. Syst. Ser. B 8(2), 417–433 (2007)

    MathSciNet  Google Scholar 

  22. Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative theory of differential equations. Transl. Math. Monogr., Amer. Math. Soc (1992)

Download references

Funding

The authors gratefully acknowledge support of the National Natural Science Foundation of China [Grant Numbers 11871415, 12271466], Youth Sustentation Fund of **nyang Normal University[Grant Numbers 2023-QN-051], and the Henan Province Distinguished Professor program.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistic, **nyang Normal University, **nyang, 464000, China

    Qian Wei & Liming Cai

Authors
  1. Qian Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liming Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liming Cai.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

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

Appendices

Appendix 1. Coefficients in the proof of Theorem 4.4

Here we give the expression of coefficients, which are used in the proof of theorem 4.4.

$$\begin{aligned} \widehat{M_{00}}&=-\frac{(1-q)^{3}}{q(1-q+q^{2})^{2}}, \quad \widehat{M_{01}}=\frac{(1-q)^{2}}{q(1-q+q^{2})^{2}}\lambda _{3}, \\ \widehat{M_{21}}&=\overline{M_{21}}+\frac{q(1-2q)^{2}}{2(1-q+q^{2})^{2}}\lambda _{3},\quad \widehat{M_{20}}=-\frac{(1-q)(6q^{4}-12q^{3}+12q^{2}-6q+1)}{2q(1-q+q^{2})^{2}}\lambda _{3},\\ \widehat{M_{30}}&=\overline{M_{30}}+\frac{(2q-1)(2q^{3}-4q^{2}+4q-1) (1-2q+2q^{2})}{2q(1-q+q^{2})^{2}}\lambda _{3},\\ \widehat{N_{00}}&=-\frac{1-q}{q(1-q+q^{2})^{2}}\left( q(1-q)(1-q+q^{2}) \lambda _{1}+q(1-q+q^{2})^{3}\lambda _{2}+(1-q)^{4}\lambda _{3}\right) ,\\ \widehat{N_{10}}&=\frac{1-q}{(1-q+q^{2})^{2}}\left( -q^{2}(1-q+q^{2}) \lambda _{1}+q(1-q+q^{2})^{3}\lambda _{2}+(2-q)(1-q)^{3}\lambda _{3}\right) ,\\ \widehat{N_{01}}&=\frac{1}{q(1-q+q^{2})^{2}}\left( q(1-q)(1-q+q^{2}) \lambda _{1}+q(1-q+q^{2})^{3}\lambda _{2}+2(1-q)^{4}\lambda _{3}\right) ,\\ \widehat{N_{20}}&=\frac{1}{2q(1-q+q^{2})^{2}}(-q(1-q+q^{2})(4q^{4}-8q^{3}+10q^{2}-6q+1) \lambda _{1}+q(1-q+q^{2})^{3}\cdot \\&\quad (4q^{3}-8q^{2}+6q-1)\lambda _{2}-(1-q)^{3}(4q^{4}-4q^{3}+2q^{2}-2q+1)\lambda _{3}),\\ \widehat{N_{11}}&=\overline{N_{11}}+\frac{1}{q(1-q+q^{2})^{2}}\left( q^{3}(1-q+q^{2}) \lambda _{1}-q^{2}(1-q+q^{2})^{3}\lambda _{2}-(1-q)^{3}\lambda _{3}\right) ,\\ \widehat{N_{02}}&=-\frac{(1-q)^{3}}{q(1-q+q^{2})^{2}}\lambda _{3},\quad \widehat{N_{12}}=\overline{N_{12}}+\frac{(1-q)^{4}}{q(1-q+q^{2})^{2}}\lambda _{3},\\ \widehat{N_{30}}&=\overline{N_{30}}+\frac{1}{2q(1-q+q^{2})^{2}}(q(2q-1)(1-q+q^{2}) (4q^{3}-6q^{2}+5q-1)\lambda _{1}+q(1-q)\cdot \\&\quad (2q^{2}-4q+1)(1-q+q^{2})^{3}\lambda _{2}-(2q-1)(1-q)^{2}(4q^{4}-6q^{3} +5q^{2}-1)\lambda _{3}),\\ \widehat{N_{21}}&=\overline{N_{21}}+\frac{1}{2q(1-q)(1-q+q^{2})^{2}}(q^{3}(1-q+q^{2})(2q^{2}-1 )\lambda _{1}-q^{2}(2q^{2}-2q+1)\cdot \\&\quad (1-q+q^{2})^{3}\lambda _{2}+(1-q)^{3}(10q^{4}-16q^{3}+15q^{2}-10q+3)\lambda _{3}),\\ \widehat{P_{00}}&=-\frac{(1-q)^{2}}{1-q+q^{2}}\lambda _{1}-(1-q)(1-q+q^{2}) \lambda _{2}-\frac{(1-q)^{5}}{q(1-q+q^{2})^{2}}\lambda _{3}+o(\lambda ),\\ \widehat{P_{10}}&=-\frac{(1-q)q^{2}}{1-q+q^{2}}\lambda _{1}+(1-q)q(1-q+q^{2}) \lambda _{2}-\frac{(1-q)^{4}(2q^{2}-4q+1)}{q(1-q+q^{2})^{2}}\lambda _{3}+o(\lambda ),\\ \widehat{P_{01}}&=\frac{1-q}{1-q+q^{2}}\lambda _{1}+(1-q+q^{2})\lambda _{2} +\frac{2(1-q)^{4}}{q(1-q+q^{2})^{2}}\lambda _{3}+o(\lambda ), \end{aligned}$$
$$\begin{aligned} \widehat{P_{20}}&=-\frac{8q^{4}-12q^{3}+11q^{2}-6q+1}{2(1-q+q^{2})}\lambda _{1}\\&\quad -\frac{1}{2(1-q)}(1-q+q^{2})(8q^{4}-16q^{3}+15q^{2}-7q+1)\lambda _{2}\\&\quad +\frac{(1-q)^{3}}{q(1-q+q^{2})^{2}}\lambda _{3}+o(\lambda ),\\ \widehat{P_{11}}&=-(1-q)+\frac{q^{2}}{1-q+q^{2}}\lambda _{1}-q(1-q+q^{2})\lambda _{2} +\frac{1}{q(1-q+q^{2})^{2}}\lambda _{3}+o(\lambda ),\\ \widehat{P_{02}}&=-\frac{(1-q)^{3}}{q(1-q+q^{2})^{2}}\lambda _{3}+o(\lambda ),\quad \widehat{P_{12}}=\frac{q^{2}(1-2q)^{2}}{(1-q)^{2}}\\&\quad +2(1-q)^{2}-\frac{q^{4}-4q^{3} +6q^{2}-4q+1}{q(1-q+q^{2})^{2}}\lambda _{3}+o(\lambda ),\\ \widehat{P_{30}}&=-q(1-q)^{3}-\frac{4q^{6}-11q^{4}+18q^{3}-15q^{2}+6q-1}{2(1-q)(1-q+q^{2})} \lambda _{1}\\&\quad +\frac{1-q+q^{2}}{2(1-q)}(4q^{5} -6q^{4}+q^{3}+5q^{2}-4q+1)\lambda _{2}\\&\quad -\frac{(1-q)^{2}}{2(1-q+q^{2})}(4q^{5}-28q^{4}+53q^{3}-52q^{2}+27q-6)\lambda _{3}+o(\lambda ),\\ \widehat{P_{21}}&=-4q^{2}+2q-\frac{1}{2}+\frac{q^{2}(2q^{2}-1)}{2(1-q)(1-q+q^{2})} \lambda _{1}-\frac{q}{2(1-q)}(1-q+q^{2})(2q^{2}-2q+1)\lambda _{2}\\&\quad +\frac{1}{2q(1-q+q^{2})^{2}}(34q^{6} -120q^{5}+201q^{4}-194q^{3}+116q^{2}-40q+6)\lambda _{3}+o(\lambda ). \end{aligned}$$

Appendix 2. Coefficients in the proof of Theorem 4.5

We provide the expression of some coefficients which are used in the proof of Theorem 4.5 in the following.

$$\begin{aligned} \overline{a_{00}}&=\frac{(1-l^2)((l^{2}+7)\lambda _{1}-(l-1)^{2} \lambda _{3})}{4((l+1)(l^{2}+7)+16\lambda _{1})},\\ \overline{a_{01}}&=\frac{l+1}{8(l-1)((l+1)(l^{2}+7)+16\lambda _{1})} (16(l-1)^{2}\lambda _{3}+(l+1)(l^{2}+7)^{2}),\\ \overline{a_{10}}&=\frac{1}{8((l+1)(l^{2}+7)+16\lambda _{1})^{2}}(512(l^{2}+3)\lambda _{1}^{2}+(16(l+1)\\&\quad (3l^{2}+5)(l^{2}+7)-256(l-1)^{2}\lambda _{3})\lambda _{1}\\&\quad +(l+1)(l-1)^{2}((l+1)(l^{2}+7)^{2}-16(l^{2}+2l+5)\lambda _{3})),\\ \overline{a_{20}}&=-\frac{1}{((l+1)(l^{2}+7)+16\lambda _{1})^{3}}\\&\quad (4096\lambda _{1}^{3}+768(l+1)(l^{2}+7)\lambda _{1}^{2}+(16(l+1)(3l+1)(l^{2}+7)^{2}\\&\quad -512(l-1)^{2}\lambda _{3})\lambda _{1}+(l+1)(l-1)((l+1)\\&\quad (l^{2}+3)(l^{2}+7)^{2}-32(l-1)(l^{2}+2l+5)\lambda _{3})),\\ \overline{a_{11}}&=-\frac{1}{(l-1)^{2}((l+1)(l^{2}+7) +16\lambda _{1})^{2}}\\&\quad ((l+1)(l^{2}+2l+5)+16\lambda _{1}) ((l+1)(l^{2}+7)^{2}+16(l-1)^{2}\lambda _{3}),\\ \overline{a_{30}}&=-\frac{32}{((l+1)(l^{2}+7)+16\lambda _{1})^{4}}\\&\quad ((l+1) (l^{2}+2l+5)+16\lambda _{1})((l+1)(l^{2}+7)^{2}+16(l-1)^{2}\lambda _{3}),\\ \overline{a_{21}}&=\frac{16}{(l-1)^{2}((l+1)(l^{2}+7)+16\lambda _{1})^{3}}\\&\quad ((l+1)(l^{2}+2l+5)+16\lambda _{1})((l+1)(l^{2}+7)^{2}+16(l-1)^{2}\lambda _{3}),\\ \overline{a_{40}}&=\frac{512}{((l+1)(l^{2}+7)+16\lambda _{1})^{5}}\\&\quad ((l+1)(l^{2}+2l+5)+16\lambda _{1})((l+1)(l^{2}+7)^{2}+16(l-1)^{2}\lambda _{3}),\\ \overline{a_{31}}&=-\frac{256}{(l-1)^{2}((l+1)(l^{2}+7) +16\lambda _{1})^{4}}\\&\quad ((l+1)(l^{2}+2l+5)+16\lambda _{1}) ((l+1)(l^{2}+7)^{2}+16(l-1)^{2}\lambda _{3}),\\ \overline{b_{00}}&=\frac{(l-1)^{2}}{64(l+3)((l+1)(l^{2}+7) +16\lambda _{1})}\\&\quad ((16(l+3)(l^{2}+7)\lambda _{2}+16(l-1)^{3})\lambda _{1}+(l+1)(l+3)(l^{2}+7)^{2}\lambda _{2}),\\ \overline{b_{10}}&=-\frac{(l-1)^{2}}{8(l+3)(l^{2}+7)((l+1)(l^{2}+7) +16\lambda _{1})^{2}}\\&\quad (256((l+3)(l^{2}+7)\lambda _{2}+2(l+1)(l-1)^{2})\lambda _{1}^{2}+16(l+3)\cdot \\&\quad (l^{2}+7)(2(l+1)(l^{2}+7)\lambda _{2}+(3l-1)(l-1)^{2})\lambda _{1}\\&\quad +(l+1)(l+3)(l^{2}+7)^{2}((l+1)(l^{2}+7)\lambda _{2}+(l-1)^{3})), \end{aligned}$$
$$\begin{aligned} \overline{b_{01}}&=\frac{1}{8(l+3)((l+1)(l^{2}+7)+16 \lambda _{1})}\\&\quad (16((l+3)(l^{2}+7)\lambda _{2}-4(l-1)^{2})\lambda _{1}+(l+1)(l+3)(l^{2}+7)((l^{2}+7)\lambda _{2}\\&\quad -(l-1)^{2})),\\ \overline{b_{11}}&=-\frac{1}{(l+3)(l^{2}+7)((l+1)(l^{2}+7)+16 \lambda _{1})^{2}}\\&\quad (256((l+3)(l^{2}+7)\lambda _{2}+2(l+1)(l-1)^{2})\lambda _{1}^{2}+16(l+3)\cdot \\&\quad (l^{2}+7)(2(l+1)(l^{2}+7)\lambda _{2}+(3l-1)(l-1)^{2})\\&\quad \lambda _{1}+(l+1)(l+3)(l^{2}+7)^{2}((l+1)(l^{2}+7)\lambda _{2}+(l-1)^{3})),\\ \overline{b_{20}}&=-\frac{2(l-1)^{4}(l^{2}+7)((l+1)(l^{2}+2l+5)+16 \lambda _{1})}{(l+3)((l+1)(l^{2}+7)+16\lambda _{1})^{3}}, \quad \overline{b_{02}}=-1,\\ \overline{b_{30}}&=\frac{32(l-1)^{4}(l^{2}+7)((l+1)(l^{2}+2l+5)+16 \lambda _{1})}{(l+3)((l+1)(l^{2}+7)+16\lambda _{1})^{4}}, \\&\quad \overline{b_{21}}=-\frac{16(l-1)^{2}(l^{2}+7)((l+1)(l^{2}+2l+5)+16 \lambda _{1})}{(l+3)((l+1)(l^{2}+7)+16\lambda _{1})^{3}}, \\ \overline{b_{40}}&=-\frac{512(l-1)^{4}(l^{2}+7)((l+1)(l^{2}+2l+5)+16 \lambda _{1})}{(l+3)((l+1)(l^{2}+7)+16\lambda _{1})^{5}}, \\&\quad \overline{b_{31}}=\frac{256(l-1)^{2}(l^{2}+7)((l+1)(l^{2}+2l+5)+16 \lambda _{1})}{(l+3)((l+1)(l^{2}+7)+16\lambda _{1})^{4}}. \end{aligned}$$

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wei, Q., Cai, L. Bifurcation analysis of an algal blooms dynamical model in trophic interaction. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02095-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12190-024-02095-3

Keywords

Search

Navigation