Skip to main content
SpringerLink
Log in

Dynamical Analysis of an Allelopathic Phytoplankton Model with Fear Effect

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper is the first to propose an allelopathic phytoplankton competition ODE model influenced by the fear effect based on natural biological phenomena. It is shown that the interplay of this fear effect and the allelopathic term cause rich dynamics in the proposed competition model, such as global stability, transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation. We also consider the spatially explicit version of the model and prove analogous results. Numerical simulations verify the feasibility of the theoretical analysis. The results demonstrate that the primary cause of the extinction of non-toxic species is the fear of toxic species compared to toxins. Allelopathy only affects the density of non-toxic species. The discussion guides the conservation of species and the maintenance of biodiversity.

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Antwi-Fordjour, K., Parshad, R.D., Beauregard, M.A.: Dynamics of a predator-prey model with generalized Holling type functional response and mutual interference. Math. Biosci. 326, 108407 (2020)

    Article  MathSciNet  Google Scholar 

  2. Bodine, E.N., Capaldi, A.: Can culling barred owls save a declining northern spotted owl population? Nat. Resour. Model. 30(3), e12131 (2017)

    Article  MathSciNet  Google Scholar 

  3. Brown, J.S., Laundré, J.W., Gurung, M.: The ecology of fear: optimal foraging, game theory, and trophic interactions. J. Mammal. 80(2), 385–399 (1999)

    Article  Google Scholar 

  4. Biswas, S., Tiwari, P.K., Pal, S.: Delay-induced chaos and its possible control in a seasonally forced eco-epidemiological model with fear effect and predator switching. Nonlinear Dyn. 104(3), 2901–2930 (2021)

    Article  Google Scholar 

  5. Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction–Diffusion Equations. Wiley, Hoboken (2004)

    Book  Google Scholar 

  6. Chen, F.: On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay. J. Comput. Appl. Math. 180, 33–49 (2005)

    Article  MathSciNet  Google Scholar 

  7. Chesson, P., Kuang, J.J.: The interaction between predation and competition. Nature 456, 235–238 (2008)

    Article  Google Scholar 

  8. Chen, F., Gong, X., Chen, W.: Extinction in two dimensional discrete Lotka–Volterra competitive system with the effect of toxic substances (II). Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 20, 449–461 (2013)

    MathSciNet  Google Scholar 

  9. Chen, F., Chen, X., Huang, S.: Extinction of a two species non-autonomous competitive system with Beddington–DeAngelis functional response and the effect of toxic substances. Open Math. 14, 1157–1173 (2016)

    Article  MathSciNet  Google Scholar 

  10. Chen, S., Chen, F., Li, Z., Chen, L.: Bifurcation analysis of an allelopathic phytoplankton model. J. Biol. Syst. 31(03), 1063–1097 (2023)

    Article  MathSciNet  Google Scholar 

  11. Chen, S., Chen, F., Srivastava, V., Parshad, R.D.: Dynamical analysis of a Lotka–Volterra competition model with both Allee and fear effect. Int. J. Biomath. (2023). https://doi.org/10.13140/RG.2.2.33238.11843

    Article  Google Scholar 

  12. Du, Y.: Effects of a degeneracy in the competition model: Part I. Classical and generalized steady-state solutions. J. Differ. Equ. 181(1), 92–132 (2002)

    Article  Google Scholar 

  13. Du, Y.: Effects of a degeneracy in the competition model: Part II. Perturbation and dynamical behaviour. J. Differ. Equ. 181(1), 133–164 (2002)

    Article  Google Scholar 

  14. Fistarol, G.O., Legrand, C., Rengefors, K., Granéli, E.: Temporary cyst formation in phytoplankton: A response to allelopathic competitors? Environ. Microbiol. 6(8), 791–798 (2004)

    Article  Google Scholar 

  15. Gilbarg, D., Trudinger, N.S., Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)

    Book  Google Scholar 

  16. Gökçe, A.: A mathematical model of population dynamics revisited with fear factor, maturation delay, and spatial coefficients. Math. Methods Appl. Sci. 45(17), 11828–11848 (2022)

    Article  MathSciNet  Google Scholar 

  17. Gökçe, A.: Dynamical behaviour of a predator–prey system encapsulating the fear affecting death rate of prey and intra-specific competition: revisited in a fluctuating environment. J. Comput. Appl. Math. 421, 114849 (2023)

    Article  MathSciNet  Google Scholar 

  18. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (2006)

    Google Scholar 

  19. Kishimoto, K., Weinberger, H.F.: The spatial homogeneity of stable equilibria of some reaction–diffusion systems on convex domains. J. Differ. Equ. 58(1), 15–21 (1985)

    Article  MathSciNet  Google Scholar 

  20. Kaur, R.P., Sharma, A., Sharma, A.K.: Impact of fear effect on plankton-fish system dynamics incorporating zooplankton refuge. Chaos Solitons Fract. 143, 110563 (2021)

    Article  MathSciNet  Google Scholar 

  21. Lam, K.Y., Lou, Y.: Introduction to Reaction–Diffusion Equations: Theory and Applications to Spatial Ecology and Evolutionary Biology. Springer, Berlin (2022)

    Book  Google Scholar 

  22. Legrand, C., Rengefors, K., Fistarol, G.O., Graneli, E.: Allelopathy in phytoplankton-biochemical, ecological and evolutionary aspects. Phycologia 42, 406–419 (2003)

    Article  Google Scholar 

  23. Long, L.L., Wolfe, J.D.: Review of the effects of barred owls on spotted owls. J. Wildl. Manag. 83, 1281–1296 (2019)

    Article  Google Scholar 

  24. Lai, L., Zhu, Z., Chen, F.: Stability and bifurcation in a predator–prey model with the additive Allee effect and the fear effect. Mathematics 8, 1280 (2020)

    Article  Google Scholar 

  25. Liu, T., Chen, L., Chen, F., Li, Z.: Stability analysis of a Leslie–Gower model with strong Allee effect on prey and fear effect on predator. Int. J. Bifurc. Chaos 32, 2250082 (2022)

    Article  MathSciNet  Google Scholar 

  26. Maynard-Smith, J.: Models in Ecology. Cambridge University Press, Cambridge (1974)

    Google Scholar 

  27. Morgan, J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20(5), 1128–1144 (1989)

    Article  MathSciNet  Google Scholar 

  28. Mulderij, G., Smolders, A.J., Van Donk, E.L.L.E.N.: Allelopathic effect of the aquatic macrophyte, Stratiotes aloides, on natural phytoplankton. Freshw. Biol. 51, 554–561 (2006)

    Article  Google Scholar 

  29. Ma, Z., Zhou, Y., Li, C.: Qualitative and Stability Methods of Ordinary Differential Equations. Science Press, Beijing (2015)

    Google Scholar 

  30. Mandal, A., Biswas, S., Pal, S.: Toxicity-mediated regime shifts in a contaminated nutrient-plankton system. Chaos Interdiscip. J. Nonlinear Sci. 33(2), 023106 (2023)

    Article  MathSciNet  Google Scholar 

  31. Mandal, A., Sk, N., Biswas, S.: Nutrient enrichment and phytoplankton toxicity influence a diversity of complex dynamics in a fear-induced plankton-fish model. J. Theor. Biol. 578, 111698 (2024)

    Article  MathSciNet  Google Scholar 

  32. Mandal, A., Tiwari, P.K., Pal, S.: A nonautonomous model for the effects of refuge and additional food on the dynamics of phytoplankton–zooplankton system. Ecol. Complex. 46, 100927 (2021)

    Article  Google Scholar 

  33. Parshad, R.D., Antwi-Fordjour, K., Takyi, E.M.: Some novel results in two species competition. SIAM J. Appl. Math. 81(5), 1847–1869 (2021)

    Article  MathSciNet  Google Scholar 

  34. Pierre, M.: Global existence in reaction–diffusion systems with control of mass: a survey. Milan J. Math. 78, 417–455 (2010)

    Article  MathSciNet  Google Scholar 

  35. Peckarsky, B.L., Abrams, P.A., Bolnick, D.I., Dill, L.M., Grabowski, J.H., Luttbeg, B., Trussell, G.C.: Revisiting the classics: considering nonconsumptive effects in textbook examples of predator-prey interactions. Ecology 89, 2416–2425 (2008)

    Article  Google Scholar 

  36. Polis, G.A., Myers, C.A., Holt, R.D.: The ecology and evolution of intraguild predation: potential competitors that eat each other. Annu. Rev. Ecol. Syst. 20, 297–330 (1989)

    Article  Google Scholar 

  37. Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (2013)

    Google Scholar 

  38. Pradhan, B., Ki, J.S.: Phytoplankton toxins and their potential therapeutic applications: a journey toward the quest for potent pharmaceuticals. Mar. Drugs 20, 271 (2022)

    Article  Google Scholar 

  39. Pringle, R.M., Kartzinel, T.R., Palmer, T.M., Thurman, T.J., Fox-Dobbs, K., Xu, C.C., et al.: Predator-induced collapse of niche structure and species coexistence. Nature 570, 58–64 (2019)

    Article  Google Scholar 

  40. Sasmal, S.K., Takeuchi, Y.: Dynamics of a predator–prey system with fear and group defense. J. Math. Anal. Appl. 481(1), 123471 (2020)

    Article  MathSciNet  Google Scholar 

  41. Saswati, B., et al.: Modeling the avoidance behavior of zooplankton on phytoplankton infected by free viruses. J. Biol. Phys. 46, 1–31 (2020)

    Article  Google Scholar 

  42. Srivastava, V., Takyi, E.M., Parshad, R.D.: The effect of “fear’’ on two species competition. Math. Biosci. Eng. 20, 8814–8855 (2023)

    Article  MathSciNet  Google Scholar 

  43. Srivastava, V., Van Lanen,N.J., Parshad, R.D.: Modeling competition co-occurrence effects between the invasive barred owl and imperiled northern spotted owl, In Preparation. (2024)

  44. Taylor, F.J.R.: The biology of dinoflagellates. Bot. Monogr. 21, 723–731 (1987)

    Google Scholar 

  45. Van Lanen, N.J., Franklin, A.B., Huyvaert, K.P., Reiser, R.F., II., Carlson, P.C.: Who hits and hoots at whom? Potential for interference competition between barred and northern spotted owls. Biol. Conserv. 144, 2194–2201 (2011)

    Article  Google Scholar 

  46. Wang, X., Walton, J.R., Parshad, R.D.: Stochastic models for the Trojan Y-Chromosome eradication strategy of an invasive species. J. Biol. Dyn. 10(1), 179–199 (2016)

    Article  MathSciNet  Google Scholar 

  47. Winder, M., Sommer, U.: Phytoplankton response to a changing climate. Hydrobiologia 698, 5–16 (2012)

    Article  Google Scholar 

  48. Wiens, J.D., Anthony, R.G., Forsman, E.D.: Competitive interactions and resource partitioning between northern spotted owls and barred owls in western Oregon. Wildl. Monogr. 185, 1–50 (2014)

    Article  Google Scholar 

  49. Wang, X., Zanette, L., Zou, X.: Modelling the fear effect in predator–prey interactions. J. Math. Biol. 73, 1179–1204 (2016)

    Article  MathSciNet  Google Scholar 

  50. Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differential Equation. Science Press, Beijing (1992)

    Google Scholar 

  51. Zhang, H., Cai, Y., Fu, S., Wang, W.: Impact of the fear effect in a prey–predator model incorporating a prey refuge. Appl. Math. Comput. 356, 328–337 (2019)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Fuzhou University, Fuzhou, 350108, Fujian, People’s Republic of China

    Shangming Chen & Fengde Chen

  2. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Vaibhava Srivastava & Rana D. Parshad

Authors
  1. Shangming Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fengde Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vaibhava Srivastava
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rana D. Parshad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rana D. Parshad.

Additional information

Publisher's Note

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

Appendix A

Appendix A

$$\begin{aligned} g_{11}&=\frac{\left( H^{2} m +1\right) Q_1 b}{Q_2^{2}}, \quad g_{02}=\frac{\left( 2 H k +1\right) \left( -1+H \right) Q_3}{Q_2^{2} \left( H k +1\right) ^{4} \left( H^{2} m +1\right) ^{2}},\quad f_{20}=-\frac{Q_4}{Q_2^{2} \left( 2 H k +1\right) ^{2}}, \\ f_{11}&=-\frac{b H \left( H k +1\right) ^{2} \left( H^{2} m +1\right) Q_5}{Q_2^{2} \left( 2 H k +1\right) },\quad f_{02}=-\frac{Q_6}{Q_2^{2} \left( H k +1\right) ^{2} \left( H^{2} m +1\right) },\\ Q_1&=4 H^{6} b k^{3} m -6 H^{5} b k^{3} m +7 H^{5} b k^{2} m -12 H^{4} bk^{2} m +4 H^{4} b k m +4 H^{4} k^{2} m \\&\quad -2 H^{3} bk^{3}-3 H^{3} bk^{2}-8 H^{3} b k m -4 H^{3} k^{2} m +H^{3} b m +4 H^{3} k^{2}+4 H^{3} k m \\&\quad -2 H^{2} bk^{2}-4 H^{2} b k -2 H^{2} b m -4 H^{2} k^{2}-4 H^{2} k m +4 H^{2} k +H^{2} m -b H \\&\quad -4 H k -H m +H -1,\\ Q_2&=H^{5} bk^{2} m +2 H^{4} b k m +H^{3} bk^{2}+H^{3} b m -2 H^{3} k m +2 H^{2} b k +2 H^{2} k m -2 H^{2} k \\&\quad -H^{2} m +b H +2 H k +H m -H +1,\\ Q_3&=3 H^{11} b^{2} k^{5} m^{3}+2 H^{10} b^{2} k^{5} m^{2}+12 H^{10} b^{2} k^{4} m^{3}+7 H^{9} b^{2} k^{5} m^{2}+9 H^{9} b^{2} k^{4} m^{2}\\&\quad +19 H^{9} b^{2} k^{3} m^{3}-4 H^{9} bk^{4} m^{3}+4 H^{8} b^{2} k^{5} m +27 H^{8} b^{2} k^{4} m^{2}+16 H^{8} b^{2} k^{3} m^{2}\\&\quad +15 H^{8} b^{2} k^{2} m^{3}-12 H^{8} b k^{4} m^{2}-12 H^{8} bk^{3} m^{3}+5 H^{7} b^{2} k^{5} m +18 H^{7} b^{2} k^{4} m \\&\quad +41 H^{7} b^{2} k^{3} m^{2}+14 H^{7} b^{2} k^{2} m^{2}+6 H^{7} b^{2} km^{3}-8 H^{7} bk^{4} m -36 H^{7} bk^{3} m^{2}\\&\quad -13 H^{7} bk^{2} m^{3}+8 H^{7} k^{3} m^{3}+2 H^{6} b^{2} k^{5}+18 H^{6} b^{2} k^{4} m +32 H^{6} b^{2} k^{3} m \\&\quad +31 H^{6} b^{2} k^{2} m^{2}-8 H^{6} b k^{4} m -8 H^{6} k^{3} m^{3}+H^{5} b^{2} k^{5}+6 H^{6} b^{2} k m^{2}+H^{6} b^{2} m^{3}\\&\quad -24 H^{6} b k^{3} m -39 H^{6} b k^{2} m^{2}-6 H^{6} b k m^{3}+24 H^{6} k^{3} m^{2}+12 H^{6} k^{2} m^{3}\\&\quad +9 H^{5} b^{2} k^{4}+25 H^{5} b^{2} k^{3} m +4 H^{5} b k^{4} m +28 H^{5} b^{2} k^{2} m +12 H^{5} b^{2} k m^{2}-8 H^{5} b k^{4}\\&\quad -24 H^{5} b k^{3} m -24 H^{5} k^{3} m^{2}-12 H^{5} k^{2} m^{3}+3 H^{4} b^{2} k^{4}+H^{5} b^{2} m^{2}-26 H^{5} b k^{2} m \\&\quad -18 H^{5} b k \,m^{2}-H^{5} b m^{3}+24 H^{5} k^{3} m +36 H^{5} k^{2} m^{2}+6 H^{5} k m^{3}+16 H^{4} b^{2} k^{3}\\&\quad +17 H^{4} b^{2} k^{2} m +4 H^{4} b \,k^{4}+12 H^{4} b \,k^{3} m +12 H^{4} b^{2} k m +2 H^{4} b^{2} m^{2}-24 H^{4} b \,k^{3}\\&\quad -26 H^{4} b \,k^{2} m -24 H^{4} k^{3} m -36 H^{4} k^{2} m^{2}-6 H^{4} k \,m^{3}+3 H^{3} b^{2} k^{3}-12 H^{4} b k m\\&\quad -3 H^{4} b \,m^{2}+8 H^{4} k^{3}+36 H^{4} k^{2} m +18 H^{4} k \,m^{2}+H^{4} m^{3}+14 H^{3} b^{2} k^{2}\\&\quad +6 H^{3} b^{2} k m +12 H^{3} b \,k^{3}+13 H^{3} b \,k^{2} m +2 H^{3} b^{2} m -26 H^{3} b \,k^{2}-12 H^{3} b k m \\&\quad -8 H^{3} k^{3}-36 H^{3} k^{2} m -18 H^{3} k \,m^{2}-H^{3} m^{3}+k^{2} b^{2} H^{2}-2 H^{3} b m +12 H^{3} k^{2}\\&\quad +18 H^{3} k m +3 H^{3} m^{2}+6 H^{2} b^{2} k +H^{2} b^{2} m +13 H^{2} b \,k^{2}+6 H^{2} b k m -12 H^{2} b k\\&\quad -2 H^{2} b m -12 H^{2} k^{2}-18 H^{2} k m -3 H^{2} m^{2}+6 H^{2} k +3 H^{2} m +H \,b^{2}+6 H b k \\&\quad +H b m-2 b H -6 H k -3 H m +H +b -1,\\ Q_4&=\left( H^{2} m +1\right) ^{3} \left( H k +1\right) ^{5} \left( -1+H \right) \left( 3 H^{2} k^{2} m +3 H k m +k^{2}+m \right) H^{2} b^{2},\\ Q_5&=H^{9} b^{2} k^{4} m^{2}+4 H^{8} b^{2} k^{3} m^{2}+2 H^{7} b^{2} k^{4} m +6 H^{7} b^{2} k^{2} m^{2}+8 H^{6} b^{2} k^{3} m -2 H^{6} bk^{3} m^{2}\\&\quad +4 H^{6} b^{2} k \,m^{2}-4 H^{6} b \,k^{3} m -3 H^{6} b \,k^{2} m^{2}+H^{5} b^{2} k^{4}+12 H^{5} b^{2} k^{2} m +4 H^{5} b \,k^{3} m \\&\quad -2 H^{5} b \,k^{2} m^{2}+H^{5} b^{2} m^{2}-10 H^{5} b \,k^{2} m -4 H^{5} b k \,m^{2}+8 H^{5} k^{2} m^{2}+4 H^{4} b^{2} k^{3}\\&\quad -4 H^{4} b \,k^{3} m +8 H^{4} b^{2} k m -4 H^{4} b \,k^{3}+4 H^{4} b \,k^{2} m -12 H^{4} k^{2} m^{2}-8 H^{4} b k m \\&\quad -H^{4} b \,m^{2}+12 H^{4} k^{2} m +8 H^{4} k \,m^{2}+6 H^{3} b^{2} k^{2}+4 H^{3} b \,k^{3}-4 H^{3} b \,k^{2} m \\&\quad +4 H^{3} k^{2} m^{2}+2 H^{3} b^{2} m -10 H^{3} b \,k^{2}-16 H^{3} k^{2} m -12 H^{3} k \,m^{2}-2 H^{2} b \,k^{3}\\&\quad -2 H^{3} b m +4 H^{3} k^{2}+12 H^{3} k m +2 H^{3} m^{2}+4 H^{2} b^{2} k +7 H^{2} b \,k^{2}+4 H^{2} k^{2} m \\&\quad +4 H^{2} k \,m^{2}-8 H^{2} b k -4 H^{2} k^{2}-16 H^{2} k m -3 H^{2} m^{2}-2 H b \,k^{2}+4 H^{2} k \\&\quad +3 H^{2} m +b^{2} H +4 H b k +4 H k m +H \,m^{2}-2 H b -4 H k -4 H m +H \\&\quad +b +m -1,\\ Q6&=H^{14} b^{3} k^{6} m^{3}+6 H^{13} b^{3} k^{5} m^{3}+3 H^{12} b^{3} k^{6} m^{2}+15 H^{12} b^{3} k^{4} m^{3}-H^{12} b^{2} k^{5} m^{3}\\&\quad +18 H^{11} b^{3} k^{5} m^{2}+H^{11} b^{2} k^{5} m^{3}+20 H^{11} b^{3} k^{3} m^{3}-2 H^{11} b^{2} k^{5} m^{2}-6 H^{11} b^{2} k^{4} m^{3}\\&\quad +3 H^{10} b^{3} k^{6} m +45 H^{10} b^{3} k^{4} m^{2}+H^{10} b^{2} k^{5} m^{2}+6 H^{10} b^{2} k^{4} m^{3}+15 H^{10} b^{3} k^{2} m^{3}\\&\quad -9 H^{10} b^{2} k^{4} m^{2}-13 H^{10} b^{2} k^{3} m^{3}+18 H^{9} b^{3} k^{5} m +H^{9} b^{2} k^{5} m^{2}+60 H^{9} b^{3} k^{3} m^{2}\\&\quad -4 H^{9} b^{2} k^{5} m +13 H^{9} b^{2} k^{3} m^{3}-4 H^{9} b \,k^{4} m^{3}+H^{8} b^{3} k^{6}+6 H^{9} b^{3} k \,m^{3}\\&\quad -16 H^{9} b^{2} k^{3} m^{2}-13 H^{9} b^{2} k^{2} m^{3}-4 H^{9} b \,k^{4} m^{2}+45 H^{8} b^{3} k^{4} m +5 H^{8} b^{2} k^{5} m \\&\quad +9 H^{8} b^{2} k^{4} m^{2}+4 H^{8} b \,k^{4} m^{3}+45 H^{8} b^{3} k^{2} m^{2}-18 H^{8} b^{2} k^{4} m -7 H^{8} b^{2} k^{3} m^{2}\\&\quad +13 H^{8} b^{2} k^{2} m^{3}-12 H^{8} b \,k^{3} m^{3}+6 H^{7} b^{3} k^{5}-H^{7} b^{2} k^{5} m+H^{8} b^{3} m^{3}\\&\quad -14 H^{8} b^{2} k^{2} m^{2}-6 H^{8} b^{2} k \,m^{3}-4 H^{8} b \,k^{4} m -12 H^{8} b \,k^{3} m^{2}+8 H^{8} k^{3} m^{3}\\&\quad +60 H^{7} b^{3} k^{3} m -2 H^{7} b^{2} k^{5}+18 H^{7} b^{2} k^{4} m +23 H^{7} b^{2} k^{3} m^{2}+12 H^{7} b \,k^{3} m^{3}\\&\quad +18 H^{7} b^{3} k \,m^{2}-32 H^{7} b^{2} k^{3} m -11 H^{7} b^{2} k^{2} m^{2}+6 H^{7} b^{2} k \,m^{3}-13 H^{7} b \,k^{2} m^{3}\\&\quad -16 H^{7} k^{3} m^{3}+15 H^{6} b^{3} k^{4}+3 H^{6} b^{2} k^{5}+4 H^{6} b \,k^{4} m^{2}-6 H^{7} b^{2} k \,m^{2}-H^{7} b^{2} m^{3}\\&\quad -12 H^{7} b \,k^{3} m -13 H^{7} b \,k^{2} m^{2}+24 H^{7} k^{3} m^{2}+12 H^{7} k^{2} m^{3}+45 H^{6} b^{3} k^{2} m \\&\quad -9 H^{6} b^{2} k^{4}+25 H^{6} b^{2} k^{3} m +25 H^{6} b^{2} k^{2} m^{2}+13 H^{6} b \,k^{2} m^{3}+8 H^{6} k^{3} m^{3}-H^{5} b^{2} k^{5}\\&\quad +3 H^{6} b^{3} m^{2}-28 H^{6} b^{2} k^{2} m -6 H^{6} b^{2} k \,m^{2}+H^{6} b^{2} m^{3}-4 H^{6} b \,k^{4}-6 H^{6} b k \,m^{3}\\&\quad -48 H^{6} k^{3} m^{2}-24 H^{6} k^{2} m^{3}+20 H^{5} b^{3} k^{3}+12 H^{5} b^{2} k^{4}+7 H^{5} b^{2} k^{3} m +4 H^{5} b \,k^{4} m \\&\quad +12 H^{5} b \,k^{3} m^{2}-H^{6} b^{2} m^{2}-13 H^{6} b \,k^{2} m -6 H^{6} b k \,m^{2}+24 H^{6} k^{3} m +36 H^{6} k^{2} m^{2}\\&\quad +6 H^{6} k \,m^{3}+18 H^{5} b^{3} k m -16 H^{5} b^{2} k^{3}+17 H^{5} b^{2} k^{2} m +12 H^{5} b^{2} k \,m^{2}+4 H^{5} b \,k^{4}\\&\quad +6 H^{5} b k \,m^{3}+24 H^{5} k^{3} m^{2}+12 H^{5} k^{2} m^{3}-3 H^{4} b^{2} k^{4}-12 H^{5} b^{2} k m -H^{5} b^{2} m^{2}\\&\quad -12 H^{5} b \,k^{3}-H^{5} b \,m^{3}-48 H^{5} k^{3} m -72 H^{5} k^{2} m^{2}-12 H^{5} k \,m^{3}+15 H^{4} b^{3} k^{2}\\&\quad +19 H^{4} b^{2} k^{3}+11 H^{4} b^{2} k^{2} m +12 H^{4} b \,k^{3} m +13 H^{4} b \,k^{2} m^{2}-6 H^{5} b k m -H^{5} b \,m^{2}\\&\quad +8 H^{5} k^{3}+36 H^{5} k^{2} m +18 H^{5} k \,m^{2}+H^{5} m^{3}+3 H^{4} b^{3} m-14 H^{4} b^{2} k^{2}\\&\quad +6 H^{4} b^{2} k m +2 H^{4} b^{2} m^{2}+12 H^{4} b \,k^{3}+H^{4} b \,m^{3}+24 H^{4} k^{3} m +36 H^{4} k^{2} m^{2}\\&\quad +6 H^{4} k \,m^{3}-3 H^{3} b^{2} k^{3}-2 H^{4} b^{2} m -13 H^{4} b \,k^{2}-16 H^{4} k^{3}-72 H^{4} k^{2} m \\&\quad -36 H^{4} k \,m^{2}-2 H^{4} m^{3}+6 H^{3} b^{3} k +15 H^{3} b^{2} k^{2}+6 H^{3} b^{2} k m +13 H^{3} b \,k^{2} m \\&\quad +6 H^{3} b k \,m^{2}-H^{4} b m +12 H^{4} k^{2}+18 H^{4} k m +3 H^{4} m^{2}-6 H^{3} b^{2} k +H^{3} b^{2} m\\&\quad +13 H^{3} b \,k^{2}+8 H^{3} k^{3}+36 H^{3} k^{2} m +18 H^{3} k \,m^{2}+H^{3} m^{3}-H^{2} b^{2} k^{2}-6 H^{3} b k\\&\quad -24 H^{3} k^{2}-36 H^{3} k m -6 H^{3} m^{2}+H^{2} b^{3}+6 H^{2} b^{2} k +H^{2} b^{2} m +6 H^{2} b k m +H^{2} b \,m^{2}\\&\quad +6 H^{3} k +3 H^{3} m -H^{2} b^{2}+6 H^{2} b k +12 H^{2} k^{2}+18 H^{2} k m +3 H^{2} m^{2}-H^{2} b -12 H^{2} k \\&\quad -6 H^{2} m +H \,b^{2}+H b m +H^{2}+H b +6 H k +3 H m -2 H +1. \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

Chen, S., Chen, F., Srivastava, V. et al. Dynamical Analysis of an Allelopathic Phytoplankton Model with Fear Effect. Qual. Theory Dyn. Syst. 23, 189 (2024). https://doi.org/10.1007/s12346-024-01047-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-024-01047-3

Keywords

Search

Navigation