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Effect of toxic substance on delayed competitive allelopathic phytoplankton system with varying parameters through stability and bifurcation analysis

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  • Pal, D.
  • Mahapatra, G.S.

Abstract

We have studied the combined effect of toxicant and fluctuation of the biological parameters on the dynamical behaviors of a delayed two-species competitive system with imprecise biological parameters. Due to the global increase of harmful phytoplankton blooms, the study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research now days. The ordinary mathematical formulation of models for two competing phytoplankton species, when one or both the species liberate toxic substances, is unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic model never predicts the sudden localized behavior of certain species. These obstacles of mathematical modeling can be overcomed if we include interval variability of biological parameters in our modeling approach. In this investigation, we construct imprecise models of allelopathic interactions between two competing phytoplankton species as a parametric differential equation model. We incorporate the effect of toxicant on the species in two different cases known as toxic inhibition and toxic stimulatory system. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Analytical findings are supported through exhaustive numerical simulations.

Suggested Citation

  • Pal, D. & Mahapatra, G.S., 2016. "Effect of toxic substance on delayed competitive allelopathic phytoplankton system with varying parameters through stability and bifurcation analysis," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 109-124.
    • Handle: RePEc:eee:chsofr:v:87:y:2016:i:c:p:109-124
    DOI: 10.1016/j.chaos.2016.03.019
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    References listed on IDEAS

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    1. Győri, István & Hartung, Ferenc & Mohamady, Nahed A., 2015. "On a nonlinear delay population model," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 909-925.
    2. Chen, Yuanyuan & Yu, Jiang & Sun, Chengjun, 2007. "Stability and Hopf bifurcation analysis in a three-level food chain system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 683-694.
    3. G. S. Mahapatra & T. K. Mandal, 2012. "Posynomial Parametric Geometric Programming with Interval Valued Coefficient," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 120-132, July.
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    Cited by:

    1. Ye, Luhong & Zhao, Hongyong & Wu, Daiyong, 2023. "Spatial movement with memory-induced cross-diffusion effect and toxin effect in predator," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 68-91.
    2. Pal, D. & Samanta, G.P. & Mahapatra, G.S., 2017. "Selective harvesting of two competing fish species in the presence of toxicity with time delay," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 74-93.

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