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Dynamical complexity in a delayed Plankton-Fish model with alternative food for predators

  • * Corresponding author: Rajinder Pal Kaur

    * Corresponding author: Rajinder Pal Kaur 

This paper is handled by Kapil Kumar Sharma as the guest editor

Abstract Full Text(HTML) Figure(8) / Table(2) Related Papers Cited by
  • The present manuscript deals with a 3-D food chain ecological model incorporating three species phytoplankton, zooplankton, and fish. To make the model more realistic, we include predation delay in the fish population due to the vertical migration of zooplankton species. We have assumed that additional food is available for both the predator population, viz., zooplankton, and fish. The main motive of the present study is to analyze the impact of available additional food and predation delay on the plankton-fish dynamics. The positivity and boundedness (with and without delay) are proved to make the system biologically valid. The steady states are determined to discuss the stability behavior of non-delayed dynamics under certain conditions. Considering available additional food as a control parameter, we have estimated ranges of alternative food for maintaining the sustainability and stability of the plankton-fish ecosystem. The Hopf-bifurcation analysis is carried out by considering time delay as a bifurcation parameter. The predation delay includes complexity in the system dynamics as it passes through its critical value. The direction of Hopf-bifurcation and stability of bifurcating periodic orbits are also determined using the centre manifold theorem. Numerical simulation is executed to validate theoretical results.

    Mathematics Subject Classification: 34C11, 34C23, 34D20, 92B05, 92D40.


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  • Figure 1.  Existence of steady states $ V_1 $, $ V_2 $ and $ V_3 $

    Figure 2.  stability of the system around $ V^* $ at $ r_5 = 0.7 $

    Figure 3.  Existence of Hopf-bifurcation at $ r_5 = 0.8 $

    Figure 4.  Bifurcation diagram for $ 0.1<r_5\leq 1 $

    Figure 5.  Existence of stability at $ \tau = 0 $ (left fig.) and $ \tau = 1.4<\tau_0 = 1.5 $ (right fig.)

    Figure 6.  Existence of Hopf-bifurcation at $ \tau = 1.5 = \tau_0 $, occurrence of limit cycles at $ \tau = 1.6, 2, 3 $(right fig)

    Figure 7.  Bifurcation diagrams for $ 1\leq\tau\leq3 $

    Figure 8.  Co-existence of all species according to table 2 with $ r_3 $ on x-axis and $ r_5 $, P(t), Z(t) and F(t) on y-axis

    Table 1.  Biological interpretation of Parameters

    Parameter Biological interpretation
    $ r_1 $ Growth rate of Phytoplankton.
    $ \alpha_1 $ Death rate phytoplankton.
    $ \beta_1 $ Maximum capture rate of phytoplankton by zooplankton.
    $ \beta_2 $ Maximum conversion rate of zooplankton.
    $ \gamma_1 $ Half saturation constant.
    $ \gamma_2 $ Half saturation constant.
    $ r_5 $ Additional food available for zooplankton.
    $ r_2 $ Death rate of zooplankton.
    $ a_1 $ Maximum capture rate of zooplankton by fish.
    $ a_2 $ Maximum conversion rate of fish.
    $ \theta $ Rate of toxin produced by TPP.
    $ r_3 $ Additional food available for fish.
    $ \alpha_2 $ Rate of quadratic harvesting.
    $ r_4 $ Natural mortality rate of fish.
     | Show Table
    DownLoad: CSV

    Table 2.  Impact of additional food ($ r_3 $ and $ r_5 $) on the co-existence of species

    $ r_3 $ $ r_5 $ P(t) Z(t) F(t)
    0.01 0.01 23.8297 3.5257 6.6514
    0.05 0.01 25.1787 2.4806 6.7049
    0.08 0.01 26.0261 1.7771 6.8449
    $ 0.1 $ 0.01 26.6036 1.2770 6.9687
    $ 0.2 $ $ 0.01 $ 27.9978 0.0000 11.1111
    $ 0.2 $ $ 0.4 $ 27.8282 0.1586 11.9851
    $ 0.2 $ $ 0.9 $ 26.0543 1.7483 20.0424
    $ 0.2 $ $ 1 $ 25.6022 2.1321 21.8148
    $ 0.2 $ $ 1.1 $ 25.0996 2.5444 23.6513
     | Show Table
    DownLoad: CSV
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