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Bifurcation analysis of a Singular Nutrient-plankton-fish model with taxation, protected zone and multiple delays

  • * Corresponding author: Xin-You Meng

    * Corresponding author: Xin-You Meng 
The first author is supported by the National Natural Science Foundation of China (Grant No.11661050, 11861044), and the HongLiu First-class Disciplines Development Program of Lanzhou University of Technology
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  • A differential algebraic nutrient-plankton-fish model with taxation, free fishing zone, protected zone and multiple delays is investigated in this paper. First, the conditions of existence and control of singularity induced bifurcation are given by regarding economic interest as bifurcation parameter. Meanwhile, the existence of Hopf bifurcations are investigated when migration rates, taxation and the cost per unit harvest are taken as bifurcation parameters respectively. Next, the local stability of the interior equilibrium, existence and properties of Hopf bifurcation are discussed in the different cases of five delays. Furthermore, the optimal tax policy is obtained by using Pontryagin's maximum principle. Finally, some numerical simulations are presented to demonstrate analytical results.

    Mathematics Subject Classification: Primary: 92D25, 34D23; Secondary: 34H05.


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  • Figure 1.  Dynamical responses of system (3) around $ X_{0} $. (a) system (7) without control under $ v = -0.1 $; (b) system (7) without control under $ v = 0 $; (c) system (7) with control

    Figure 2.  Dynamical responses of system (7) with different $ \delta_{1} $. (a) N(t); (b) P(t); (c) Z(t); (d) $ F_{1}(t) $; (e)$ F_{2}(t) $

    Figure 3.  Phase diagram of system (7) around $ \bar{X} $: (a)$ \delta_{1} = 1 $; (b)$ \delta_{1} = 2.5 $; (c) $ \delta_{1} = 7 $

    Figure 4.  Dynamical responses of system (7) with different $ \delta_{2} $: (a) N(t); (b) P(t); (c) Z(t); (d) $ F_{1}(t) $; (e)$ F_{2}(t) $; (f)-(h) $ \delta_{2} = 0.06,0.3 $

    Figure 5.  Dynamical responses of system (7) with different $ c $: (a) N(t); (b) P(t); (c) Z(t); (d) $ F_{1}(t) $; (e)$ F_{2}(t) $; (f)$ c = 0.2 $; (g)$ c = 1 $; (h)$ c = 2.6 $

    Figure 6.  Dynamical responses of system (7) with different $ T $: (a) N(t); (b) P(t); (c) Z(t); (d) $ F_{1}(t) $; (e)$ F_{2}(t) $; (f)-(h) $ T = 0.85,2 $

    Figure 7.  Dynamical response and phase plot of system at $ \bar{X} $. (a)and(d): system (17) with $ \tau_{2} = 0.8,\tau_{4} = 1<\tau^{*}_{4} $; (b)and(e): system (17) with $ \tau_{2} = 0.8,\tau_{4} = 1.38>\tau^{*}_{4} $; (c)and(f): the control system with $ \tau_{2} = 0.8,\tau_{4} = 1.38>\tau^{*}_{4} $

    Figure 8.  Dynamical response of system (3) with $ \tau = 0.003<\tau_{0} $ and $ \tau = 0.09>\tau_{0} $ at $ \bar{X} $.(a): dynamical response of system; (b)and(c): phase plot of system

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