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A spatial food chain model for the Black Sea Anchovy, and its optimal fishery

  • * Corresponding author: Mahir Demir

    * Corresponding author: Mahir Demir 
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  • We present a spatial food chain model on a bounded domain coupled with optimal control theory to examine harvesting strategies. Motivated by the fishery industry in the Black Sea, the anchovy stock and two more trophic levels are modeled using nonlinear parabolic partial differential equations with logistic growth, movement by diffusion and advection, and Neumann boundary conditions. Necessary conditions for the optimal harvesting control are established. The objective for the problem is to find the spatial optimal harvesting strategy that maximizes the discounted net value of the anchovy population. Numerical simulations using data from the Black Sea are presented. We discuss spatial features of harvesting and the effects of diffusion and advection (migration speed) on the anchovy population. We also present the landing of anchovy and its net profit by applying two different harvesting strategies.

    Mathematics Subject Classification: Primary: 35B30, 93C20; Secondary: 49K20.

    Citation:

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  • Figure 1.  Optimal harvesting rate applied for 9 months. Left graph obtained with baseline parameters given in Table 1 and right graph obtained with different cost of anchovy fishing ($ \mu_1 $) and the other parameters from Table 1.

    Figure 2.  Optimal harvesting rate applied for 9 months. Left graph obtained with baseline parameters given in Table 1 and right graph obtained with a different initial condition and the other parameters from Table 1

    Figure 3.  Initial Biomass of anchovy (blue), jellyfish (red), and zooplankton (green).

    Figure 4.  Comparison of seasonal optimal harvesting strategy and the constant harvesting strategy applied for three months with the baseline parameters given in Table 1. The left plots show the populations with the optimal harvesting strategy with maximum harvest rate $ 0.35 $, and the right plots show the populations with constant harvesting strategy $ h = 0.35 $.

    Figure 5.  3D plot of optimal harvesting strategy given in Figure 4

    Figure 6.  Seasonal optimal harvesting rate applied for 3 months with different advection rates, $ b_i $ for $ i = 1,2,3 $

    Figure 7.  Seasonal optimal harvesting rate applied for 3 months with different diffusion rates, $ D_i $ for $ i = 1,2,3 $

    Table 1.  Parameter descriptions, units and numerical values

    Parameters Descriptions Unit Value
    $ r_1 $ Intrinsic growth rate of anchovy, $ A $ days$ ^{-1} $ 0.3
    $ r_2 $ Intrinsic growth rate of jelly fish, $ P $ days$ ^{-1} $ 0.75
    $ r_3 $ Intrinsic growth rate of zoo-plankton, $ Z $ days$ ^{-1} $ 0.9
    $ K_1 $ Carrying capacity of anchovy, $ A $ Tonnes 3.5$ e^{+5} $
    $ K_2 $ Carrying capacity of jelly fish, $ P $ Tonnes 7.5$ e^{+3} $
    $ K_3 $ Carrying capacity of zoo-plankton, $ Z $ Tonnes 4$ e^{+4} $
    $ m_0 $ Growth rate of $ A $ due to predation of $ Z $ (days x Tonnes)$ ^{-1} $ 1.4$ e^{-6} $
    $ m_1 $ Consumption rate of $ A $ by its predator, $ P $ (days x Tonnes)$ ^{-1} $ $ 0.66e^{-5} $
    $ m_2 $ Growth rate of $ P $ due to predation of $ A $ (days x Tonnes)$ ^{-1} $ $ 4.95e^{-6} $
    $ m_3 $ Growth rate of $ P $ due to predation of $ Z $ (days x Tonnes)$ ^{-1} $ $ 5.7e^{-6} $
    $ m_4 $ Consumption rate of $ Z $ due to its predator $ A $ (days x Tonnes)$ ^{-1} $ $ 0.2e^{-5} $
    $ m_5 $ Consumption rate of $ Z $ due to its predator $ P $ (days x Tonnes)$ ^{-1} $ $ 1e^{-5} $
    $ m_6 $ Consumption rate of $ P $ due to its predators days$ ^{-1} $ 0.2
    $ \mu_1 $ Coefficient of linear part of the cost function US Dollar $44,750,000
    $ \mu_2 $ Coefficient of quadratic part of the cost function US Dollar x days 0.1
    $ p $ Price of anchovy per tonnes US Dollar x (Tonnes)$ ^{-1} $ $1000
    $ \alpha $ The interest rate of the discount rate year$ ^{-1} $ 0.01
    $ b_i $ Advection Coefficient, ($ b_i > 0 $ for $ i=1,2,3 $) km x days$ ^{-1} $ 0.18
    $ D_i $ Diffusion Coefficient, ($ D_i > \theta > 0 $ for $ i=1,2,3 $) km$ ^{2} $ x days$ ^{-1} $ 0.05
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