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The multi-patch logistic equation

  • * Corresponding author: Tewfik Sari

    * Corresponding author: Tewfik Sari

The authors where supported by CNRS-PICS project CODYSYS 278552

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  • The paper considers a $ n $-patch model with migration terms, where each patch follows a logistic law. First, we give some properties of the total equilibrium population. In some particular cases, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might be greater or smaller than the sum of the $ n $ carrying capacities. Second, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic law with a carrying capacity which in general is different from the sum of the $ n $ carrying capacities. Finally, for the three-patch model we show numerically that the increase in number of patches from two to three gives a new behavior for the dynamics of the total equilibrium population as a function of the migration rate.

    Mathematics Subject Classification: Primary: 37N25, 92D25; Secondary: 34D23, 34D15.

    Citation:

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  • Figure 1.  Qualitative properties of model (4) when (16) holds. In $ \mathcal{J}_0 $, patchiness has a beneficial effect on total equilibrium population. This effect is detrimental in $ \mathcal{J}_2 $. In $ \mathcal{J}_1 $, the effect is beneficial for $ \beta<\beta_0 $ and detrimental for $ \beta>\beta_0 $

    Figure 2.  Total equilibrium population $ X_{T}^{\ast} $ of the system (4) $ (n = 3) $ as a function of migration rate $ \beta $. The parameter values are given in Table 1

    Figure 3.  Total equilibrium population $ X_{T}^{\ast} $ of the system (4) $ (n = 3) $ as a function of migration rate $ \beta $. The figure on the right is a zoom, near the origin, of the figure on the left. The parameter values are given in Table 1

    Figure 4.  Total equilibrium population $ X_{T}^{\ast} $ of the system (4) $ (n = 3) $ as a function of migration rate $ \beta $. The parameter values are given in Table 1

    Figure 5.  The intersection point $ (x^{\ast}, x_n^{\ast}) $, between Ellipse $ \mathcal{E} $ and Parabola $ \mathcal{P} $, lies in the interior of triangle $ ABC $. (a): the case $ K<K_n $. (b): the case $ K>K_n $

    Table 1.  The numerical values of the parameters for the logistic growth function of the model (4), with $ n = 3 $, used in Fig.k__ge 2, 3, 4. All migration coefficients satisfy $ \gamma_{ij} = 1 $. The derivative of the total equilibrium population at $ \beta = 0 $ is computed with Eq. (48), and the perfect mixing total equilibrium population $ X_{T}^{\ast}(+\infty) $ is computed with Eq. (24)

    Figure $ r_{1} $ $ r_{2} $ $ r_{3} $ $ K_{1} $ $ K_{2} $ $ K_{3} $ $ \frac{dX^{\ast}_{T}(0)}{d \beta} $ $ X_{T}^{\ast}(+\infty) $
    2 $ 0.12 $ $ 18 $ $ 0.02 $ $ 0.5 $ $ 1.5 $ $ 2 $ $ -79.19 $ $ 4.44> \sum K_{i}=4 $
    3 $ 0.04 $ $ 3 $ $ 0.2 $ $ 0.5 $ $ 6 $ $ 9.5 $ $ 299.33 $ $ 16.17> \sum K_{i}=16 $
    4 $ 4 $ $ 0.7 $ $ 0.06 $ $ 5 $ $ 1 $ $ 4 $ $ -24.58 $ $ 9.42< \sum K_{i}=10 $
     | Show Table
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