The multi-patch logistic equation

Bilel Elbetch; Tounsia Benzekri; Daniel Massart; Tewfik Sari

doi:10.3934/dcdsb.2021025

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2021, Volume 26, Issue 12: 6405-6424. Doi: 10.3934/dcdsb.2021025 open access

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The multi-patch logistic equation

1.
Department of Mathematics, University Dr. Moulay Tahar of Saida, Algeria
2.
Department of Mathematics, USTHB, Bab Ezzouar, Algiers, Algeria
3.
IMAG, Univ Montpellier, CNRS, Montpellier, France
4.
ITAP, Univ Montpellier, INRAE, Institut Agro, Montpellier, France

^* Corresponding author: Tewfik Sari
^* Corresponding author: Tewfik Sari

Received: July 2020

Revised: December 2020

Early access: January 2021

Published: December 2021

The authors where supported by CNRS-PICS project CODYSYS 278552

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Abstract

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.

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

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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 $

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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

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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

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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

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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 $

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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 $

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