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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

Received  July 2020 Revised  December 2020 Published  January 2021

Fund Project: The authors where supported by CNRS-PICS project CODYSYS 278552

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.

Citation: Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021025
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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 $\beta>\beta_0$
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
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
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
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 $K>K_n$
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 $4$ $0.7$ $0.06$ $5$ $1$ $4$ $-24.58$ $9.42< \sum K_{i}=10$
 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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