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December  2021, 26(12): 6405-6424. doi: 10.3934/dcdsb.2021025

The multi-patch logistic equation

Bilel Elbetch 1, , Tounsia Benzekri 2, , Daniel Massart 3, and  Tewfik Sari 4,,

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  December 2021 Early access  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.

Keywords: Fragmented habitat, Migration rate, Logistic growth, Carrying capacity, Perfect mixing.
Mathematics Subject Classification: Primary: 37N25, 92D25; Secondary: 34D23, 34D15.
Citation: Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6405-6424. doi: 10.3934/dcdsb.2021025
References:
[1]

R. Arditi, L.-F Bersier and R. P. Rohr, The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka, Ecosphere, 7 (2016), e01599. doi: 10.1002/ecs2.1599.

[2]

R. Arditi, C. Lobry and T. Sari, Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation, Theoretical Population Biology, 106 (2015), 45-59.  doi: 10.1016/j.tpb.2015.10.001.

[3]

R. Arditi, C. Lobry and T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theoretical Population Biology, 120 (2018), 11-15.  doi: 10.1016/j.tpb.2017.12.006.

[4]

A. Cvetković, Stabilizing the Metzler matrices with applications to dynamical systems, Calcolo, 57 (2020), Paper No. 1, 34 pp. doi: 10.1007/s10092-019-0350-3.

[5]

D. L. DeAngelis, C. C. Travis and W. M. Post, Persistence and stability of seed-dispersel species in a patchy environment, Theoretical Population Biology, 16 (1979), 107-125.  doi: 10.1016/0040-5809(79)90008-X.

[6]

D. L. DeAngelis and B. Zhang, Effects of dispersal in a non-uniform environment on population dynamics and competition: a patch model approach, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3087-3104.  doi: 10.3934/dcdsb.2014.19.3087.

[7]

D. L. DeAngelis, We i-Ming Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453.  doi: 10.1007/s12080-016-0302-3.

[8]

H. I. Freedman, B. Rai and P. Waltman, Mathematical models of population interactions with dispersal II: Differential survival in a change of habitat, Journal of Mathematical Analysis and Applications, 115 (1986), 140-154.  doi: 10.1016/0022-247X(86)90029-6.

[9]

H. I. Freedman and P. Waltman, Mathematical Models of Population Interactions with Dispersal I: Stabilty of two habitats with and without a predator, SIAM Journal on Applied Mathematics, 32 (1977), 631-648.  doi: 10.1137/0132052.

[10]

F. Gantmacher, The Theory of Matrices, Volume 2, AMS Chelsea Publishing, 2000.

[11]

R. D. Holt, Population dynamics in two patch environments: Some anomalous consequences of an optimal habitat distribution, Theoretical Population Biology, 28 (1985), 181-201.  doi: 10.1016/0040-5809(85)90027-9.

[12]

S. A. Levin, Dispersion and population interactions, Amer. Natur, 108 (1974), 207-228.  doi: 10.1086/282900.

[13]

S. A. Levin, Spatial patterning and the structure of ecological communities, in Some Mathematical Questions in Biology, VII, Lectures on Math. in the Life Sciences, Amer. Math. Soc., Providence, R.I., 8 (1976), 1–35.

[14]

C. Lobry, T. Sari and S. Touhami, On Tykhonov's theorem for convergence of solutions of slow and fast systems, Electron. J. Differential Equations, 19 (1998), 22pp. https://ejde.math.txstate.edu/Volumes/1998/19/Lobry.pdf

[15]

Z. Lu and Y. Takeuchi, Global asymptotic behavior in single-species discrete diffusion systems, J. Math. Biol., 32 (1993), 67-77.  doi: 10.1007/BF00160375.

[16]

Y. Nesterov and V. Y. Protasov, Computing closest stable nonnegative matrix, SIAM Journal on Matrix Analysis and Applications, 41 (2020), 1-28.  doi: 10.1137/17M1144568.

[17]

H. G. Othmer, A Continuum Model for Coupled Cells, J. Math. Biology, 17 (1983), 351-369.  doi: 10.1007/BF00276521.

[18]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of microbial competition, Cambridge Studies in Mathematical Biology, 13. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.

[19]

Y. Takeuchi, Cooperative systems theory and global stability of diffusion models, Acta Applicandae Mathematicae, 14 (1989), 49-57.  doi: 10.1007/BF00046673.

[20]

A. N. Tikhonov, Systems of differential equations containing small parameters in the derivatives, Mat. Sb. (N.S.), 31 (1952), 575–586. http://www.mathnet.ru/links/9e00b6540bb8ca1fdb5147771c7d98d4/sm5548.pdf

[21]

W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Robert E. Krieger Publishing Company, Huntington, NY, 1976.

[22]

B. P. Yurk and C. A. Cobbold, Homogenization techniques for population dynamics in strongly heterogeneous landscapes, Journal of Biological Dynamics, 12 (2018), 171-193.  doi: 10.1080/17513758.2017.1410238.

[23]

N. Zaker, L. Ketchemen and F. Lutscher, The effect of movement behavior on population density in patchy landscapes, Bulletin of Mathematical Biology, 82 (2020), 24pp. doi: 10.1007/s11538-019-00680-3.

[24]

B. Zhang, X. Liu, D. L. DeAngelis, W. M. Ni and G. G. Wang, Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment, Math. Biosci., 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.

show all references

References:
[1]

R. Arditi, L.-F Bersier and R. P. Rohr, The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka, Ecosphere, 7 (2016), e01599. doi: 10.1002/ecs2.1599.

[2]

R. Arditi, C. Lobry and T. Sari, Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation, Theoretical Population Biology, 106 (2015), 45-59.  doi: 10.1016/j.tpb.2015.10.001.

[3]

R. Arditi, C. Lobry and T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theoretical Population Biology, 120 (2018), 11-15.  doi: 10.1016/j.tpb.2017.12.006.

[4]

A. Cvetković, Stabilizing the Metzler matrices with applications to dynamical systems, Calcolo, 57 (2020), Paper No. 1, 34 pp. doi: 10.1007/s10092-019-0350-3.

[5]

D. L. DeAngelis, C. C. Travis and W. M. Post, Persistence and stability of seed-dispersel species in a patchy environment, Theoretical Population Biology, 16 (1979), 107-125.  doi: 10.1016/0040-5809(79)90008-X.

[6]

D. L. DeAngelis and B. Zhang, Effects of dispersal in a non-uniform environment on population dynamics and competition: a patch model approach, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3087-3104.  doi: 10.3934/dcdsb.2014.19.3087.

[7]

D. L. DeAngelis, We i-Ming Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453.  doi: 10.1007/s12080-016-0302-3.

[8]

H. I. Freedman, B. Rai and P. Waltman, Mathematical models of population interactions with dispersal II: Differential survival in a change of habitat, Journal of Mathematical Analysis and Applications, 115 (1986), 140-154.  doi: 10.1016/0022-247X(86)90029-6.

[9]

H. I. Freedman and P. Waltman, Mathematical Models of Population Interactions with Dispersal I: Stabilty of two habitats with and without a predator, SIAM Journal on Applied Mathematics, 32 (1977), 631-648.  doi: 10.1137/0132052.

[10]

F. Gantmacher, The Theory of Matrices, Volume 2, AMS Chelsea Publishing, 2000.

[11]

R. D. Holt, Population dynamics in two patch environments: Some anomalous consequences of an optimal habitat distribution, Theoretical Population Biology, 28 (1985), 181-201.  doi: 10.1016/0040-5809(85)90027-9.

[12]

S. A. Levin, Dispersion and population interactions, Amer. Natur, 108 (1974), 207-228.  doi: 10.1086/282900.

[13]

S. A. Levin, Spatial patterning and the structure of ecological communities, in Some Mathematical Questions in Biology, VII, Lectures on Math. in the Life Sciences, Amer. Math. Soc., Providence, R.I., 8 (1976), 1–35.

[14]

C. Lobry, T. Sari and S. Touhami, On Tykhonov's theorem for convergence of solutions of slow and fast systems, Electron. J. Differential Equations, 19 (1998), 22pp. https://ejde.math.txstate.edu/Volumes/1998/19/Lobry.pdf

[15]

Z. Lu and Y. Takeuchi, Global asymptotic behavior in single-species discrete diffusion systems, J. Math. Biol., 32 (1993), 67-77.  doi: 10.1007/BF00160375.

[16]

Y. Nesterov and V. Y. Protasov, Computing closest stable nonnegative matrix, SIAM Journal on Matrix Analysis and Applications, 41 (2020), 1-28.  doi: 10.1137/17M1144568.

[17]

H. G. Othmer, A Continuum Model for Coupled Cells, J. Math. Biology, 17 (1983), 351-369.  doi: 10.1007/BF00276521.

[18]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of microbial competition, Cambridge Studies in Mathematical Biology, 13. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.

[19]

Y. Takeuchi, Cooperative systems theory and global stability of diffusion models, Acta Applicandae Mathematicae, 14 (1989), 49-57.  doi: 10.1007/BF00046673.

[20]

A. N. Tikhonov, Systems of differential equations containing small parameters in the derivatives, Mat. Sb. (N.S.), 31 (1952), 575–586. http://www.mathnet.ru/links/9e00b6540bb8ca1fdb5147771c7d98d4/sm5548.pdf

[21]

W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Robert E. Krieger Publishing Company, Huntington, NY, 1976.

[22]

B. P. Yurk and C. A. Cobbold, Homogenization techniques for population dynamics in strongly heterogeneous landscapes, Journal of Biological Dynamics, 12 (2018), 171-193.  doi: 10.1080/17513758.2017.1410238.

[23]

N. Zaker, L. Ketchemen and F. Lutscher, The effect of movement behavior on population density in patchy landscapes, Bulletin of Mathematical Biology, 82 (2020), 24pp. doi: 10.1007/s11538-019-00680-3.

[24]

B. Zhang, X. Liu, D. L. DeAngelis, W. M. Ni and G. G. Wang, Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment, Math. Biosci., 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.

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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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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