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

## 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  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.

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.
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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