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On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions
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 |
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.
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. |


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