Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients
DCDS-B Home
On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions
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,,
Department of Mathematics, University Dr. Moulay Tahar of Saida, Algeria
Department of Mathematics, USTHB, Bab Ezzouar, Algiers, Algeria
IMAG, Univ Montpellier, CNRS, Montpellier, France
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
Figure(5) / Table(1)
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 & Continuous Dynamical Systems - B, 2021, 26 (12) : 6405-6424. doi: 10.3934/dcdsb.2021025
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
F. Gantmacher, The Theory of Matrices, Volume 2, AMS Chelsea Publishing, 2000. Google Scholar
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. Google Scholar
S. A. Levin, Dispersion and population interactions, Amer. Natur, 108 (1974), 207-228. doi: 10.1086/282900. Google Scholar
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. Google Scholar
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 Google Scholar
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. Google Scholar
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. Google Scholar
H. G. Othmer, A Continuum Model for Coupled Cells, J. Math. Biology, 17 (1983), 351-369. doi: 10.1007/BF00276521. Google Scholar
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. Google Scholar
Y. Takeuchi, Cooperative systems theory and global stability of diffusion models, Acta Applicandae Mathematicae, 14 (1989), 49-57. doi: 10.1007/BF00046673. Google Scholar
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 Google Scholar
W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Robert E. Krieger Publishing Company, Huntington, NY, 1976. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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 $
Table 1">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
Table 1">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
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 $
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 $
Download as excel
Xing Liang, Lei Zhang. The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2055-2065. doi: 10.3934/dcdsb.2020280
Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315
J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054
Hanwu Liu, Lin Wang, Fengqin Zhang, Qiuying Li, Huakun Zhou. Dynamics of a predator-prey model with state-dependent carrying capacity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4739-4753. doi: 10.3934/dcdsb.2019028
François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
Qiaoling Chen, Fengquan Li, Feng Wang. A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 13-35. doi: 10.3934/dcdsb.2016.21.13
Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259
Yanan Wang, Tao Xie, Xiaowen Jie. A mathematical analysis for the forecast research on tourism carrying capacity to promote the effective and sustainable development of tourism. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 837-847. doi: 10.3934/dcdss.2019056
Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1763-1781. doi: 10.3934/dcdsb.2021005
Tong Li, Jeungeun Park. Traveling waves in a chemotaxis model with logistic growth. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6465-6480. doi: 10.3934/dcdsb.2019147
E. Trofimchuk, Sergei Trofimchuk. Global stability in a regulated logistic growth model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 461-468. doi: 10.3934/dcdsb.2005.5.461
Suqi Ma, Qishao Lu, Shuli Mei. Dynamics of a logistic population model with maturation delay and nonlinear birth rate. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 735-752. doi: 10.3934/dcdsb.2005.5.735
Steffen Eikenberry, Sarah Hews, John D. Nagy, Yang Kuang. The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 2009, 6 (2) : 283-299. doi: 10.3934/mbe.2009.6.283
Xinyue Fan, Claude-Michel Brauner, Linda Wittkop. Mathematical analysis of a HIV model with quadratic logistic growth term. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2359-2385. doi: 10.3934/dcdsb.2012.17.2359
Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934/dcdsb.2020218
Markus Thäter, Kurt Chudej, Hans Josef Pesch. Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth. Mathematical Biosciences & Engineering, 2018, 15 (2) : 485-505. doi: 10.3934/mbe.2018022
Xianhua Tang, Xingfu Zou. A 3/2 stability result for a regulated logistic growth model. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 265-278. doi: 10.3934/dcdsb.2002.2.265
Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199
Urszula Ledzewicz, James Munden, Heinz Schättler. Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 415-438. doi: 10.3934/dcdsb.2009.12.415
Qian Zhao, Bin Liu. Global generalized solutions to the forager-exploiter model with logistic growth. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021273
Bilel Elbetch Tounsia Benzekri Daniel Massart Tewfik Sari