In this paper, we consider a general single species model in a heterogeneous environment of patches (), where each patch follows a generalized logistic law. First, we prove the global stability of the model. Second, in the case of perfect mixing, i.e. when the migration rate tends to infinity, the total population follows a generalized logistic law with a carrying capacity which in general is different from the sum of the carrying capacities. Next, we give some properties of the total equilibrium population and we compute its derivative at no dispersal. 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 carrying capacities. Finally, we study an example of two-patch model where the first patch follows a logistic law and the second a Richard’s law, we give a complete classification of the model parameter space as to whether dispersal is beneficial or detrimental to the sum of two carrying capacities.
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Bilel Elbetch 1
@article{CRMATH_2023__361_G5_911_0, author = {Bilel Elbetch}, title = {Generalized logistic equation on {Networks}}, journal = {Comptes Rendus. Math\'ematique}, pages = {911--934}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.460}, language = {en}, }
Bilel Elbetch. Generalized logistic equation on Networks. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 911-934. doi : 10.5802/crmath.460. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.460/
[1] In dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation, Theor. Popul. Biol., Volume 106 (2015), pp. 45-59 | DOI | Zbl
[2] Asymmetric dispersal in the multi-patch logistic equation, Theor. Popul. Biol., Volume 120 (2018), pp. 11-15 | DOI | Zbl
[3] Diseases in metapopulations, Modeling and Dynamics of Infectious Diseases (Zhien Ma; Yicang Zhou; Jianhong Wu, eds.) (Series in Contemporary Applied Mathematics), Volume 11, World Scientific, 2009, pp. 64-122 | DOI
[4] Number of Source Patches Required for Population Persistence in a Source-Sink Metapopulation with Explicit Movement, Bull. Math. Biol., Volume 81 (2019), pp. 1916-1942 | DOI | Zbl
[5] A quantitative theory of organic growth, Hum. Biol., Volume 10 (1938) no. 2, pp. 181-213 | DOI
[6] Logistic Growth Rate Functions, J. Theor. Biol., Volume 21 (1968) no. 1, pp. 42-44 | DOI
[7] The effects of human movement on the persistence of vector-borne diseases, J. Theor. Biol., Volume 258 (2009), pp. 550-560 | DOI
[8] Stabilizing the Metzler matrices with applications to dynamical systems, Calcolo, Volume 57 (2020) no. 1 | DOI | Zbl
[9] Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theor. Ecol., Volume 9 (2016), pp. 443-453 | DOI
[10] Persistence and stability of seed-dispersel species in a patchy environment, Theor. Popul. Biol., Volume 16 (1979) no. 2, pp. 107-125 | DOI | Zbl
[11] Effects of dispersal in a non-uniform environment on population dynamics and competition: a patch model approach, Discrete Contin. Dyn. Syst., Volume 19 (2014) no. 10, pp. 3087-3104 | DOI | Zbl
[12] Effect of dispersal in Two-patch environment with Richards growth on population dynamics, J. Innov. Appl. Math. Comput. Sci., Volume 2 (2022) no. 3, pp. 41-68 | DOI
[13] Effects of rapid population growth on total biomass in Multi-patch environment (2022) (https://hal.science/hal-03698445)
[14] The multi-patch logistic equation, Discrete Contin. Dyn. Syst., Ser. B, Volume 26 (2021) no. 12, pp. 6405-6424 | DOI | Zbl
[15] The multi-patch logistic equation with asymmetric migration, Rev. Integr., Volume 40 (2022) no. 1, pp. 25-57 | DOI | Zbl
[16] Mathematical Models of Population Interactions with Dispersal II: Differential Survival in a Change of Habitat, J. Math. Anal. Appl., Volume 115 (1986), pp. 140-154 | DOI | Zbl
[17] Global stability and predator dynamics in a model of prey dispersal in a patchy environment, Nonlinear Anal., Theory Methods Appl., Volume 13 (1989) no. 8, pp. 993-1002 | DOI | Zbl
[18] Mathematical Models of Population Interactions with Dispersal I: Stabilty of two habitats with and without a predator, SIAM J. Appl. Math., Volume 32 (1977), pp. 631-648 | DOI | Zbl
[19] The Theory of Matrices, Volume 2, AMS Chelsea Publishing, 2000
[20] How does dispersal affect the infection size?, SIAM J. Appl. Math., Volume 80 (2020) no. 5, pp. 2144-2169 | DOI | Zbl
[21] Fast diffusion inhibits disease outbreaks, Proc. Am. Math. Soc., Volume 148 (2020) no. 4, pp. 1709-1722 | DOI | Zbl
[22] A multipatch malaria model with logistic growth, SIAM J. Appl. Math., Volume 72 (2012) no. 3, pp. 819-841 | DOI | Zbl
[23] Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., Volume 14 (2006), pp. 259-284 | Zbl
[24] Modelling Biological Systems: Principles and Applications, Chapman & Hall, 1996 | DOI
[25] Population dynamics in two patch environments: some anomalous consequences of an optimal habitat distribution, Theor. Popul. Biol., Volume 28 (1985) no. 2, pp. 181-201 | DOI | Zbl
[26] Dispersion and population interactions, Am. Natur., Volume 108 (1974) no. 960, pp. 207-228 | DOI
[27] Spatial patterning and the structure of ecological communities, Some Mathematical Questions in Biology VII (Lectures on Mathematics in the Life Sciences), Volume 8, American Mathematical Society, 1976 | Zbl
[28] On Tykhonov’s theorem for convergence of solutions of slow and fast systems, Electron. J. Differ. Equ., Volume 1998 (1998), 19, 22 pages | Zbl
[29] Elements of Mathematical Biology, Dover Publications, 1956
[30] Global asymptotic behavior in single-species discrete diffusion systems, J. Math. Biol., Volume 32 (1993), pp. 67-77 | DOI | Zbl
[31] Population Parameters: Estimation for Ecological Models, Blackwell Science, 2000 | DOI
[32] Computing closest stable nonnegative matrix, SIAM J. Matrix Anal. Appl., Volume 41 (2020) no. 1, pp. 1-28 | DOI | Zbl
[33] A Flexible Growth Function for Empirical Use, J. Exp. Bot., Volume 10 (1959) no. 29, pp. 290-300 | DOI
[34] The theory of the chemostat. Dynamics of microbial competition, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, 1995
[35] Cooperative systems theory and global stability of diffusion models, Acta Appl. Math., Volume 14 (1989) no. 1-2, pp. 49-57 | DOI
[36] Systems of differential equations containing small parameters in the derivatives, Mat. Sb., Volume 31 (1952) no. 3, pp. 575-586
[37] Analysis of Logistic Growth Models, Math. Biosci., Volume 179 (2002) no. 1, pp. 21-55 | DOI | Zbl
[38] A Theory of Growth, Math. Biosci., Volume 29 (1976), pp. 367-373 | DOI
[39] Notice sur la loi que la population suit dans son accroissement, Corr. Math. Physics, Volume 10 (1838) no. 113 | DOI
[40] Asymptotic Expansions for Ordinary Differential Equations, Robert E. Krieger Publishing Company, 1976
[41] Dispersal asymmetry in a two-patch system with source-sink populations, Theor. Popul. Biol., Volume 131 (2020), pp. 54-65 | DOI | Zbl
[42] Effects of dispersal on total biomass in a patchy, heterogeneous system: analysis and experiment, Math. Biosci., Volume 264 (2015), pp. 54-62 | DOI | Zbl
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