Effect of dispersal in two-patch environment with Richards growth on population dynamics
Article Sidebar
Main Article Content
Abstract
In this paper, we consider a two-patch model coupled by migration terms, where each patch follows a Richards law. First, we prove the global stability of the model. Second, in the case when the migration rate tends to infinity, the total carrying capacity is given, which in general is different from the sum of the two carrying capacities and depends on the parameters of the growth rate and also on the migration terms. Using the theory of singular perturbations, we give an approximation of the solutions of the system in this case. Finally, 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 two carrying capacities and we give a complete classification for all possible cases. The total equilibrium population formula for a large migration rate plays an important role in this classification. We show that this choice of local dynamics has an influence on the effect of dispersal. Comparing the dynamics of the total equilibrium population as a function of the migration rate with that of the logistic model, we obtain the same behavior. In particular, we have only three situations that the total equilibrium population can occur: it is always greater than the sum of two carrying capacities, always smaller, and a third case, where the effect of dispersal is beneficial for lower values of the migration rate and detrimental for the higher values. We end by examining the two-patch model where one growth rate is much larger than the second one, we compare the total equilibrium population with the sum of the two carrying capacities.
Downloads
Metrics
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
- Authors keep the rights and guarantee the Journal of Innovative Applied Mathematics and Computational Sciences the right to be the first publication of the document, licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License that allows others to share the work with an acknowledgement of authorship and publication in the journal.
- Authors are allowed and encouraged to spread their work through electronic means using personal or institutional websites (institutional open archives, personal websites or professional and academic networks profiles) once the text has been published.
References
R. Arditi, C. Lobry and T. Sari, In dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation, Theor. Popul. Biol., 106 (2015), 45-59.
R. Arditi, C. Lobry and T.Sari, Asymmetric dispersal in the multi-patch logistic equation, Theor. Popul. Biol., 1206 (2018), 11-15.
L. Von. Bertalanffyi, A quantitative theory of organic growth, Human Biology., 10(2) (1938), 181-213.
A. A. Blumberg, Logistic Growth rate functions, J. Theor. Biol., 21 (1968), 42-443.
L. Chen, T. Liu and F. Chen, Stability and bifurcation in a two-patch model with additive Allee effect, AIMS Math., 7(1) (2021), 536-551.
D. L. DeAngelis, C. C. Travis and W. M. Post, Persistence and stability of seed-dispersel species in a patchy environment. Theor. Popul. Biol., 16 (1979), 107-125.
B. Elbetch, T. Benzekri, D. Massart and T. Sari, The multi-patch logistic equation. Discrete Contin. Dyn. Syst. - B., 26 (12) (2020), 6405-6424.
B. Elbetch, T. Benzekri, D. Massart and T. Sari, The multi-patch logistic equation with asymmetric migration, Rev. Integr. Temas Mat., 40(1) (2022), 25-57.
H. I. Freedman and P. Waltman, Mathematical Models of Population Interactions with Dispersal I: stability of two habitats with and without a predator. SIAM J. Appl Math., 32 (1977), 631-648.
H. I. Freedman, B. Rai and P. Waltman, Mathematical Models of Population Interactions with Dispersal II: Differential Survival in a Change of Habitat. J. Math. Anal. Appl., 115 (1986), 140-154.
D. Gao and Y. Lou, Total biomass of a single population in two-patch environments. Theor. Popul. Biol., (146) (2022), 1-14.
J. W. Haefner, Modelling Biological Systems: Principles and Applications, ITP Chapman and Hall, New York, 1996.
I. A. Hanski and M. E. Gilpin, Metapopulation Biology: Ecology, Genetics, and Evolution. Academic Press, 1997.
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, 1998.
R. D. Holt, Population dynamics in two patch environments: some anomalous consequences of an optimal habitat distribution. Theor. Popul. Biol., 28 (1985), 181-201.
Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects, J. Math. Biol., 62 (2011), 925-973.
C. Lobry, T. Sari and S. Touhami, On Tykhonov’s theorem for convergence of solutions of slow and fast systems. Electron. J. Differ. Equ., 19(1998), 1-22.
S. A. Levin, Dispersion and population interactions. Amer. Natur., 108 (1974), 207-228.
S. A. Levin, Spatial patterning and the structure of ecological communities, in Some Mathematical Questions in Biology VII. American Mathematical Society, 1976.
S. A. Levin, T. M. Powell and J. H. Steele, Patch Dynamics, volume 96 of Lecture Notes in Biomathematics. Springer-Verlag, 1993.
A. J. Lotka, Elements of Mathematical Biology. DOVER, New York, 1956.
Y. Y. Lv, L. J. Chen and F. D. Chen, Stability and bifurcation in a single species logistic model with additive Allee effect and feedback control, Adv. Differ. Equ., 8 (2020), 2686-2697.
Y. Y. Lv, L. J. Chen, F. D. Chen and Z. Li, Stability and bifurcation in an SI epidemic model with additive Allee effect and time delay, Int. J. Bifurcat. Chaos, 31(2021), 2150060.
H. McCallum, Population Parameters: Estimation for Ecological Models, Blackwell Science, UK 2008.
D. Pal and G. P. Samanta, Effects of dispersal speed and strong Allee effect on stability of a two-patch predator-prey model, Int. J. Dyn. Control, 6 (2018), 1484-1495.
J.-C. Poggiale, P. Auger, D. Nérini, C. Manté and F. Gilbert, Global production increased spatial heterogeneity in a population dynamics model. Acta Biotheor., 53 (2005), 359-370.
F. J. Richards, A Flexible Growth Function for Empirical Use, J. Exp. Bot., 10(29) (1959), 290-300.
S. Saha and G. P. Samanta, Influence of dispersal and strong Allee effect on a two-patch predator-prey model, Int. J. Dyn. Control, 7 (2009), 1321-1349.
H. L. Smith and P. Waltman, The Theory of the Chemostat : Dynamics of Microbial Competition, ambridge Studies in Mathematical Biology, 13 (1995).
A. N. Tikhonov, Systems of differential equations containing small parameters in the derivatives. at. Sb. (N.S.), 31(73):3 (1952), 575-586.
A. Tsoularis and J. Wallace, Analysis of Logistic Growth Models. Math. Biosci., 179 (2002), 21-55.
M. E. Turner, E. Bradley , K. Kirk and K. Pruitt, A Theory of Growth. Math. Biosci., 29 (1976), 367-373.
P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement. Corr. Math. Physics, 10(1838), 113.
W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations. Robert E. Krieger Publishing Company, Huntington, New York, 1976.
H. Wu, Y. Wang, Y. Li and D. DeAngelis, Dispersal asymmetry in a two-patch system with source-sink populations. Theor. Popul. Biol., 131 (2020), 54-65