Geodesic
A hybrid model of population dynamics with refuge-regime: regularization and self-organization
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 3, pp. 467-482 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A mathematical model of the dynamics of the predator and prey populations in the form of a hybrid dynamical system consisting of two two-dimensional systems switching between each other is proposed. Switching of the systems allows us to reproduce a special refuge-regime when the prey number is very small and predators have complications to find preys. The sliding modes are studied using Filippov approach. Regularization of the system by using two switching lines to avoid chattering is provided. For the regularized model the limit sets are established. A scenario of the system self-organization preventing the unbounded populations' growth is proposed. A sensitivity study is carried out with respect to a parameter defining the switching lines. An important result of the research is that sufficiently small changing of the switching lines does not change the qualitative behavior of the system.
Keywords: hybrid systems, regularization, limit sets. 
@article{VUU_2023_33_3_a5,
     author = {A. N. Kirillov and A. M. Sazonov},
     title = {A hybrid model of population dynamics with refuge-regime: regularization and self-organization},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {467--482},
     year = {2023},
     volume = {33},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2023_33_3_a5/}
}
TY  - JOUR
AU  - A. N. Kirillov
AU  - A. M. Sazonov
TI  - A hybrid model of population dynamics with refuge-regime: regularization and self-organization
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2023
SP  - 467
EP  - 482
VL  - 33
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VUU_2023_33_3_a5/
LA  - ru
ID  - VUU_2023_33_3_a5
ER  - 
%0 Journal Article
%A A. N. Kirillov
%A A. M. Sazonov
%T A hybrid model of population dynamics with refuge-regime: regularization and self-organization
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2023
%P 467-482
%V 33
%N 3
%U http://geodesic.mathdoc.fr/item/VUU_2023_33_3_a5/
%G ru
%F VUU_2023_33_3_a5
A. N. Kirillov; A. M. Sazonov. A hybrid model of population dynamics with refuge-regime: regularization and self-organization. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 3, pp. 467-482. http://geodesic.mathdoc.fr/item/VUU_2023_33_3_a5/

[1] Deng Jiawei, Tang Sanyi, Lai Choi-Hong, “Non-smooth ecological systems with a switching threshold depending on the pest density and its rate of change”, Nonlinear Analysis: Hybrid Systems, 42 (2021), 101094 | DOI | MR | Zbl

[2] Zhang Yunhu, Xiao Yanni, “Global dynamics for a Filippov epidemic system with imperfect vaccination”, Nonlinear Analysis: Hybrid Systems, 38 (2020), 100932 | DOI | MR | Zbl

[3] Aihara Kazuyuki, Suzuki Hideyuki, “Theory of hybrid dynamical systems and its applications to biological and medical systems”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 4893–4914 | DOI | Zbl

[4] Perkins T.J., Wilds R., Glass L., “Robust dynamics in minimal hybrid models of genetic networks”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 4961–4975 | DOI | MR | Zbl

[5] Imura Jun-Ichi, Kashima Kenji, Kusano Masami, Ikeda Tsukasa, Morohoshi Tomohiro, “Piecewise affine systems approach to control of biological networks”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 4977–4993 | DOI | MR | Zbl

[6] Osborne J.M., Walter A., Kershaw S.K., Mirams G.R., Fletcher A.G., Pathmanathan P., Gavaghan D., Jensen O.E., Maini P.K., Byrne H.M., “A hybrid approach to multi-scale modelling of cancer”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 5013–5028 | DOI | MR | Zbl

[7] Tanaka Gouhei, Hirata Yoshito, Goldenberg S.L., Bruchovsky N., Aihara Kazuyuki, “Mathematical modelling of prostate cancer growth and its application to hormone therapy”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 5029–5044 | DOI | MR | Zbl

[8] Suzuki Taiji, Bruchovsky N., Aihara Kazuyuki, “Piecewise affine systems modelling for optimizing hormone therapy of prostate cancer”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 5045–5059 | DOI | MR | Zbl

[9] Izhikevich E.M., “Hybrid spiking models”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 5061–5070 | DOI | MR | Zbl

[10] Cao Hongjun, Ibarz Borja, “Hybrid discrete-time neural networks”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 5071–5086 | DOI | MR | Zbl

[11] Proctor J., Kukillaya R.P., Holmes P., “A phase-reduced neuro-mechanical model for insect locomotion: feed-forward stability and proprioceptive feedback”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368:1930 (2010), 5087–5104 | DOI | MR | Zbl

[12] Filippov A.F., Differential equations with discontinuous righthand sides, Kluwer Academic Publishers, Dordrecht, 1988 | DOI | MR | MR

[13] Chen Xiaoyan, Huang Lihong, “A Filippov system describing the effect of prey refuge use on a ratio-dependent predator–prey model”, Journal of Mathematical Analysis and Applications, 428:2 (2015), 817–837 | DOI | MR | Zbl

[14] Utkin V.I., Sliding modes and their applications to the systems with variable structure, Nauka, Moscow, 1974 | MR

[15] Acary V., Brogliato B., Orlov Yu.V., “Chattering-free digital sliding-mode control with state observer and disturbance rejection”, IEEE Transactions on Automatic Control, 57:5 (2012), 1087–1101 | DOI | MR | Zbl

[16] Rothe F., “The periods of the Volterra–Lotka system”, Journal für die reine und angewandte Mathematik (Crelles Journal), 1985:355 (1985), 129–138 | DOI | MR | Zbl

[17] Waldvogel J., “The period in the Lotka–Volterra system is monotonic”, Journal of Mathematical Analysis and Applications, 114:1 (1986), 178–184 | DOI | MR | Zbl

[18] Hausrath A.R., Manasevich R.F., “Periodic solutions of a periodically perturbed Lotka–Volterra equation using the Poincaré–Birkhoff theorem”, Journal of Mathematical Analysis and Applications, 157:1 (1991), 1–9 | DOI | MR | Zbl