Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory

Mirela Garić-Demirović; Dragana Kovačević; Mehmed Nurkanović; Mirela Garić-Demirović; Dragana Kovačević; Mehmed Nurkanović

doi:10.3934/math.2024753

AIMS Mathematics

2024, Volume 9, Issue 6: 15584-15609. doi: 10.3934/math.2024753

Previous Article Next Article

Research article Special Issues

Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory

1.
Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina
2.
Department of Computer Science, Sarajevo School of Science and Technology, 71210 Sarajevo, Bosnia and Herzegovina

Received: 19 February 2024 Revised: 16 April 2024 Accepted: 18 April 2024 Published: 29 April 2024
MSC : 39A30, 39A60, 37G05, 92D25, 65P20

We use the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions of a system of difference equations, a certain class of a generalized May's host-parasitoid model. We show the existence of the extinction, interior, and boundary equilibrium points and examine their stability. When the rate of increase of hosts is less than one, the zero equilibrium is globally asymptotically stable, which means that both populations are extinct. We thoroughly describe the dynamics of 1:1 non-isolated resonance fixed points and have used the KAM theory to determine the stability of interior equilibrium point. Also, we have conducted several numerical simulations to support our findings by using the software package Mathematica.
- difference equations,
- fixed point,
- area-preserving map,
- KAM theory
Citation: Mirela Garić-Demirović, Dragana Kovačević, Mehmed Nurkanović. Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory[J]. AIMS Mathematics, 2024, 9(6): 15584-15609. doi: 10.3934/math.2024753

Related Papers:

Abstract

We use the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions of a system of difference equations, a certain class of a generalized May's host-parasitoid model. We show the existence of the extinction, interior, and boundary equilibrium points and examine their stability. When the rate of increase of hosts is less than one, the zero equilibrium is globally asymptotically stable, which means that both populations are extinct. We thoroughly describe the dynamics of 1:1 non-isolated resonance fixed points and have used the KAM theory to determine the stability of interior equilibrium point. Also, we have conducted several numerical simulations to support our findings by using the software package Mathematica.

References

[1]	M. Tabor, Chaos and integrability in nonlinear dynamics. An Introduction, Wiley, 1989.
[2]	A. J. Nicholson, V. A. Bailey, The balance of animal populations: Part Ⅰ, Proc. Zool. Soc. Lond., 105 (1935), 551–598. https://doi.org/10.1111/j.1096-3642.1935.tb01680.x doi: 10.1111/j.1096-3642.1935.tb01680.x
[3]	R. M. May, Host-parasitoid systems in patchy environments: A phenomenological model, J. Anim. Ecol., 47 (1978), 833–843.
[4]	S. Kalabušić, E. Pilav, Stability of May's Host-Parasitoid model with variable stocking upon parasitoids, Int. J. Biomath., 15 (2021), 2150072. https://doi.org/10.1142/S1793524521500728 doi: 10.1142/S1793524521500728
[5]	G. Ladas, G. Tzanetopoulos, A. Tovbis, On May's host parasitoid model, J. Differ. Equ. Appl., 2 (1996), 195–204. https://doi.org/10.1080/10236199608808054 doi: 10.1080/10236199608808054
[6]	W. T. Jamieson, On the global behaviour of May's host-parasitoid model, J. Differ. Equ. Appl., 25 (2019), 583–596. https://doi.org/10.1080/10236198.2019.1613387 doi: 10.1080/10236198.2019.1613387
[7]	S. Jašarević Hrustić, Z. Nurkanović, M. R. S. Kulenović, E. Pilav, Birkhoff normal forms, KAM Theory and symmetries for certain second order rational difference equation with quadratic term, Int. J. Differ. Equ., 10 (2015), 181–199. Available from: https://campus.mst.edu/ijde/contents/v10n2p4.pdf.
[8]	M. Garić-Demirović, M. Nurkanović, Z. Nurkanović, Stability, periodicity and symmetries of certain second-order fractional difference equation with quadratic terms via KAM theory, Math. Meth. Appl. Sci., 40 (2017), 306–318. https://doi.org/10.1002/mma.4000 doi: 10.1002/mma.4000
[9]	M. R. S. Kulenović, Z. Nurkanović, E. Pilav, Birkhoff normal forms and KAM theory for Gumowski-Mira equation, The Scientific World J., 2014 (2014), 819290. https://doi.org/10.1155/2014/819290 doi: 10.1155/2014/819290
[10]	M. Nurkanović, Z. Nurkanović, Birkhoff normal forms, KAM theory, periodicity and symmetries for certain rational difference equation with cubic terms, Sarajevo J. Math., 12 (2016), 217–231. https://doi.org/10.5644/SJM.12.2.08 doi: 10.5644/SJM.12.2.08
[11]	S. R. J. Jang, Discrete-time host-parasitoid models with Allee effects: Density dependence versus parastism, J. Differ. Equ. Appl., 2 (1996), 195–204.
[12]	L. G. Ginzburg, D. E. Tanehill, Population cycles of forest Lepidoptera: A maternal effect hypothesis, J. Anim. Ecol., 63 (1994), 79–92.
[13]	M. Gidea, J. D. Meiss, I. Ugarcovici, H. Weiss, Applications of KAM theory to population dynamics, J. Biol. Dyn., 5 (2011), 44–63. https://doi.org/10.1080/17513758.2010.488301 doi: 10.1080/17513758.2010.488301
[14]	J. K. Hale, H. Kocak, Dynamics and bifurcation, Springer, New York, 1991.
[15]	J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött., 2 (1962), 1–20. https://doi.org/10.1080/17513758.2010.488301 doi: 10.1080/17513758.2010.488301
[16]	C. L. Siegel, J. K. Moser, Lectures on celestial mechanics, Springer, New York, 1971.
[17]	W. T. Jamieson, O. Merino, Local dynamics of planar maps with a non-isolated fixed point exhibiting 1–1 resonance, Adv. Differ. Equ., 2018 (2018). https://doi.org/10.1186/s13662-018-1595-x doi: 10.1186/s13662-018-1595-x
[18]	M. R. S. Kulenović, O. Merino, Discrete dynamical systems and difference equations with mathematica, Chapman & HALL/CRC, Boca Raton-New York, 2000.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)