Bifurkasi Periode Ganda dan Neimark-Sacker pada Model Diskret Leslie-Gower dengan Fungsi Respon Ratio-Dependent

Reza Mokodompit, Nurwan Nurwan, Emli Rahmi

Abstract


Dinamika model Leslie-Gower dengan fungsi respon ratio-dependent yang didiskretisasi menggunakan skema Euler maju adalah fokus utama pada artikel ini. Analisis diawali dengan mengidentifikasi eksistensi dari titik ekuilibrium dan kestabilan lokalnya. Diperoleh empat titik ekuilibrium yaitu titik kepunahan kedua populasi dan titik kepunahan predator yang selalu tidak stabil, dan titik kepunahan prey dan eksistensi kedua populasi yang stabil kondisional. Selanjutnya dipelajari eksistensi dari bifurkasi periode ganda dan Neimark-Sacker di sekitar titik eksistensi kedua populasi sebagai akibat perubahan parameter h (time-step). Dari hasil analisis ditemukan bahwa bifurkasi periode ganda terjadi setelah melewati h=h_a atau h=h_c dan bifurkasi Neimark-Sacker terjadi setelah melewati h=hb. Di akhir pembahasan, diberikan simulasi numerik yang mendukung hasil analisis sebelumnya.


Keywords


model predator-prey; Leslie-Gower; proses diskretisasi; bifurkasi

Full Text:

PDF

References


A. A. Berryman, "The Orgins and Evolution of Predator-Prey Theory," Ecology, vol. 73, no. 5, pp. 1530–1535, Oct. 1992, doi: 10.2307/1940005.

J. D. Murray, Mathematical Biology: An Introduction, 3rd ed. New York, NY: Springer New York, 2002.

P. H. Leslie, "Some Further Notes on the Use of Matrices in Population Mathematics," Biometrika, vol. 35, no. 3–4, pp. 213–245, 1948, doi: 10.1093/biomet/35.3-4.213.

M. A. Aziz-Alaoui and M. Daher Okiye, "Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes," Appl. Math. Lett., vol. 16, no. 7, pp. 1069–1075, 2003, doi: 10.1016/S0893-9659(03)90096-6.

C. S. Holling, "Some Characteristics of Simple Types of Predation and Parasitism," Can. Entomol., vol. 91, no. 07, pp. 385–398, 1959, doi: 10.4039/Ent91385-7.

R. Arditi and L. R. Ginzburg, "Coupling in predator-prey dynamics: Ratio-Dependence," J. Theor. Biol., vol. 139, no. 3, pp. 311–326, 1989, doi: 10.1016/S0022-5193(89)80211-5.

I. Hanski, "The functional response of predators: Worries about scale," Trends Ecol. Evol., vol. 6, no. 5, pp. 141–142, May 1991, doi: 10.1016/0169-5347(91)90052-Y.

M. J. Bishop, B. P. Kelaher, M. P. L. Smith, P. H. York, and D. J. Booth, "Ratio-dependent response of a temperate Australian estuarine system to sustained nitrogen loading," Oecologia, vol. 149, no. 4, pp. 701–708, Oct. 2006, doi: 10.1007/s00442-006-0481-5.

A. Suryanto, I. Darti, H. S. Panigoro, and A. Kilicman, "A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting," Mathematics, vol. 7, no. 11, p. 1100, Nov. 2019, doi: 10.3390/math7111100.

N. Hasan, R. Resmawan, and E. Rahmi, "Analisis Kestabilan Model Eko-Epidemiologi dengan Pemanenan Konstan pada Predator," J. Mat. Stat. dan Komputasi, vol. 16, no. 2, p. 121, Dec. 2019, doi: 10.20956/jmsk.v16i2.7317.

H. S. Panigoro, "Analisis Dinamik Sistem Predator-Prey Model Leslie-Gower dengan Pemanenan Secara Konstan terhadap Predator," EULER, vol. 2, no. 1, pp. 1–12, 2014.

P. D. Ernawati and I. Darti, "Stability analysis of the Euler discretization for the harvesting Leslie-Gower predator-prey model," Int. J. Pure Apllied Math., vol. 105, no. 2, pp. 213–221, Nov. 2015, doi: 10.12732/ijpam.v105i2.8.

H. S. Panigoro, E. Rahmi, N. Achmad, and S. L. Mahmud, "The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model," Jambura J. Math., vol. 2, no. 2, pp. 87–96, Mar. 2020, doi: 10.34312/jjom.v2i2.4566.

H. S. Panigoro, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, "Dynamics of a Fractional-Order Predator-Prey Model with Infectious Diseases in Prey," Commun. Biomath. Sci., vol. 2, no. 2, pp. 105–117, 2019, doi: 10.5614/cbms.2019.2.2.4.

M. Renisagayaraj, A. G. M. Selvam, and M. Meganathan, "Dynamics in a Discrete Prey-Predator System," Int. J. Eng. Res. Dev., vol. 6, no. 5, pp. 1–5, 2013.

M. Zhao, Z. Xuan, and C. Li, "Dynamics of a discrete-time predator-prey system," Adv. Differ. Equations, vol. 2016, no. 1, p. 191, Dec. 2016, doi: 10.1186/s13662-016-0903-6.

L. Edelstein-Keshet, "Mathematical models in biology," in Classics in applied mathematics, 2005.

H. S. Panigoro and E. Rahmi, "Modifikasi sistem predator-prey: dinamika model Leslie-Gower dengan daya dukung yang tumbuh logistik," in SEMIRATA MIPAnet, 2017, pp. 94–103.

S. N. Elaydi, Discrete Chaos: With Applications in Science and Engineering, 2nd ed., vol. 2007. San Antonio, Texas: Chapman & Hall/ CRC, 2007.

M. A. M. Abdelaziz, A. I. Ismail, F. A. Abdullah, and M. H. Mohd, "Bifurcations and chaos in a discrete SI epidemic model with fractional order," Adv. Differ. Equations, vol. 2018, no. 1, p. 44, Dec. 2018, doi: 10.1186/s13662-018-1481-6.

L. G. Yuan and Q. G. Yang, "Bifurcation, invariant curve and hybrid control in a discrete-time predator–prey system," Appl. Math. Model., vol. 39, no. 8, pp. 2345–2362, Apr. 2015, doi: 10.1016/j.apm.2014.10.040.




DOI: http://dx.doi.org/10.12962/limits.v17i1.6809

Refbacks

  • There are currently no refbacks.


Jumlah Kunjungan:

Creative Commons License
Limits: Journal Mathematics and its Aplications by Pusat Publikasi Ilmiah LPPM Institut Teknologi Sepuluh Nopember is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://iptek.its.ac.id/index.php/limits.