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Abstract
This study aims to analyze the dynamical behavior of a predator–prey model incorporating a Holling type II functional response and an anti-predator mechanism. The methods employed include dynamical systems analysis, namely equilibrium point determination, stability analysis using the Jacobian matrix, and bifurcation analysis. This analytical approach is supported by numerical simulations to construct phase portraits and illustrate system trajectories around equilibrium points. The results reveal various complex behaviors, particularly changes in stability and the emergence of bifurcation phenomena, which are influenced by variations in key parameters such as the saturation parameter, anti-predator parameter, and predator–prey interaction rate. Specifically, three types of bifurcations are identified: Hopf, transcritical, and saddle-node bifurcations. Hopf bifurcation leads to stable periodic oscillations, transcritical bifurcation results in an exchange of stability between equilibrium points, while saddle-node bifurcation causes the appearance or disappearance of equilibrium points. Numerical simulations further support these findings by illustrating system behavior before, at, and after bifurcation. Overall, the interaction between functional response saturation and anti-predator mechanisms plays a crucial role in determining the stability and dynamics of predator–prey populations.
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References
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References
F. R. Giordano, P. F. William, and S. B. Horton, A First Course in Mathematical Modeling Fifth edition. Richard Stratton, 2014. https://vdoc.pub/documents/a-first-course-in-mathematical-modeling-4pk4d8g4tpj0.
S. Maisaroh, R. Resmawan, and E. Rahmi, “Analisis kestabilan model predator-prey dengan infeksi penyakit pada prey dan pemanenan proporsional pada predator,” Jambura Journal of Biomathematics (JJBM), vol. 1, no. 1, pp. 8–15, 2020. https://doi.org/10.34312/jjbm.v1i1.5948.
N. S. Rahman, H. A. Fatahillah, and D. Aldila, “Analisis kemunculan solusi periodik dari model matematika penyebaran demam berdarah dengue dengan laju infeksi tidak standar,” Jurnal Matematika Integratif, vol. 21, no. 1, pp. 1–14, 2025. https://doi.org/10.24198/jmi.v21.n1.60353.1-14.
D. Maknun, EKOLOGI: POPULASI, KOMUNITAS, EKOSISTEM, Mewujudkan Kampus Hijau, Asri, Islami dan Ilmiah, vol. 1. Nujati Press, 2017.
H. Hazisyah, A. R. Putri, and S. Bahri, “Analisis kestabilan model prey - predator holling tipe iii,” Jurnal Matematika UNAND, vol. 10, no. 1, pp. 29–37, 2021. https://doi.org/10.25077/jmu.10.1.29-37.2021.
N. Nurrachmawati and A. Kusnanto, “Bifurkasi hopf pada model silkus bisnis kaldor-kalecki tanpa waktu tunda,” MILANG Journal of Mathematics and Its Applications, vol. 9, no. 2, pp. 13–18, 2010. https://doi.org/10.29244/jmap.9.2.13-18.
P. Blanchard, R. L. Devaney, and G. R. Hall, Differential Equations. Richard Stratton, 2012. https://studylib.net/doc/26282142/differential-equations--4th-edition.
L. Perko, Differential equations and dynamical systems, vol. 7. Springer, 2001. https://doi.org/10.1007/978-1-4613-0003-8.
W. E. Boyce and R. C. DiPrima, Elementary differential equations and boundary value problems. Laurie Rosatone, 2009. https://www.academia.edu/39005387/Elementary_Differential_Equations_9th.
H. Anton and C. Rorres, Elementary linear algebra: applications version. John Wiley & Sons, 2014. https://studylib.net/doc/26033624/037-elementary-linear-algebra-applications-version-howard.
M. Marwan and J. M. Tuwankotta, “Infinitely many equilibria and some codimension one bifurcations in a subsystem of a two-preys one-predator dynamical system,” Journal of Physics: Conference Series, vol. 1245, no. 1, pp. 1–8, 2019. https://doi.org/10.1088/1742-6596/1245/1/012063.