# Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior

• Joko Harianto Program Studi Matematika, Fakultas MIPA, Universitas Cendrawasih
• Titik Suparwati Program Studi Matematika, Fakultas MIPA, Universitas Cendrawasih
• Alfonsina Lisda Puspa Dewi Program Studi Matematika, Fakultas MIPA, Universitas Cendrawasih

### Abstract

This article describes the dynamics of local stability equilibrium point models of interaction between prey populations and their predators. The model involves response functions in the form of Holling type III and anti-predator behavior. The existence and stability of the equilibrium point of the model can be obtained by reviewing several cases. One of the factors that affect the existence and local stability of the model equilibrium point is the carrying capacity (k) parameter. If x3, y3  > 0 is a constant solution of the model and ∈ (0,x3), then there is a unique boundary equilibrium point Ek (k , 0). Whereas, if k ∈ (x4, y4], then E(k, 0) is unstable and E(x3, y3) is stable. Furthermore, if k ∈ ( x4, ∞), then E( k, 0) remains stable and E(x4, y4) is unstable, but the stability of the equilibrium point E(x3, y3) is branching. The equilibrium point E(x3, y3) can be stable or unstable depending on all parameters involved in the model. Variations of k parameter values are given in numerical simulation to verify the results of the analysis. Numerical simulation indicates that if k = 0,92 then nontrivial equilibrium point E(0,92 ; 0) stable. If k = 0,93 then Ek (0,93 ; 0) unstable and E3(0,929; 0,00003) stable. If k = 23,94, then Ek (23,94 ; 0) and E3(0,929; 0,143) stable, but E4(23,93 ; 0,0005) unstable. If k = 38 then Ek(38,0) stable, but E3(0,929; 0,145) and E4(23,93 ; 0,739) unstable.
Keywords: anti-predator behavior, carrying capacity, and holling type III.

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Published
2021-07-22
How to Cite
HARIANTO, Joko; SUPARWATI, Titik; DEWI, Alfonsina Lisda Puspa. Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior. Jurnal ILMU DASAR, [S.l.], v. 22, n. 2, p. 153-160, july 2021. ISSN 2442-5613. Available at: <https://jurnal.unej.ac.id/index.php/JID/article/view/23991>. Date accessed: 31 jan. 2023. doi: https://doi.org/10.19184/jid.v22i2.23991.
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