Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior
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 m...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Fakultas MIPA Universitas Jember
2021-07-01
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| Series: | Jurnal Ilmu Dasar |
| Online Access: | https://jurnal.unej.ac.id/index.php/JID/article/view/23991 |
| _version_ | 1818924289009647616 |
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| author | Joko Harianto Titik Suparwati Alfonsina Lisda Puspa Dewi |
| author_facet | Joko Harianto Titik Suparwati Alfonsina Lisda Puspa Dewi |
| author_sort | Joko Harianto |
| collection | DOAJ |
| description | 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 Ek (k, 0) is unstable and E3 (x3∗, y3∗) is stable. Furthermore, if k ∈ ( x4∗, ∞), then Ek ( k, 0) remains stable and E4 (x4∗, y4∗) is unstable, but the stability of the equilibrium point E3 (x3∗, y3∗) is branching. The equilibrium point E3 (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 Ek (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. |
| first_indexed | 2024-12-20T02:22:57Z |
| format | Article |
| id | doaj.art-1e89534895444290b240c568033b7f42 |
| institution | Directory Open Access Journal |
| issn | 1411-5735 2442-5613 |
| language | English |
| last_indexed | 2024-12-20T02:22:57Z |
| publishDate | 2021-07-01 |
| publisher | Fakultas MIPA Universitas Jember |
| record_format | Article |
| series | Jurnal Ilmu Dasar |
| spelling | doaj.art-1e89534895444290b240c568033b7f422022-12-21T19:56:46ZengFakultas MIPA Universitas JemberJurnal Ilmu Dasar1411-57352442-56132021-07-0122215316010.19184/jid.v22i2.2399123991Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator BehaviorJoko Harianto0Titik Suparwati1Alfonsina Lisda Puspa Dewi2Program Studi Matematika, Fakultas MIPA, Universitas CendrawasihProgram Studi Matematika, Fakultas MIPA, Universitas CendrawasihProgram Studi Matematika, Fakultas MIPA, Universitas CendrawasihThis 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 Ek (k, 0) is unstable and E3 (x3∗, y3∗) is stable. Furthermore, if k ∈ ( x4∗, ∞), then Ek ( k, 0) remains stable and E4 (x4∗, y4∗) is unstable, but the stability of the equilibrium point E3 (x3∗, y3∗) is branching. The equilibrium point E3 (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 Ek (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.https://jurnal.unej.ac.id/index.php/JID/article/view/23991 |
| spellingShingle | Joko Harianto Titik Suparwati Alfonsina Lisda Puspa Dewi Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior Jurnal Ilmu Dasar |
| title | Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior |
| title_full | Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior |
| title_fullStr | Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior |
| title_full_unstemmed | Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior |
| title_short | Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior |
| title_sort | local stability dynamics of equilibrium points in predator prey models with anti predator behavior |
| url | https://jurnal.unej.ac.id/index.php/JID/article/view/23991 |
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