Sistem Dinamika Dasar Model Lotka Volterra ( Mangsa Pemangsa)
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Transcript of Sistem Dinamika Dasar Model Lotka Volterra ( Mangsa Pemangsa)
![Page 1: Sistem Dinamika Dasar Model Lotka Volterra ( Mangsa Pemangsa)](https://reader031.fdocuments.net/reader031/viewer/2022013108/55adf8691a28aba55f8b464b/html5/thumbnails/1.jpg)
ANALISIS KESTABILAN MODEL LOTKA-
VOLTERRA TIPE MANGSA-PEMANGSA
Ervina Marviana (G54100015)
Vivianisa Wahyuni (G54100035)
Lola Oktasari (G54100054)
Novia Yuliani (G54100075)
Bilyan Ustazila (G54100101)
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OUTLINE
LATAR BELAKANG
MODELANALISIS
KESTABILANSIMULASI
KESIMPULAN
DAFTAR PUSTAKA
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LATAR BELAKANG
POPULASI DAN INTERAKSI
PREDASI Alfred Lotka (1925) dan Volterra Vito (1927)
ANALISIS KESTABILAN DAN
KEBIJAKAN
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MODEL
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ANALISIS KESTABILAN
Titik Tetap
Kestabilan
Bifurkasi
Simulasi
LANGKAH KERJA :
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ANALISIS KESTABILAN (CONT’D....)
Titik
Tetap
T1(0,0)
T2(1,0)
T3(x*,y*)
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ANALISIS KESTABILAN (CONT’D....)
Jacobi :
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ANALISIS KESTABILAN (CONT’D....)
Nilai eigen :
T1(0,0) :
T2(1,0) :
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ANALISIS KESTABILAN (CONT’D....)
Titik
Tetap
T1(0,0)
T2(1,0)
T3(x*,y*)
1. Jika r > 3α, maka T(x*,y*) titik simpul stabil
2. Jika r < 3α, maka T(x*,y*) titik spiral stabil
3. Jika r = 3α, maka T(x*,y*) degenerate node
Titik Saddle
Titik tetap tak terisolasi
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BIFURKASI
Karena, semua parameter berniali positif, maka kedua nilai eigen di
atas tidak mungkin berbentuk imaginer murni. Sehingga, tidak
terdapat bifurkasi Hopf.
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SIMULASI
Kondisi 1 : r > 3a, dimana r = 6, a = 1
Jacobi T1(0,0) :
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SIMULASI
Kondisi 1 : r > 3a, dimana r = 6, a = 1
Jacobi T1(1,0) :
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SIMULASI
Kondisi 1 : r > 3a, dimana r = 6, a = 1
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SIMULASI (CONT’D....)
Kondisi 1 : r > 3a, dimana r = 6, a = 1
Titik
Tetap
T1(0,0)
T2(1,0)
Titik Simpul Stabil
Titik Saddle
Titik tetap tak terisolasi
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SIMULASI (CONT’D....)
Plot Bidang Fase
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Bidang Solusi
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SIMULASI (CONT’D....)
Kondisi 2 : r < 3a, dimana r = 2 , a = 2
Jacobi T1(0,0) :
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SIMULASI (CONT’D....)
Kondisi 2 : r < 3a, dimana r = 2 , a = 2
Jacobi T2(1,0) :
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SIMULASI (CONT’D....)
Kondisi 2 : r < 3a, dimana r = 2 , a = 2
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SIMULASI (CONT’D....)
Kondisi 2 : r < 3a, dimana r = 2 , a = 2
Titik
Tetap
T1(0,0)
T2(1,0)
Titik Spiral Stabil
Titik Saddle
Titik tetap tak terisolasi
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SIMULASI (CONT’D....)
Plot Bidang Fase
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Bidang Solusi
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SIMULASI (CONT’D....)
Kondisi 3 : r = 3a, dimana r = 6 , a = 2
Jacobi T1(0,0) :
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SIMULASI (CONT’D....)
Kondisi 3 : r = 3a, dimana r = 6 , a = 2
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SIMULASI (CONT’D....)
Kondisi 3 : r = 3a, dimana r = 6 , a = 2
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SIMULASI (CONT’D....)
Kondisi 3 : r = 3a, dimana r = 6 , a = 2
Titik
Tetap
T1(0,0)
T2(1,0)
Titik Degenerate
Titik Saddle
Titik tetap tak terisolasi
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SIMULASI (CONT’D....)
Plot Bidang Fase
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Bidang Solusi
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KESIMPULAN
• Jika r > 3a, maka T(x*,y*) titik simpul stabil
• Jika r < 3a, maka T(x*,y*) titik spiral stabil
• Jika r = 3a, maka T(x*,y*) degenerate node
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KESIMPULAN (CONT’D....)T1 T2 T3
r > 3 a Titik tetap tak terisolasi Titik sadel Titik simpul stabil
r < 3 a Titik tetap tak terisolasi Titik sadel Titik spiral stabil
r = 3 a Titik tetap tak terisolasi Titik sadel Degenerate node
• Kondisi titik tetap T3 akan stabil jika proporsi laju kelahiran dari
populasi mangsa lebih besar dari laju kelah iran dari populasi
pemangsa
Dari ketiga kondisi, titik spiral stabil kemudian titik degenerate node dan akan menuju titik stabil ( jumlah kelahiran populasi mangsapemangsa stabil ) seperti yang direpresentasikan oleh grafik bidang solusi.
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DAFTAR PUSTAKA
Merdan, Huseyin.2010. “Stability Analysis of A Lotka-Volterra
Type Predator-Prey System Involving Allee Effects”,
Journal of ANZIAM J. 52. 139-145
Strogatz SH. 1994. Nonlinear Dynamics and Chaos, With
Application to Physics, Biology, Chemistry, and
Engineering. Addison-Wesley Publishing Company,
Reading, Massachusetts.
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TERIMAKASIH