pptklmpok9-131230090302-phpapp01.ppt
TRANSCRIPT
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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
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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
LANGKAH KERJA :
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ANALISIS KESTABILAN (CONT’D....)
TitikTetap
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....)
TitikTetap
T1(0,0)
T2(1,0)
T3(x*,y*)1. Jika r > 3α, maka T(x*,y*) titik simpul stabil2. Jika r < 3α, maka T(x*,y*) titik spiral stabil3. Jika r = 3α, maka T(x*,y*) degenerate node
1. Jika r > 3α, maka T(x*,y*) titik simpul stabil2. Jika r < 3α, maka T(x*,y*) titik spiral stabil3. Jika r = 3α, maka T(x*,y*) degenerate node
Titik SaddleTitik Saddle
Titik tetap tak terisolasiTitik 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
TitikTetap
T1(0,0)
T2(1,0)
Titik Simpul StabilTitik Simpul Stabil
Titik SaddleTitik Saddle
Titik tetap tak terisolasiTitik 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
TitikTetap
T1(0,0)
T2(1,0)
Titik Spiral StabilTitik Spiral Stabil
Titik SaddleTitik Saddle
Titik tetap tak terisolasiTitik 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
TitikTetap
T1(0,0)
T2(1,0)
Titik DegenerateTitik Degenerate
Titik SaddleTitik Saddle
Titik tetap tak terisolasiTitik 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....)
• 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 mangsa pemangsa 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