Dynamics of Infectious Diseases. Using Lotka-Volterra equations? PredatorPrey VS.
An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic...
-
Upload
julianna-veronica-newman -
Category
Documents
-
view
266 -
download
5
Transcript of An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic...
![Page 1: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/1.jpg)
An introduction to prey-predator Models
• Lotka-Volterra model• Lotka-Volterra model with prey logistic growth• Holling type II model
![Page 2: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/2.jpg)
• Generic Model
),()(
),()(
yxehygdtdy
yxhxfdtdx
• f(x) prey growth term• g(y) predator mortality term• h(x,y) predation term• e prey into predator biomass conversion coefficient
![Page 3: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/3.jpg)
• Lotka-Volterra Model
bxymydtdy
axyrxdtdx
• r prey growth rate : Malthus law• m predator mortality rate : natural mortality• Mass action law • a and b predation coefficients : b=ea• e prey into predator biomass conversion coefficient
![Page 4: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/4.jpg)
• Lotka-Volterra nullclines
![Page 5: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/5.jpg)
Direction field for Lotka-Volterra model
![Page 6: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/6.jpg)
Local stability analysis
• Jacobian at positive equilibrium
0
0*
*
*
by
axJ
• detJ*>0 and trJ*=0 (center)
![Page 7: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/7.jpg)
• Linear 2D systems (hyperbolic)
![Page 8: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/8.jpg)
Local stability analysis
• Proof of existence of center trajectories (linearization theorem)
• Existence of a first integral H(x,y) :
aybxyrxmyxH )ln()ln(),(
![Page 9: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/9.jpg)
Lotka-Volterra model
![Page 10: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/10.jpg)
Lotka-Volterra model
![Page 11: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/11.jpg)
Hare-Lynx data (Canada)
![Page 12: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/12.jpg)
• Logistic growth (sheep in Australia)
![Page 13: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/13.jpg)
• Lotka-Volterra Model with prey logistic growth
bxymydtdy
axyKx
rxdtdx
1
![Page 14: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/14.jpg)
Nullclines for the Lotka-Volterra model with prey logistic growth
![Page 15: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/15.jpg)
• Lotka-Volterra Model with prey logistic growth
bxymydtdy
axyKx
rxdtdx
1
• Equilibrium points : (0,0) (K,0) (x*,y*)
![Page 16: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/16.jpg)
Local stability analysis
• Jacobian at positive equilibrium
0*
**
*
by
axKrx
J
• detJ*>0 and trJ*<0 (stable)
![Page 17: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/17.jpg)
• Condition for local asymptotic stability
![Page 18: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/18.jpg)
Lotka-Volterra model with prey logistic growth : coexistence
![Page 19: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/19.jpg)
Lotka-Volterra with prey logistic growth : predator extinction
![Page 20: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/20.jpg)
• Transcritical bifurcation
*xK
*xK (K,0) stable and (x*,y*) unstable and negative
(K,0) and (x*,y*) same
*xK (K,0) unstable and (x*,y*) stable and positive
![Page 21: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/21.jpg)
• Loss of periodic solutions
bxymydtdy
axyKx
rxdtdx
1
x-y
0 0,3 0,6 0,9 1,2 1,5
x
0
1,6
3,2
4,8
6,4
8
y
x-y
0 0,3 0,6 0,9 1,2 1,5
x
0
4
8
12
16
20
y
coexistence Predator extinction
![Page 22: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/22.jpg)
Functional response I and II
![Page 23: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/23.jpg)
• Holling Model
xDbxy
mydtdy
xDaxy
Kx
rxdtdx
1
![Page 24: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/24.jpg)
• Existence of limit cycle (Supercritical Hopf bifurcation)
22
22
yxyxdtdy
yxxydtdx
• Polar coordinates
1
2
dtd
rrdtdr
![Page 25: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/25.jpg)
• Stable equilibrium
![Page 26: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/26.jpg)
• At bifurcation
![Page 27: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/27.jpg)
• Existence of a limit cycle
![Page 28: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/28.jpg)
• Supercritical Hopf bifurcation
![Page 29: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/29.jpg)
Poincaré-Bendixson Theorem
A bounded semi-orbit in the plane tends to :• a stable equilibrium• a limit cycle• a cycle graph
![Page 30: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/30.jpg)
Trapping region
![Page 31: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/31.jpg)
Trapping region : Annulus
![Page 32: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/32.jpg)
Example of a trapping region
xdtdy
xxy
dtdx
3
3
• Van der Pol model (>0)
![Page 33: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/33.jpg)
• Holling Model
xDbxy
mydtdy
xDaxy
Kx
rxdtdx
1
![Page 34: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/34.jpg)
Nullclines for Holling model
![Page 35: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/35.jpg)
Poincaré box for Holling model
![Page 36: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/36.jpg)
Holling model with limit cycle
![Page 37: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/37.jpg)
Paradox of enrichment
When K increases :
• Predator extinction• Prey-predator coexistence (TC)• Prey-predator equilibrium becomes unstable (Hopf)• Occurrence of a stable limit cycle (large variations)
![Page 38: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/38.jpg)
Other prey-predator models• Functional responses (Type III, ratio-dependent …)• Prey-predator-super-predator…• Trophic levels
![Page 39: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/39.jpg)
Routh-Hurwitz stability conditions
0... 1
1
2
2
1
1
n
n
n
nnn aaaa
00)(, ik HRk
11 aH
• Characteristic equations
• Stability conditions : M* l.a.s.
2
31
2 1 a
aaH
31
42
531
3
0
1
aa
aa
aaa
H
![Page 40: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/40.jpg)
Routh-Hurwitz stability conditions
032
2
1
3 aaa
011 trAaH
• Dimension 2
• Dimension 3
0det2 AtrA
0det3212 AaaaH
011 aH
03212 aaaH
033 aH
![Page 41: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/41.jpg)
3-trophic example
dyzzzdtdz
cyzbxymydtdy
axyrxdtdx
)1(
![Page 42: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/42.jpg)
• Interspecific competition Model
2
121
2
222
2
1
212
1
111
1
1
1
Kx
aKx
xrdtdx
Kx
aKx
xrdtdx
• Transformed system
buvvddv
avuuddu
1
1
![Page 43: An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model.](https://reader033.fdocuments.in/reader033/viewer/2022061523/56649cd65503460f9499cf05/html5/thumbnails/43.jpg)
Competition model