An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic...
-
Upload
julianna-veronica-newman -
Category
Documents
-
view
285 -
download
5
Transcript of An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic...
An introduction to prey-predator Models
• Lotka-Volterra model• Lotka-Volterra model with prey logistic growth• Holling type II model
• 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
• 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
• Lotka-Volterra nullclines
Direction field for Lotka-Volterra model
Local stability analysis
• Jacobian at positive equilibrium
0
0*
*
*
by
axJ
• detJ*>0 and trJ*=0 (center)
• Linear 2D systems (hyperbolic)
Local stability analysis
• Proof of existence of center trajectories (linearization theorem)
• Existence of a first integral H(x,y) :
aybxyrxmyxH )ln()ln(),(
Lotka-Volterra model
Lotka-Volterra model
Hare-Lynx data (Canada)
• Logistic growth (sheep in Australia)
• Lotka-Volterra Model with prey logistic growth
bxymydtdy
axyKx
rxdtdx
1
Nullclines for the Lotka-Volterra model with prey logistic growth
• Lotka-Volterra Model with prey logistic growth
bxymydtdy
axyKx
rxdtdx
1
• Equilibrium points : (0,0) (K,0) (x*,y*)
Local stability analysis
• Jacobian at positive equilibrium
0*
**
*
by
axKrx
J
• detJ*>0 and trJ*<0 (stable)
• Condition for local asymptotic stability
Lotka-Volterra model with prey logistic growth : coexistence
Lotka-Volterra with prey logistic growth : predator extinction
• 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
• 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
Functional response I and II
• Holling Model
xDbxy
mydtdy
xDaxy
Kx
rxdtdx
1
• Existence of limit cycle (Supercritical Hopf bifurcation)
22
22
yxyxdtdy
yxxydtdx
• Polar coordinates
1
2
dtd
rrdtdr
• Stable equilibrium
• At bifurcation
• Existence of a limit cycle
• Supercritical Hopf bifurcation
Poincaré-Bendixson Theorem
A bounded semi-orbit in the plane tends to :• a stable equilibrium• a limit cycle• a cycle graph
Trapping region
Trapping region : Annulus
Example of a trapping region
xdtdy
xxy
dtdx
3
3
• Van der Pol model (>0)
• Holling Model
xDbxy
mydtdy
xDaxy
Kx
rxdtdx
1
Nullclines for Holling model
Poincaré box for Holling model
Holling model with limit cycle
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)
Other prey-predator models• Functional responses (Type III, ratio-dependent …)• Prey-predator-super-predator…• Trophic levels
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
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
3-trophic example
dyzzzdtdz
cyzbxymydtdy
axyrxdtdx
)1(
• 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
Competition model