The Evolution of Stability in a Stochastic Predator-Prey System
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
n
n + 1
n
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
-
5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
http:///reader/full/the-evolution-of-stability-in-a-stochastic-predator-prey-sys
G
N
G = N /N
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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N=fN(N, P)
P =fP(N, P)
N
P
100
2
1
2
fN fP
fN fP
fN =N[r(C) kN CP (C, N)]
fP =P[BCN(C, N) d].
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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C
C >0
r(C) =R + qC
R,q >0
(C, N) = 1/(1+hCN)
h
CN (C, N)
CP (C, N)
B
r(C) N
d P
N P
N =N
R+ qC kN
CP
1 + hCN
P =P
BC N
1 + hCN d
,
(C, N) = 1
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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C
C
d
h
q
B
k
R
C C
d d
h h
q q
C =B
kC, d =
1
Rd, h =
R
Bh,
q =
k
BRq
N
P
N =N R/k
P = PBR/k
t = t/R
N=N
1 + qC N
CP
1 + hCN
P =P CN
1 + hCN d ,
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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G
C
G(C, C) =N(C)
N(C)= 1 + qC N(C)
CP(C)
1 + hCN(C).
C
C
G(C, C) = 1 + qC N(C) CP(C)
1 + hCN(C).
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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G(C, C)
G(C, C)
C = C
C > C
C < C
G(C=C, C)
G(C, C)
C
C
G/C
g
g(C) =d C(1 dh)
C2(1 dh) .
C
g(C)
x
C = d
1 dh,
G(C, C)
C
2G
C2
C=C
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dg
dC
C=C
=
1 dh
d
2,
C
C
C
C
C
C
C
C < C
C =C
C > C
g(C) > 0
C
C
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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0
The invasion fitness evaluated at a specific C
Vulnerability, C
G
(C,
C
)
C
G
C < C
g(C) > 0
0
The invasion fitness evaluated at an ESS
Vulnerability, C
G(C,
C
)
ESS
C
ESS
G C =C g(C) = 0
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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G
g
G= 1 + qC N( )
CP( )
1 + hCN( )
g= q P(
)
1 + hCN( )
.
N( ) P( )
P(t)
1/(1 +hCN(t))
C
C
=
P( )
1 + hCN( )
.
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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(Ni, P
i)
(N1 , P1 ) =
d
C(1 dh),1 + hCN
1
C (1 + qC N1 )
,
(N2 , P2 ) = (1 + qC, 0)
(N3 , P3 ) = (0, 0).
dh < 1
1 + qC > N1
d
d < C(1 + qC)1 + hC+ hqC2 ,
N
P
(N, P)
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5/19/2018 The Evolution of Stability in a Stochastic Predator-Prey System
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(N, P) =
1 + qC 2N
CP
(1 + hCN)2
CN
1 + hCNCP
(1 + hCN)2CN
1 + hCN d
.
(N, P)
J22
C
Ci
Ci = 1
2q1 +
4q
h
1 + dh
1 dh 1 .
C Ci (N, P)
1 2
C
Cd
Ch
Cq
(N, P)
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0 5 10 15 2010
5
0
5
10
Eigenvalues of the Jacobian as a function of the prey vulnerability
Vulnerability, C
Re{1}Im{1}
Re{2}Im{2}
C
1.2 1.4 1.6 1.8 2
2
1.5
1
0.5
0
0.5
1
genva ues o t e aco an as a unct on o t e prey vu nera ty
Vulnerability, C
Re{1}Im{1}Re{2}Im{2}
C
{2}
1.05
C
C = Ci
C > 1.05
1.05 < C 1.35
1.35 < C Ci
Ci C 9.45
C >9.45 {1}
d= 1.0
h= 0.5
q= 0.85
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w(t)
w= w(t + t) w(t)
w
t
w
2t
w(t)
t 0
dw
2dt
dt
t
dw
dw= AN,
0< A 0.2
dt
N
dw
dN=fNdt + dw
dP =fPdt.
x1 x2
x1= N N
x2= P P.
d =
dt + d
= [x1 x2]
T
d
= [dw 0]T
g= g0+ gNx1+ g
Px2+
1
2gN,Nx
21+ g
N,Px1x2+
1
2gP,Px
22.
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g
g= g0+ gNx1+ g
Px2+
1
2gN,NV(x1)
+ gN,PC(x1, x2) +1
2gP,PV(x2).
dx1 dx2
d
+
T
+ = 0.
V11 = V(x1)V22 = V(x2) V12 = V21 = C(x1, x2)
211 dw
V(x1) = 2
11
2J11,
V(x2) = J21
J12V(x1)
C(x1, x2) = 0.
V(x1) J11
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0 =J11x1+ J12x2+1
2(fN)
N,NV(x1)+
+ (fN)N,PC(x1, x2) +
1
2(fN)
P,PV(x2)
0 =J21x1+ J22x2+1
2(fP)
N,NV(x1)+
+ (fP)N,PC(x1, x2) +12(fP)P,PV(x2).
J22 = (fN)
P,P = (fP)
P,P = 0
J11x1+ J12x2+
hC2P
(1 + hCN)3 1
V(x1) = 0
J21x1 hC2P
(1 + hCN)3V(x1) = 0.
x1= C2hP
(1 + hCN)3V(x1)
J21
x2= 1
J12
1
hC2P
(1 + hCN)3
V(x1) J11x1
.
g= q P
1 + hCN+
hCP
(1 + hCN)2x1
1
1 + hCNx2
2h2C2P
(1 + hCN)3V(x1).
x21 x
22 x1x2
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Ni+1= Ni+ Ni
1 + qC Ni
CPi1 + hCNi
dt + dwi
Pi+1= Pi+ Pi
CNi
1 + hCNi d
dt.
N(
)
P(
)
gj Cj
gj =q P( )
1 + hCjN( ).
C
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d= 1.0
h= 0.5
q= 0.85
dt= 103
C
C
Ci = 2.1329
C Ci
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0 20 40 60 80 1000.5
1
1.5
2
Population densities as functions of time
Time, t
N(t)P(t)
1.3 1.4 1.5 1.6 1.70
0.5
1
1.5
Phase portrait
N(t)
P(t)
C= 1.2
0 10 20 30 400.5
1
1.5
2
Population densities as functions of time
Time, t
N(t)P(t)
0.8 1 1.2 1.4 1.6
1.4
1.6
1.8
2
Phase portrait
N(t)
P(t)
C= 1.9
0 20 40 600
1
2
3
4
Population densities as functions of time
Time, t
N(t)P(t)
0 1 2 30
1
2
3
4
Phase portrait
N(t)
P(t)
C= 2.4
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3 4 5 60
0.5
1
1.5
2
2.5
3
Mean and equilibrium densities as functions of the vulnerability
Vulnerability, C
N(t)N
P(t)P
Ci
C
[1.055, 6.4]
N
P
N(
)
P(
)
2 3 4 5 60
0.2
0.4
0.6
0.8
1
The MPP as a function of prey vulnerability
Vulnerability, C
MPP(N(t), P(t))MPP(N, P)
C Ci
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C
C >5.92
C= 2.70
C
C
= 2.3
C
N
P
C
C
C
C < C
C < C
C
C
C =C
C
G(C, C)
C
CC
C
C = C
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1.2 1.4 1.6 1.8 2
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25 The invasionfi
tness evaluated atfi
xed C
s
Vulnerability, C
G(C,
C
=
constant)
C =1.05
C =
1.10
C =
1.15
C =
1.20
C =
1.25
C =
1.30
C =
1.35
C =1
.45
C =1
.50
C =1
.40
C =1.5
5
C =1.6
0
C =1.65
C
= 1.70
C
= 1.75
C
= 1.80
C
= 1.85
C = 1.90
C
= 1.95
C
= 2.00
C
= 2.05
C
= 2.10
C
C
C
1.5 2 2.5 3
1.5
2
2.5
3
C =C
C
=
CESS
g(C
)=0
g(C
)=0
The invasion fitness in the CC-plane
Resident vulnerability, C
Mu
tantvulnerability,C
+ - +
- + -
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2 3 4 5 60.4
0.2
0
0.2
0.4
0.6
0.8
1
The selection gradient as a function of the prey vulnerability
Vulnerability, C
g(N(t), P(t))g(N, P)
x
C= C
C
C Ci
g(C)
C < C
C > Ci
g(C)
x
C
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0 1 2 3 4 5 60
0.5
0.85
1
1.5
2
2.5
The stability areas of the different parameter spaces
Vulnerability, C
Cq-borderCd-borderCh-border
Cd
Ch
Cq
y
q= 0.85
d= 1.0
h= 0.5
C= 2
C = 2
C
2.1329
{2} = 0 C = 1.0546 C
C > 1.0546
d
h
q
d
C
h
q
Cd
Ch
Cq
q
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C
q
Ci
C
C < Ci
q
C
qi
C =Ci
qi =1 dh
d2h ,
d h C
Ci
d
h
C
Ci
d
h
C
Ci
C
C
Ci
g
C =Ci
q
A
A
A
A
A
C
A
A
C
C
A
0.0 A 0.12
0 t 103
t= 103
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Ci
0.0 A 0.10
A 0.16 A 0.17 A= 0.12
Ci
A
Ci
C
Ci
C
A = 0.12
C
0.555 105 Ci
2.257 105
C
Ci
g
C
C Ci
gP g
N,N g
N
gN,P
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A C < C i C Ci C Ci
1.8 C 2.3
0 t 103
t = 103
Ci
A
Ci 105
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1.
8
1.
9
2
2.
1
2
.2
2.
3
0.
04
0.
020
0.
02
0.
04
0.
06
0.
08
0.
1
Theselec
tiongradientwithd
ifferentamountofw
hitenoise
Vulnera
bility,
C
Theselectiongradient
A
=
0.22
A
=
0.20
A
=
0.18
A
=
0.16
A
=
0.14
A
=
0.12
A
=
0.10
A
=
0.08
A
=
0.06
A
=
0.04
A
=
0.02
A
=
0
A
A
-
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C
Ci
C
Ch
h
h
-
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h
h
Ch
C
-
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A
C
0.01 A 0.06
0.07 A 0.10 0.11 A 0.14
0.15 A 0.17
0.18 A 0.22
A
A Ci
g(C)< 0
-
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C
C = 2.3
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