The Evolution of Stability in a Stochastic Predator-Prey System

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



G

N

G = N /N



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].



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



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 ,



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).



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



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



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



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( )

.



(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)



(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)



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



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.



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



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



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



d= 1.0

h= 0.5

q= 0.85

dt= 103

C

C

Ci = 2.1329

C Ci



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


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


Time, t

N(t)P(t)

0 1 2 30

1

2

3

4

Phase portrait

N(t)

P(t)

C= 2.4



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



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



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

+ - +

- + -



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



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



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



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



A C < C i C Ci C Ci

1.8 C 2.3

0 t 103

t = 103

Ci

A

Ci 105



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



C

Ci

C

Ch

h

h



h

h

Ch

C



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



C

C = 2.3

Documents

The Evolution of Stability in a Stochastic Predator-Prey System