Chapter 9: Competition From: Gause 1934 Competitive exclusion and - - PowerPoint PPT Presentation

From: Gause 1934

Chapter 9: Competition

Competitive exclusion and co-existence

Synedra ulna Together Asterionella formosa

Competitive exclusion: several consumers using 1 resource

F = K −

n

X

i

eiNi , dNi dt = Ni(biF − di) , for i = 1, 2, . . . , n ,

Closed system with fixed amount of resource K: Since for each species ¯

F = di/bi = K/R0i they have to exclude each other

R0i = biK di

bi ¯ F − di > 0

• r

bi d1 b1 − di > 0

• r

bi di d1 b1 > 1

• r

bi di > b1 d1 , (9.3)

Competitive exclusion: several consumers using 1 resource

F = K −

n

X

i

eiNi , dNi dt = Ni(biF − di) , for i = 1, 2, . . . , n ,

Closed system with fixed amount of resource K: Carrying capacity of one species:

Ki = ¯ Ni = K − di/bi ei = K(1 − 1/R0i) ei with ¯ F = K − ei ¯ Ni =

Nullclines for 2-D closed system

F = K −

n

X

i

eiNi , dNi dt = Ni(biF − di) , for i = 1, 2, . . . , n , (9.1)

F = K - e1N1 - e2N2

N2 = K − d1/b1 e2 − e1 e2 N1 = K(1 − 1/R01) e2 − e1 e2 N1 and N2 = K(1 − 1/R02) e2 − e1 e2 N1 , (9.4)

Nullclines for 2-D closed system

F = K −

n

X

i

eiNi , dNi dt = Ni(biF − di) , for i = 1, 2, . . . , n , (9.1)

F = K - e1N1 - e2N2

Time Density

N1 N2

• N1

N2

K1 K2

N2 = K − d1/b1 e2 − e1 e2 N1 = K(1 − 1/R01) e2 − e1 e2 N1 and N2 = K(1 − 1/R02) e2 − e1 e2 N1 , (9.4)

Competitive exclusion when birth rate is saturated (closed)

F = K −

n

X

i

eiNi , dNi dt = Ni ✓ biF hi + F − di ◆

Carrying capacity of one species, and the corresponding steady state for F: Thus the consumer with the lowest hi over R0-1 ratio depletes the resource most.

bj ¯ F hj + ¯ F > dj

• r

¯ F > hj R0j − 1

At the lowest F the other species cannot invade: _

¯ Ni = K(R0i − 1) − hi ei(R0i − 1) with ¯ F = hi R0i − 1

Competition in open systems (one resource)

dR dt = s − dR − R

n

X

i=1

ciNi with dNi dt = Ni ✓ biciR hi + ciR − di ◆

• r

dR dt = s − dR − R

n

X

i=1

ciNi hi + R with dNi dt = Ni ✓ biR hi + R − di ◆

• r

dR dt = rR(1 − R/K) − R

n

X

i=1

ciNi with dNi dt = Ni ✓ biciR hi + ciR − di ◆

• r

dR dt = rR(1 − R/K) − R

n

X

i=1

ciNi hi + R with dNi dt = Ni ✓ biR hi + R − di ◆ ,

R⇤

i =

hi/ci R0i − 1

• r

R⇤

i =

hi R0i − 1 , where R0i = bi di ,

Exclusion because

R N1 N2

• (0,0,0)

%

R∗

1

%

R∗

2

R N1 N2

• (0,0,0)

%

R∗

1

%

R∗

2

(c) (d) — R — N1 — N2

Quasi steady state to reveal interactions: resource with source

dR dt = s − dR − R

n

X

i=1

ciNi with dNi dt = Ni ✓ biciR hi + ciR − di ◆

n

✓ ◆

ˆ R = s d + P ciNi ,

dNi dt = Ni ⇣ bis s + (hi/ci)(d + P cjNj) di ⌘ = Ni ⇣ βi 1 + P Nj/kj di ⌘

Ki = s hi ⇣ R0i 1 ⌘ d ci = s ciR∗

i

d ci

Quasi steady state to reveal interactions: logistic resource

X ✓ ◆ dR dt = rR(1 − R/K) − R

n

X

i=1

ciNi with dNi dt = Ni ✓ biciR hi + ciR − di ◆ ✓ ◆

ˆ R = K ✓ 1 1 r X ciNi ◆

6 ¯ Ni = r ci ⇣ 1 R∗

i

K ⌘

dNi dt = Ni ⇣ bi(r P cjNj) (hi/ci)(r/K) + r P cjNj di ⌘

dNi dt = riNi ⇣ 1

n

X

j=1

AijNj ⌘

• (a)

N2 1

1 A21

1

1 A12

• (b)

1

1 A21

1

1 A12

• (c)

N1 N2 1

1 A21

1

1 A12

• (d)

N1 1

1 A21

1

1 A12

Lotka-Volterra competition model

N2 = 1 A12 − A11 A12 N1 = 1 A12 (1 − N1) N2 = 1 A22 − A21 A22 N1 = (1 − A21N1)

Several consumers on two substitutable resources

dNi dt = ⇣ βi P

j cijRj

hi + P

j cijRj

δi ⌘ Ni , dRj dt = sj djRj X

i

cijNiRj

R2 = hi ci2(R0i 1) ci1 ci2 R1

Consumer nullcline depends on resources only: where R0i = βi/δi Starting and ending at critical resource density: 0i i i

, R∗

ij = hi cij(R0i−1)

R = R∗ c

i−

R2 = R∗

i2 ci1 ci2 R1

Simplified nullcline: Straight line with slope -ci1/ci2

N1 N2 N3

(a)

R1 R2 R∗

11

R∗

21

R∗

31

R∗

12

R∗

22

R∗

31

N1 N2 N3

• (b)

(0,0,0)

— N1 — N2 — N3

Several consumers with same diet ci1 and ci2.

Tilman diagram QSSA

• ci1/ci2

h1 < h2 < h3

Several consumers having different diets ci1 and ci2.

QSSA QSSA QSSA

Generically only one intersection point between all nullclines: maximally two co-existing species on two resources. Lowest intersection not invadable by other consumers (but no guarantee that this is a steady state).

N1 N2 N3

• (a)

R1 R2 R∗

11

R∗

31

R∗

21

R∗

22

R∗

32

N1 N2

• (b)

N1 N2 K1 K2

N1 N3

• (c)

N1 N3 K1 K3

N2 N3

(d)

N2 N3 K2 K3

Tilman diagram

Essential resources

dNi dt = ⇣ βi Y

j

cijRj hij + cijRj − δi ⌘ Ni , dRj dt = sj − djRj − X

i

cijNiRj

dN1 dt = ⇣ β1 c11R1 h11 + c11R1 c12R2 h12 + c12R2 − δ1 ⌘ N1 dN2 dt = ⇣ β2 c21R1 h21 + c21R1 c22R2 h22 + c22R2 − δ2 ⌘ N2

Two consumers using two resources: Several consumers:

Essential resources

N1 N2 N3

• (a)

R1 R2

s1 d1 s2 d2 ↑ R∗

11

R∗

21

R∗

31

R∗

12

R∗

22

R∗

32

N1 N2 N3

• (b)

R1 R2

s1 d1 s2 d2 ↑ R∗

21

R∗

11

R∗

31

R∗

22

R∗

12

R∗

32

N1 N2

• (c)

N1 N2 K1 K2

N1 N2

• (d)

N1 N2 K1 K2

dN1 dt = ⇣ β1 c11R1 h11 + c11R1 c12R2 h12 + c12R2 − δ1 ⌘ N1 dN2 dt = ⇣ β2 c21R1 h21 + c21R1 c22R2 h22 + c22R2 − δ2 ⌘ N2

Asymptotes defined by letting

• r R1 ! 1 or R2 ! 1, i

t c11 > c12, c22 > c21 and c31 ' c32, r two on resource two, and consumer th

QSSA of (a) QSSA of (b)

Local steepness defines stability

4-dimensional Jacobian

J =    ∂R1R0

1

. . . ∂N2R0

1

. . . ... ∂R1N 0

2

. . . ∂N2N 0

2

   =    

d1c11 ¯ N1c21 ¯ N2 c11 ¯ R1 c21 ¯ R1 d2c12 ¯ N1c22 ¯ N2 c12 ¯ R2 c22 ¯ R2 Φ1c11 Φ1c12 Φ2c21 Φ2c22

    where Φ1 = β1h1 ¯ N1 (h1 + c11 ¯ R1 + c12 ¯ R2)2 and Φ2 = β2h2 ¯ N2 (h2 + c21 ¯ R1 + c22 ¯ R2)2

dR1 dt = s1 − d1R1 − c11N1R1 − c21N2R1 , dR2 dt = s2 − d2R2 − c12N1R2 − c22N2R2 , dN1 dt = ⇣ β1 c11R1 + c12R2 h1 + c11R1 + c12R2 − δ1 ⌘ N1 , dN2 dt = ⇣ β2 c21R1 + c22R2 h2 + c21R1 + c22R2 − δ2 ⌘ N2 ,

J =     −ρ1 −γ11 −γ21 −ρ2 −γ12 −γ22 φ11 φ12 φ21 φ22    

λ4 + a3λ3 + a2λ2 + a1λ + a0 = 0

a0 = (γ11γ22−γ12γ21)(φ11φ22−φ12φ21)

et c11 > c12 and c22 > c21,

4-dimensional Jacobian: essential resources

J =     −ρ1 −γ11 −γ21 −ρ2 −γ12 −γ22 φ11 φ12 φ21 φ22    

λ4 + a3λ3 + a2λ2 + a1λ + a0 = 0

a0 = (γ11γ22−γ12γ21)(φ11φ22−φ12φ21)

et c11 > c12 and c22 > c21,

φ11φ22 − φ12φ21

Unknown sign: If negative, steady state will be unstable.

✓∂R1N0

1

∂R2N0

1

∂R1N0

2

∂R2N0

2

◆ = Φ1

¯ R2 1+ ¯ R1/H11

Φ1

¯ R1 1+ ¯ R2/H12

Φ2

¯ R2 1+ ¯ R1/H21

Φ2

¯ R1 1+ ¯ R2/H22

! = ✓φ11 φ12 φ21 φ22 ◆ where Hij = hij/cij and Φ1 = β1 ¯ N1 (H11 + ¯ R1)(H12 + ¯ R2) and Φ2 = β2 ¯ N2 (H21 + ¯ R1)(H22 + ¯ R2) .

dN1 dt = ⇣ β1 c11R1 h11 + c11R1 c12R2 h12 + c12R2 δ1 ⌘ N1 dN2 dt = ⇣ β2 c21R1 h21 + c21R1 c22R2 h22 + c22R2 δ2 ⌘ N2