comes freely av R T = R + cN , w d N h + R N = b ( R T cN ) N d t - - PowerPoint PPT Presentation

Chapter 5: Killing and Consumption

dN dt = bRN h + R − δN = b(RT − cN)N h + RT − cN − δN = bN ⇣ 1 − h h + RT − cN ⌘ − δN ,

comes freely av RT = R + cN, w

Bacteria in a chemostat: birth rate proportional to consumption aR

dR dt = s − wR − aRN

dN dt = caRN − (w + d)N = caRN − δN ,

N = s aR − w a

nullclines: R ’=0:

heir R0 = ca ¯

R δ

= cas

δw >

g R = δ ca, ¯ N = sc δ − w a

N = 0

• r

R = δ ca

and: N ’=0:

Bacteria in a chemostat: birth rate proportional to consumption aR

• (a)

R N

δ ca s w

¯ N

• (b)

R N

(c)

Time Density ¯ N ¯ R

dR dt = s − wR − aRN

dN dt = caRN − (w + d)N = caRN − δN ,

N = s aR − w a

nullclines

N = 0

• r

R = δ ca

• (a)

R N

δ ca s w

¯ N

dR dt = s − wR − aRN

dN dt = caRN − (w + d)N = caRN − δN ,

J = ✓∂Rf ∂Nf ∂Rg ∂Ng ◆

|( ¯ R, ¯ N)

= ✓w a ¯ N a ¯ R ca ¯ N ca ¯ R δ ◆ = ✓w a ¯ N δ/c ca ¯ N ◆ = ✓α β +γ ◆ (

x, tr = α < 0,

, det = 0 βγ = βγ > 0,

Saturated consumption in a chemostat, birth rate proportional to consumption

dR dt = s − wR − aRN h + R and dN dt = caRN h + R − (w + d)N = caRN h + R − δN .

ivial N = e R0 = ca

δ . T

h

an R0 =

cas δ(wh+s)

e fact that the

¯ R = hδ ca − δ = h R0 − 1

dN dt = 0

gives

• r

s wR = aRN h + R $ N = (h + R)(s wR) aR = h + R a ⇣ s R w ⌘

R N

• (a)

R N

h R0−1 s w

¯ N

• (b)

R N

(c)

Time Density ¯ N ¯ R

dR dt = 0

gives

dR dt = s − wR − aRN h + R and dN dt = caRN h + R − (w + d)N = caRN h + R − δN .

−w a R

dR dt = s − wR − aRN h + R and dN dt = caRN h + R − (w + d)N = caRN h + R − δN .

R N

• (a)

R N

h R0−1 s w

¯ N

• (b)

R N

(c)

Time Density ¯ N ¯ R

J = ✓ ∂Rf ∂Nf ∂Rg ∂Ng ◆

|( ¯ R, ¯ N)

= ✓−α −β +γ ◆

Replicating resource: Lotka-Volterra model, birth rate proportional to consumption

dR dt = rR(1 R/K) aRN , dN dt = caRN δN .

N = r

a(1 − R K )

axis, the nullc

( ¯ R, ¯ N) = (0, 0), (K, 0) and ⇣ δ ca, r a h 1 − δ caK i⌘

Steady states:

¯ R = δ ca

dN dt = 0

gives

dR dt = 0

gives

dR dt = rR(1 R/K) aRN , dN dt = caRN δN .

• (a)

R N

δ ca

K

r a

¯ N

• (b)

R N

(c)

Time Density ¯ N ¯ R

N = r

a(1 − R K )

axis, the nullc

• (a)

R N

δ ca

K

r a

¯ N

• (b)

R N

(c)

Time Density ¯ N ¯ R J = ✓∂Rf ∂Nf ∂Rg ∂Ng ◆

|( ¯ R, ¯ N)

= ✓r − 2r

K ¯

R − a ¯ N −a ¯ R ca ¯ N ca ¯ R − δ ◆ = ✓ − rδ

caK

−δ/c ca ¯ N ◆ = ✓−α −β +γ ◆

dR dt = rR(1 R/K) aRN , dN dt = caRN δN .

Generalized Lotka-Volterra model, birth rate proportional to consumption

dR dt = rR(1 − (R/K)m) − aRN , dN dt = caRN − δN ,

• (a)

R N

δ ca

K

r a

• (b)

R N

δ ca

K

r a

N = r a(1 − (R/K)m)

dR dt = 0

gives

dR dt = [f(R) − aN] R

N = f(R)/c

dR dt = 0

gives

dR dt = rR(1 − (R/K)m) − aRN , dN dt = caRN − δN ,

• (a)

R N

δ ca

K

r a

• (b)

R N

δ ca

K

r a

J = ✓ ∂Rf ∂Nf ∂Rg ∂Ng ◆

|( ¯ R, ¯ N)

= ✓−α −β +γ ◆

• (a)

R N

δ ca

K

r a

• (b)

R N

δ ca

K

r a

g(aR) = aR H + aR = R h + R

h R0 − 1 h R0 − 1

dR dt = rR(1 − (R/K)m) − aRN , dN dt = caRN − δN ,

dN dt =  βR h + R − δ

• N = [βf(R) − δ] N

gives

J = ✓ −α −β +γ ◆

When are horizontal and vertical nullclines a robust result?

dR dt = rR − aRN and dN dt = caRN − δN

N = r a and R = δ ca

J = ✓ r − a ¯ N −a ¯ R ca ¯ N ca ¯ R − δ ◆ = ✓ 0 −δ/c cr ◆

λ± = ± √ −δr = ± i √ δr

• (a)

N

δ ca r a

✓ − − + ◆ tr < 0

• (b)

δ ca r a

✓ − + ◆ tr = 0

• (c)

δ ca r a

✓ + − + ◆ tr > 0

• (d)

R N

δ ca r a

✓ − + + ◆ tr > 0

• (e)

R

δ ca r a

✓ − + − ◆ tr < 0

• (f)

R

δ ca r a

✓ − − + + ◆ tr =?

• (a)

R N

δ ca s1 w s2 w

• (b)

R N

δ ca

K1 K2

r a

Enrichment for resource affects consumer steady state only