Modeling Nutrient Limitation: Ecosystem Consequences of

Resource Optimization

Nature should weed out sub-optimal strategies of acquiring resources from

the environment

E.B. RastetterThe Ecosystems CenterMarine Biological Laboratory Woods Hole, MA 02543

Captiva Island Meeting March 2011

U1 = g B f(R1) f(T)Uncoupled:

U1 = g B min{f(R1), f(R2), f(R3)...} f(T)U2 = q2 U1

U3 = q3 U1...

Liebig Limitation:

U1 = g B {f(R1) f(R2) f(R3)...} f(T)U2 = q2 U1

U3 = q3 U1...

Concurrent Limitation:

Strategies for modeling resource acquisition:

Drew 1975

Control (HHH) PO4 (LHL) NO3 (LHL)

NH4 (LHL) K (LHL)

Barley roots

0

100

200

0

100

200

0

100

200

Len

gth

of

pri

mar

y la

tera

l ro

ots

(%

co

ntr

ol)

PO4 NO3 K NH4

Nutrient limiting in all layers

Nutrient supplied to

top and bottom layers

Nutrient supplied to

middle layer

0 - 4 cm

4 - 8 cm

8 - 12 cm

Data from Drew 1975

0

0.25

0.5

0.75

1

0.2 0.4 0.6 0.8 1

N P S

0

0.25

0.5

0.75

1

0.2 0.4 0.6 0.8 1

N P S K Mg Mn

Data from Wikström and Ericsson 1995

1

0.25

0.5

0.75

00.2 0.4 0.6 0.8 1

Concentration of nutrient in the plant(fraction of optimum)

Ro

ot:

sho

ot

rati

o

Response of birch seedlings to element limitation

0

10

20

30

40

Wet sedge Toolik inlet

Wet sedge Toolik outlet

Tussock

Root

NPP

as %

of t

otal

NPP

Control

fertilized

Nadelhoffer et al. 2002

0

5

10

15

20

25

30

170 340 510 680

CO2 Concentration (ppm)

Ph

oto

syn

thes

is (

m

ol m

-2 s

-1)

680 ppm

340 ppm

Response of an arctic cotton grass to elevated CO2

data from Tissue & Oechel 1987

U1 = g1 B V1 f(R1) f(T)Optimized:

U2 = g2 B V2 f(R2) f(T)U3 = g3 B V3 f(R3) f(T)

...

V1 + V2 + V2 ... = 1

U2 = q2 U1

U3 = q3 U1...

Maximize U1 under the constraints that

Ui = gi B Vi f(Ri) f(T)

Adaptive (Optimizing):

dVi/dt = a ln{Φ qi U1/Ui } Vi

Select Φ so that ∑dVi/dt = 0 (i.e., ∑Vi = 1):

q1 ≡ 1

Φ = π(Ui/(qi U1))Vi

Uncoupled: U1U = g B f(R1) f(T)

Liebig: UiL = qi g B min{f(R1), f(R2), f(R3)} f(T)

Concurrent: UiC = qi g B {f(R1) f(R2) f(R3)} f(T)

Optimized: UiO = gi B Vi f(Ri) f(T)

If f(R1) doubles: U1U 2×UiL ≥ 0×, ≤ 2×UiC 2×UiO > 0×, < 2×

Comparison of responses for 3-resource models:

U1U 2×UiL 2×UiC 8×UiO 2×

If all three f(Ri) double:

Uncoupled: U1U = g B f(R1) f(T)

Liebig: UiL = qi g B min{f(R1), f(R2), f(R3)} f(T)

Concurrent: UiC = qi g B {f(R1) f(R2) f(R3)} f(T)

Optimized: UiO = gi B Vi f(Ri) f(T)

Comparison of responses for 3-resource models:

PlantsBC: 43550

BN: 74BP: 11

InorganicEN: 2.6EP:0.26

Microbes and soil organic

matterDC: 19960

DN: 420DP: 42

UN: 6.5UP: 1.2

INF: 0.28IND: 0.2IP: 0.05

LC: 770LN: 6.5LP: 1.2

QN: 0.014QP: 0.025

QOC: 255QON: 0.47QOP: 0.025

UmN: 19.98UmP: 1.387

MN: 26MP: 2.6

Rm: 515

Pn: 770UNfix: 0

Based on Sollins et al. 1980HJ Andrews Forest

Rastetter 2011

1010T

aC

amaxAC Q

Ck

CPBU

Uncoupled:

ignored NU

ignored PU

1010min T

iP

iPmaxP

iN

iNmaxN

aC

amax

AC Q

Pk

PUq

Nk

NUq

Ck

CP

BU

N

nN q

PU

P

nP q

PU

Liebig:

1010T

iP

i

iN

i

aC

aCmaxAC Q

Pk

P

Nk

N

Ck

CPBU

N

nN q

PU

P

nP q

PU

Concurrent:

1010T

aC

aCmaxAC Q

Ck

CVPBU

1010T

iN

iNNmaxAN Q

Nk

NVUBU

1010T

iP

iPPmaxAN Q

Pk

PVUBU

ii

ii VU

Ra

td

Vd

ln

PNC V

P

P

V

N

N

V

C

C

R

U

R

U

R

U

Adaptive:

ii

i

R

U

At steady state:

1.18

1.20

1.22

1.24

0 20 40 60 80 100

5.8

5.9

6.0

6.1

0 20 40 60 80 100

600

700

800

900

1000

1100

0 20 40 60 80 100

Uncoupled Liebig

Concurrent Acclimating

Years

NP

P(g

C m

-2 y

r-1 )

Net

N

min

eral

izat

ion

(g

N m

-2 y

r-1 )

Net

P

min

eral

izat

ion

(g P

m-2

yr-1

)

2 x CO2

600

700

800

900

1000

1100

0 20 40 60 80 100

5.8

5.9

6.0

6.1

0 20 40 60 80 100

1.18

1.20

1.22

1.24

0 20 40 60 80 100

5.8

5.9

6.0

6.1

0 20 40 60 80 100

600

700

800

900

1000

1100

0 20 40 60 80 100

Uncoupled Liebig

Concurrent Acclimating

Years

NP

P(g

C m

-2 y

r-1 )

Net

N

min

eral

izat

ion

(g

N m

-2 y

r-1 )

Net

P

min

eral

izat

ion

(g P

m-2

yr-1

)

2 x CO2

600

700

800

900

1000

1100

0 20 40 60 80 100

600

700

800

900

1000

1100

0 20 40 60 80 100

1.18

1.20

1.22

1.24

0 20 40 60 80 100

5.8

5.9

6.0

6.1

0 20 40 60 80 100

1.18

1.20

1.22

1.24

0 20 40 60 80 100

5.8

5.9

6.0

6.1

0 20 40 60 80 100

600

700

800

900

1000

1100

0 20 40 60 80 100

Uncoupled Liebig

Concurrent Acclimating

Years

NP

P(g

C m

-2 y

r-1 )

Net

N

min

eral

izat

ion

(g

N m

-2 y

r-1 )

Net

P

min

eral

izat

ion

(g P

m-2

yr-1

)

2 x CO2

600

700

800

900

1000

1100

0 20 40 60 80 100

600

700

800

900

1000

1100

0 20 40 60 80 100

1.18

1.20

1.22

1.24

0 20 40 60 80 100

1.18

1.20

1.22

1.24

0 20 40 60 80 100

5.8

5.9

6.0

6.1

0 20 40 60 80 100

600

700

800

900

1000

1100

0 20 40 60 80 100

Uncoupled Liebig

Concurrent Acclimating

Years

NP

P(g

C m

-2 y

r-1 )

Net

N

min

eral

izat

ion

(g

N m

-2 y

r-1 )

Net

P

min

eral

izat

ion

(g P

m-2

yr-1

)

2 x CO2

600

700

800

900

1000

1100

0 20 40 60 80 100

600

700

800

900

1000

1100

0 20 40 60 80 100

600

700

800

900

1000

1100

0 20 40 60 80 100

5.8

5.9

6.0

6.1

0 20 40 60 80 100

5.8

5.9

6.0

6.1

0 20 40 60 80 100

1.18

1.20

1.22

1.24

0 20 40 60 80 100

1.18

1.20

1.22

1.24

0 20 40 60 80 100

0

1

2

3

0 20 40 60 80 100

0

5

10

15

0 20 40 60 80 100

0

500

1000

1500

2000

0 20 40 60 80 100

Uncoupled Liebig

Concurrent Acclimating

Years

NP

P(g

C m

-2 y

r-1 )

Net

N

min

eral

izat

ion

(g

N m

-2 y

r-1 )

Net

P

min

eral

izat

ion

(g P

m-2

yr-1

)0

500

1000

1500

2000

0 20 40 60 80 100

2 x CO2 + 4oC

0

1

2

3

0 20 40 60 80 100

0

1

2

3

0 20 40 60 80 100

0

5

10

15

0 20 40 60 80 100

0

5

10

15

0 20 40 60 80 100

0

500

1000

1500

2000

0 20 40 60 80 100

Uncoupled Liebig

Concurrent Acclimating

Years

NP

P(g

C m

-2 y

r-1 )

Net

N

min

eral

izat

ion

(g

N m

-2 y

r-1 )

Net

P

min

eral

izat

ion

(g P

m-2

yr-1

)

2 x CO2 + 4oC

0

500

1000

1500

2000

0 20 40 60 80 100

0

500

1000

1500

2000

0 20 40 60 80 100

66

02

0

1

2

3

0 20 40 60 80 100

0

5

10

15

0 20 40 60 80 100

0

500

1000

1500

2000

0 20 40 60 80 100

Uncoupled Liebig

Concurrent Acclimating

Years

NP

P(g

C m

-2 y

r-1 )

Net

N

min

eral

izat

ion

(g

N m

-2 y

r-1 )

Net

P

min

eral

izat

ion

(g P

m-2

yr-1

)0

500

1000

1500

2000

0 20 40 60 80 100

2 x CO2 + 4oC

0

1

2

3

0 20 40 60 80 100

0

5

10

15

0 20 40 60 80 100

0

500

1000

1500

2000

0 20 40 60 80 100

0

1

2

3

0 20 40 60 80 100

0

5

10

15

0 20 40 60 80 100

0

500

1000

1500

2000

0 20 40 60 80 100

Uncoupled Liebig

Concurrent Acclimating

Years

NP

P(g

C m

-2 y

r-1 )

Net

N

min

eral

izat

ion

(g

N m

-2 y

r-1 )

Net

P

min

eral

izat

ion

(g P

m-2

yr-1

)0

500

1000

1500

2000

0 20 40 60 80 100

2 x CO2 + 4oC

0

1

2

3

0 20 40 60 80 100

0

5

10

15

0 20 40 60 80 100

0

500

1000

1500

2000

0 20 40 60 80 100

0

1

2

3

0 20 40 60 80 100

0

5

10

15

0 20 40 60 80 100

0

500

1000

1500

2000

0 20 40 60 80 100

66

02

Conclusions:

1. Acclimation toward optimal resource use adds additional dynamics that propagate through and interact with ecosystem resource cycles.

2. These dynamics reflect adjustments within the biotic components of the ecosystem to maintain a metabolic balance and meet stoichiometric constraints

3. These dynamics are not represented in either “Liebig’s Law of the minimum” or “Concurrent” models of resource acquisition.

4. Because of these additional dynamics, initial responses are not likely to reflect long-term responses, making long-term projections from short-term experiments or observations difficult.

5. The optimization of resource use will tend to synchronize ecosystem resource cycles in the long term unless disturbance resets the system.

6. These “acclimation” responses act on many time scales and include activation/deactivation of enzyme systems, allocation of resources to individual tissues, replacement of suboptimal species with other species with more “optimal” uptake characteristics, and even natural selection of more “optimal” uptake characteristics.

Conclusions:

6. Resource optimization in my model is simulated through the reallocation of an abstract quantity I call “effort” (Vi)

7. Because the allocation of “effort” represents many processes within the vegetation, it is difficult to quantify except in terms of the observed long-terms dynamics of the ecosystem; this is the model’s biggest weakness.

8. Currently the allocation of “effort” is tied to a single rate constant. Because of the many processes involved in acclimation, a formulation with several rate constants might be more realistic

9. Model description in Rastetter EB. 2011. Modeling coupled biogeochemical cycles. Frontiers in Ecology and the Environment 9:68-73.

10. Executable code, sample data files, and instructions are available at http://dryas.mbl.edu/Research/Models/frontiers/welcome.html