Ecological Stoichiometry Hao Wang MATH 570 University of Alberta

Ecological Stoichiometry

Hao WangMATH 570

University of Alberta

What is STOICHIOMETRY?

Stoichiometry in biology: Focuses on the balance of elements and energy within biological systems. This includes metabolism, growth, etc.

Stoichiometry in ecology: Focuses on the balance of elements and energy within ecological systems. This includes competition, commensalisms, etc.

Stoichiometry in chemistry: Focuses on the balance of elements within chemical equations.

Why need STOICHIOMETRY?Carbon (C), nitrogen (N), and phosphorus (P) are vital constitutes in biomass: C supplies energy to cells, N is essential to build proteins, and P is an essential component of nucleic acids. The scarcity of any of these elements canSeverely restrict organism and population growth.

What would happen to cows if there wasn't so much sunlight?

Expectations from a 9-year old ecologist

James J. Elser’sson

What would happen to secondary production if solar radiation were reduced?

Expectations from single-currency ecological theory

Solar RadiationHighLow

Sec

on

dar

y P

rod

uct

ion

,H

erb

ivo

re B

iom

ass

Solar Radiation

Pri

mar

y P

rod

uct

ion

,A

uto

tro

ph

Bio

mas

s

HighLow

Terrestrial herbivore (Pieris)Freshwater herbivore (Daphnia)

0

10

20

30

40

50

60

10% 90%

0 200 400 600 800 1000

Biomass C:P in Food

mean

Daphnia magna

0

5

10

15

20

25

0 10 20 30 40Biomass C:N in Food

10%

mean90% (59)Pieris rapae

From: Elser, J.J., W.F. Fagan, R.F. Denno, D.R. Dobberfuhl, A. Folarin, A. Huberty, S. Interlandi, S.S. Kilham, E. McCauley, K.L. Schulz, E.H. Siemann, and R.W. Sterner. 2000. Nutritional constraints in terrestrial and freshwater food webs. Nature 408: 578-580.

Stoichiometric Imbalance Impairs HerbivoresIn Freshwater and Terrestrial Ecosystems

Mass of New Biomass ProducedGGE(gross growth efficiency)=

Mass Ingested

Solar RadiationHighLow

Sec

on

dar

y P

rod

uct

ion

,H

erb

ivo

re B

iom

ass

Very High

Starvation Junk food

(CXN

YP

Z)

prey + (C

XN

YP

Z)

predator -> Q (C

XN

YP

Z)

predator + (C

XN

YP

Z)’

waste

From: Elser, J.J., and J. Urabe. 1999. The stoichiometry of consumer-driven nutrient recycling: theory, observations, and consequences. Ecology 80: 735-751.

(CXN

YP

Z)inorganic

+ (CXN

YP

Z)

autotroph + light -> Q (C

XN

YP

Z)'

autotroph + (C

XN

YP

Z)’ inorganic

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100C : 1P

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1000C : 1P

C, N, and P are three of the main constitutes in biological structural molecules. HOWEVER, C,N, and P are not particularly abundant on Earth or in the universe as a whole and thus it seems that living things made a very discriminating selection of elements from the environment.

S0.11

S0.14

C0.18

C20.2

Ca3.22

Ca2.5

Fe4.18

Fe0.01

Si25.80

O50.02

O63.0

Al7.30

Na2.36

Na0.10

K2.28

K0.11

H0.25

H9.9

P0.11

P1.14

N0.03

N2.5

Composition of Earth’s Crust

Composition of Human Body

Laws and Hypotheses

• Conservation Law of Matter

• Homeostasis

• Liebig’s Minimum Law

• Growth Rate Hypothesis

• Light:Nutrient Hypothesis

Conservation Law of Matter

In an ordinary chemical reaction, matter

and component elements are neither

created nor destroyed.

Homeostasis

Consumers have fixed elemental composition in biomass. This is often called “strict homeostasis”. However, plants have highly variable nutrient contents because of physiological plasticity in relation to environmental conditions such as light, nutrient supply, CO2, etc.

(examined via bifurcations in Hao Wang, Robert W. Sterner, and James J. Elser. On the "strict homeostasis" assumption in ecological stoichiometry, Ecological Modelling, Vol. 243: 81-88, 2012).

[Strict] Homeostasis (animals, heterotrophic bacteria, etc.)

Nonhomeostasis (plants, etc.)

Producer C:nutrient

Her

bivo

re C

:nut

rient

0

Substrate C:nutrient

Pro

duce

r C

:nut

rient

0

Droop Equation

Plant growth rate as

a function of cell

quota (P:C or N:C

ratio) is modeled and

experimentally

verified by Michael

Droop in 1973-1974.max min/ (1 / ) ,

where A is the plant biomass,

is the cell quota.

dA dt Q Q A

Q

Liebig’s Minimum Law

The growth of an organism is controlled

by the most limiting element.

min, min, min,

The specific biomass growth rate

min(1 ,1 ,1 )P N C

P N C

Q Q Q

Q Q Q

From: C.A. Klausmeier et al. 2004. Optimal nitrogen-to-phosphorus stoichiometry of phytoplankton. Nature.

Growth Rate Hypothesis

Rapid growth requires

increased allocation to

P-rich ribosomal RNA

to meet the protein

synthesis demands of

rapid growth.

Based on: Elser, J.J., D.R. Dobberfuhl, N.A. MacKay, and J.H. Schampel. Organism size, life history, and N:P stoichiometry: toward a unified view of cellular and ecosystem processes. BioScience 46: 674-684.

Body C:N:P

food quality constraints on growth /

reproduction

resource competition

trophic efficiency

nutrient recycling

natural selection on growth rate

cellular investment (ribosome content)

biochemical investment (RNA:protein)

Autotroph C : Nutrient Ratio Increases(Nutrient Content Declines)

With Increasing Light Intensity

From: Sterner, R.W. and J.J. Elser . 2002. Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere. Princeton University Press, Princeton, NJ.

N:C

(µ

g :

mg

)P

:C

(µg

: m

g)

N:P

(µ

g :

µg

)

(no relationships)

300

200

100

030

20

12840

0 0.4 0.8 1.2

Specific Growth Rate (d )-1

N-limited GrowthC

ellu

lar

Ra

tio

s (

by

ma

ss

)

P:C

(µ

g :

mg

)N

:C

(µg

: m

g)

N:P

(µ

g :

µg

)15

10

5

300

200

10050

30

increasing light

P-limited Growth

Specific Growth Rate (d )-1

0 0.4 0.8 1.2

0.0

0.1

0.2

0.3

0.4

High LowLight

% P a

ba

ab

a aa

a

% N a

abb b

aba

bb

0.0

1.0

2.0

3.0

4.0

High LowLight

HN / HP

LN / LP

HN / LPLN / HP

Nutrient-limitedCyanobacterium(Synechococcus linearis)

From Healey (1985)

Nutrient-limitedRed pine(Pinus resinosa)

From Elliot and White (1994)

Light:Nutrient Hypothesis

Aquatic ecosystems with low

light:nutrient ratios should have several

trophic levels simultaneously carbon

(or energy) limited, while ecosystems

with high light:nutrient ratios should

have several trophic levels

simultaneously limited by nutrients.

Evaluating The Light:Nutrient Hypothesis:A Mix of Strategies

SimpleControlledReplicatedShortSmallArtificial

ComplexUncontrolledUnreplicated

LongLarge

Natural

Analytical model

Simulationmodel

Labflask

Small indoormicrocosms

Fieldmicrocosms

Whole-ecosystemmanipulation

Fieldsampling

Big indoormicrocosms

Theoretical Test of Light:Nutrient Effects

Model of Loladze, Kuang and Elser (modified from model of T. Andersen)

Producer

Grazer

x' (t) = bx (1 - min[K, (P - y)/ q]

x ) - f (x)y

y' (t) = min (1,

(P - y) / x ) f (x)y - dy

From: Loladze, I, Y. Kuang, and J.J. Elser. 2000. Stoichiometry in producer-grazer systems: linking energy flow and element cycling. Bull. Math. Biol. 62: 1137-1162.

(carbon biomass)

'( ) (1 ) ( )min[ , ( ) / ]

( ) /'( ) min(1, ) ( )

'( ) min(1 ,1 ) ( )( ) /

( ) /'( ) min(1, ) ( )

'( ) min(1 ,1 ) ( )( ) /

( ) /'( ) min(1, ) ( )

xx t bx f x y

K P y q

P y xy t f x y dy

x xx t bx f x y

K P y q

P y xy t f x y dy

x qx t bx f x y

K P y x

P y xy t f x y dy

Logistic growth (light-dependent)

Droop equation (nutrient-dependent)

Plant P:C ratio

Herbivore P:C ratio

Assumption 1Fixed total mass of phosphorus, P, in the entire system.

Assumption 2Plant P:C varies with a minimum q; herbivore P:C is a constant, θ.

Assumption 3All phosphorus of the system is either in plants or in herbivores.

Liebig’s Law

light

light

light





light


Grazer


Experimental Test of the Light:Nutrient Hypothesis

"Aquatron" Experiment (summer 2000) by Urabe and Elser

Aquatron Dynamics

0

0.5

1

1.5

Co

nsu

me

r b

iom

ass

(m

gC

l-1)

0 30 60 90

A

0

2

4

6

Fo

od

bio

ma

ss (

mg

C l-1

) P:C

ratio

of fo

od

(x 10 -3)

0 30 60 90

B

0 30 60 90

C

Days

1

10

50

5

Low Light(40 µE / sq m / s)

High Light (310 µE / sq m / s)

(Extra)High Light

(380 µE / sq m / s)

AlgalP:C

Algal C

Daphnia

C transferefficiency:

~30%

C transferefficiency:

~7%

Urabe, J., J.J. Elser, M. Kyle, T. Sekino and Z. Kawabata. 2002. Herbivorous animals can mitigate unfavorable ratios of energy and material supplies by enhancing nutrient recycling. Ecology Letters: in press.

Daphnia

Algal C

Algal C

Daphnia

Extinction?

100 % treatment 25 % treatment

The Light : Nutrient Project

Field Test of the Light:Nutrient Hypothesis

The Light:Nutrient Hypothesis:Consequences

low high

I : nutrient supplym

low high

Particulate C:P

lowhigh

"Ecological efficiency" (2°/1°)

weak strong

Phytoplankton-bacteria competition

strong weak

Strength of trophic cascade

high low

Efficiency of P recycling

Based on: Sterner, R.W., J.J. Elser, E.J. Fee, S.J. Guildford, and T.H. Chrzanowski. 1997. The light:nutrient balance in lakes: the balance of energy and materials affects ecosystem structure and process. Am. Nat. 150: 663-684.

Under future climate scenarios in the continental boreal regions (Schindler 1998), runoff to lakes will likely decrease. Effects of such shifts on lakes remain unclear.

However, such climate changes will likely lower external nutrient supply while simultaneously raising light intensity (due to lower DOC inputs). Light:nutrient supply may become increasingly unbalanced.

These effects are analogous to effects of elevated pCO2 in terrestrial systems and appear to be driven by similar mechanisms.

Light:Nutrient Balance and Global Change

Daphnia-Algae Experiment

Study organisms

Daphnia pulex: a widespread and important planktonic herbivore in N. America

Daphnia lumholtzi: a daphnia native to Africa but now invasive in N. America.

Scenedesmus obliquus: a Chlorophyte (green algae) found in many lakes and easily grown in the laboratory.

Pictures courtesy of: Paul Hebert; aslo.org; and www.biol.tsukuba.ac.jp/

Methods

Population sizes and species composition were measured twice weekly, while algal carbon and phosphorus data and Daphnia body sizes and egg numbers were measured once weekly.

Experimental Design High Light Low Light

Daphnia pulex alone

Daphnia lumholtzi alone

No Daphnia

D. pulex and D. lumholtzi together

n = 3

n = 3n = 3

n = 3n = 3

n = 3n = 3

n = 3

3-L Jars 218 uE/m2/s21.8 uE/m2/s

Hi gh Li ght - Compet i t i on

0%

20%

40%

60%

80%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13

Ti me (hal f week)

Percentage

Lumhol tziPul ex

Low Li ght - Competi ti on

0%

20%

40%

60%

80%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13

Ti me (hal f week)

Percentage

Lumhol tziPul ex

Main Experimental Results

Competitive Exclusion in both high light or low light!

Competition Model

1 1 2 2

11 1 1 1 1

1

22 2 2 2 2

2

1 1 2 2 1 1

1 ( ) ( )min{ , / }

/ ˆˆ min 1, ( ) (D. pulex)

/ ˆˆ min 1, ( ) (D. lumholtzi)

( ) ( )

dx xrx f x y f x y

dt K p q

dy p xe f x y d y

dt

dy p xe f x y d y

dt

dp pg T p y y x f x y

dt x

2 2( )p

f x y dpx

Algal C

Algal P

Pulex C

Lumholtzi C

High Light Low Lightmeasured in carbon biomass

HypothesisD. lumholtzi has higher requirements for C (energy)

while D. pulex has higher requirements for P (nutrient).

Rich Dynamics

Low light intensityin the experiment

High light intensityin the experiment

A chaotic attractor

The Light:Nutrient Hypothesis:Consequences

low high

I : nutrient supplym

low high

Particulate C:P

lowhigh

"Ecological efficiency" (2°/1°)

weak strong

Phytoplankton-bacteria competition

strong weak

Strength of trophic cascade

high low

Efficiency of P recycling

Based on: Sterner, R.W., J.J. Elser, E.J. Fee, S.J. Guildford, and T.H. Chrzanowski. 1997. The light:nutrient balance in lakes: the balance of energy and materials affects ecosystem structure and process. Am. Nat. 150: 663-684.

Dynamics of Stoichiometric Bacteria-Algae Interactions in the Epilimnion

Question 1

What is the relationship between cyanobacteria and algae?

How do light and nutrient availability regulate relative abundances of bacteria and algae in the Epilimnion?

A Lake System

mz quickly well mixed

input

water exchange

Epilimnion

Hypolimnion

quiescent

brookriver

input input

Scenario The Epilimnion is quickly well mixed in the sense that it is well mixed over night.

Assumption The Epilimnion is well mixed all the time.

Algae Competitive System

0

0

0

1 ( , )1

( , )

1 ( , )( , ) 1 (3.1)

( , )

( ) ( , )

where

( , ) exp[ ( ) ]

( , ) 1

( , )

m

m

m

zm

A mm m

zm

Am

inm

in bg

z

QdA I s A DA ds l A A

dt Q z I s A H z

QdQ I s AQ P Q ds

dt Q z I s A H

dP DP P Q P A

dt z

I s A I kA K s

I s Ads

I s A H kA K

max

ln( , )

( , )

in

bg m

M

M m

H I

H I z A

Q Q PQ P

Q Q M P

This system is modeled for Epilimnion following‘quickly well mixed’.

Algae

Cell Quota

DIP Flexible stoichiometry of algae using Droop’s

form as a nutrient limitng factor

Average sunlight uptake efficiency

of algae in the mixing layer

Respiration loss

Algal sinking and water exchangePhosphorus input

and water exchange

Phosphorus consumption

by algae

The replenishment rate of cell quota is the per unit consumption rate

The cell quota depletion rate is proportional to

algal growth rate

Hao Wang, Hal L. Smith, Yang Kuang, and James J. Elser. 2007. Dynamics of Stoichiometric Bacteria-Algae Interactions in the Epilimnion. SIAM J. Appl. Math, Vol. 68, pp. 503-522.

. , whenever (0) .

. ( ), ( ) 0, whenever (0), (0) 0 .

. Let , which is the total phosphorus of the

system (3.1), then limsup ( ) .

In conclusion,

m M m M

int

Q Q Q Q Q Q

A t P t A P

S AQ P

S t P

Result 1

Result 2

Result 3

we show the boundedness of the system.

Rationality of Model

Exercise 30: Prove these results mathematically to obtain a forward invariant set

0

0

0

. The basic reproductive number for algae is

(0)(1 / )

1 ( , )where ( ) .

( , )

ˆThen the algae extinction steady state E (0, , ) is

(1) locally asymptotical

m

m

A m

Dm z

z

m

in

h Q QR

l

I s Ah A ds

z I s A H

Q P

Local Stability

0

0

ly stable when 1;

(2) unstable when 1.

R

R

0 0

0

*

. 1 implies E is globally asymptotically

stable. 1 implies there exists a unique positive steady

state E and algae uniformly persist: there exists 0 such

that liminf ( )t

R

R

A t

Global Stability

for all solutions with (0) 0.

Proof: Apply theory of monotone dynamical systems

to prove the global stability.

A

P

A Q

+

+

_

competitive system

System Types of Two Species

• Predator-prey

• Cooperative

• Competitive

The latter two are called monotonedynamical systems.How about the three species case? Hal Smith’s TMDS book

+

-Species 2Species 1

+Species 2Species 1

+

-Species 2Species 1

-

Algae love shallower epilimnions, becausesunlight is more sufficient on average.

A-B story

DIP

DOC

Competition

Stimulation

Sunlight

P input

Scenario Algae have flexible stoichiometry whereas bacteria have relatively fixed stoichiometry.

Assumption (P:C) in bacterial cells is fixed.W. Makino, J. B. Cotner, R. W. Sterner and J. J. Elser, Are bacteria more like plants or animals? Growth rate and resource dependence of bacterial C:N:P stoichiometry, Functional Ecology, 17 (2003), pp. 121-130.

Homeostasis

DOC Exudation by Algae

total OCproduced

OC used foralgal growth

extra OCexuded from algae

Energy

0

The exudation rate of DOC by algae is the difference

between the potential growth rate attained when

growth is not mineral nutrient limited,

1 ( , ) , and actual growth rate,

( , )

1

mz

Am

mA

I s AA ds

z I s A H

QA

Q

0

0

1 ( , ) .

( , )

Therefore, the exudation rate is

1 ( , ) .

( , )

m

m

z

m

zm

Am

I s Ads

z I s A H

Q I s AA ds

Q z I s A H

Algae-Bacteria System

0

0

1 ( , )1

( , )

1 ( , )( , ) 1

( , )

( ) ( , ) ( ) ( ) (2.2)

( ) ( ) ( )

m

m

zm

A mm m

zm

Am

in Bm

B r gm

mA


dt Q z I s A H z

QdQ I s AQ P Q ds

dt Q z I s A H

dP DP P Q P A Bf P g C

dt z

dB DBf P g C B B

dt z

QdCA

dt Q

0

1 ( , )( ) ( )

( , )

mzB

m m

I s A Dds Bf P g C C

z I s A H r z

Consumptionsby bacteria

Bacterial growthRespiration and grazing

Water exchange

Exudationby Algae

*1

The basic reproductive number for bacteria is

( ) ( ), where , are components of E

in the system (3.1). This is calculated from the linear

approximation (Jacobian matrix).

m

B

Dr g z

f P g CR P C

Result 1.

R

A

Positivity and Dissipativity (or boundedness).

When (0) , both algae and bacteria go extinct.m

Dh

z

esult 2.

Result 3.

Data fitting

(c)(f) are from 1999 ponds experiment, Roberts and Howarth, Limnol. Oceanogr., 2006

Balance of light and nutrient availability

Bacterial Strains

• LNA: bacterial strain with low nucleic acid contents.

• HNA: bacterial strain with high nucleic acid contents.

• VHNA: bacterial strain with very high nucleic acid contents.

Body C:N:P

food quality constraints on growth /

reproduction

resource competition

trophic efficiency

nutrient recycling

natural selection on growth rate

cellular investment (ribosome content)

biochemical investment (RNA:protein)

(Nishimura et al 2005)

thermocline

Epilimnion

Hypolimnion

(Nishimura et al 2005, Applied and Environmental Microbiology)

VHNA + HNA vs LNA

Previous claim LNA is less active, or dormant.

Observation In late August of Lake Biwa, LNA grows faster than VHNA and HNA.

Hypothesis Probably P-limitation is severe at that time and LNA has higher nutrient uptake efficiency or lower requirements for P.

Question 2

Why do LNA bacteria dominate other two strainsin late August of Lake Biwa?

Competing Bacterial Strains

0

0

1 1 1 1 2 2 2 2

11 1 1 1 1

22 2

1 ( , )1

( , )

1 ( , )( , ) 1

( , )

( ) ( , ) [ ( ) ( )] ( )

( ) ( ) ( )

m

m

zm

A mm m

zm

Am

inm

r gm


dt Q z I s A H z

QdQ I s AQ P Q ds

dt Q z I s A H

dP DP P Q P A B f P B f P g C

dt z

dB DB f P g C B B

dt z

dBB

dt

2 2 2

1 1 1 2 2 20

( ) ( ) ( )

1 ( , ) 1[ ( ) ( )] ( )

( , )

m

r gm

zm

Am m

Df P g C B B

z

QdC I s A DA ds B f P B f P g C C

dt Q z I s A H r z

HNA

LNA

(Nishimura et al 2005, Applied and Environmental Microbiology)

VHNA + HNA vs LNA

Previous claim LNA is less active, or dormant.

Observation In late August of Lake Biwa, LNA grows faster than VHNA and HNA.

Hypothesis Probably P-limitation is severe at that time and LNA has higher nutrient uptake efficiency or lower requirements for P.

Hypothesis 1“higher nutrient uptake efficiency” “lower half-saturation constant for P”

Hypothesis 2“lower requirements for P” “smaller cell quota”

Hypothesis 2Hypothesis 2 Hypothesis 1Hypothesis 1

Lower P requirement higher uptake efficiency

Both of those seemingly conflict views can be true under different nutrient status.

Stoichiometry Discussion

Mathematics inside ES

…

…

Xiong Li, Hao Wang, Yang Kuang,J. Math. Biol. (2011) DOI 10.1007/s00285-010-0392-2

C, N, and P are three of the main constitutes in biological structural molecules. HOWEVER, C,N, and P are not particularly abundant on Earth or in the universe as a whole and thus it seems that living things made a very discriminating selection of elements from the environment.

S0.11

S0.14

C0.18

C20.2

Ca3.22

Ca2.5

Fe4.18

Fe0.01

Si25.80

O50.02

O63.0

Al7.30

Na2.36

Na0.10

K2.28

K0.11

H0.25

H9.9

P0.11

P1.14

N0.03

N2.5

Composition of Earth’s Crust

Composition of Human Body

Significant concentrations of C and N in Earth’s atmosphere?

What features of C, N, and P are so important for biological functionthat these elements have been so intensively distilled from the nonliving world?

C and Si are in the same column of the periodic table, and are almostequally abundant in the solar system, and Si is even more abundantthan C in the Earth’s crust. Why C, not Si to form living things?(Possible answers: C has very high binding energy store energy; high degree of bonding flexibility of C considerable architecturalflexibility)

Why N, P as main nutrient elements, not other elements?

A chemostat-type stoichiometric model

organism

Rin D

R

Q

min

min

( )

( ) (1 )

(1 )

in

dRR DR f R A

dtQdQ

f R rQdt Q

QdArA DA

dt Q

Exercise 31:Interpret this model term byterm and performstability analysis,and discuss viasimulations howRin and D affectthe dynamics

Documents

Ecological Stoichiometry Hao Wang MATH 570 University of Alberta