Stoichiometric producer-grazer models
Modeling the effects of food quality on grazer dynamics
Angela Peace
NIMBioS Seminar Series
September 23, 2014
Stoichiometric producer-grazer models
Angela Peace
1/49
Outline
Motivating example: Quality is important
Introduce the framework: Ecological Stoichiometry
Incorporating nutrient deficiency into modeling: LKE model
Incorporating nutrient excess into modeling: Stoichiometric
knife-edge models
Further applications using this modeling framework
Stoichiometric producer-grazer models
Angela Peace
2/49
Lotka-Volterra Producer Grazer Model
Rosenzweig MacArthur variation
dx
x
= bx 1 −
− f (x)y
dt
K
dy
= ef (x)y − dy
dt
x(t) biomass of the producer
y (t) biomass of the grazer
b max producer growth rate
K producer carrying capacity
Stoichiometric producer-grazer models
e production efficiency
d grazer loss rate
f (x) grazer ingestion rate
Angela Peace
3/49
Lotka-Volterra Producer Grazer Model
x = Algae population density
Units: mg C/L
Stoichiometric producer-grazer models
Angela Peace
4/49
Lotka-Volterra Producer Grazer Model
y = Daphnia population density
Units: mg C/L
Stoichiometric producer-grazer models
Angela Peace
5/49
Lotka-Volterra Producer Grazer Model
Rosenzweig MacArthur variation
dx
x
= bx 1 −
− f (x)y
dt
K
dy
= ef (x)y − dy
dt
x(t) mg C/L of the producer
y (t) mg C/L of the grazer
b max producer growth rate
K producer carrying capacity
Stoichiometric producer-grazer models
e production efficiency
d grazer loss rate
f (x) grazer ingestion rate
Angela Peace
6/49
Ecological Stoichiometry
Urabe, J. et al. 2002
organisms are composed of several chemical elements
single currency vs. multiple currency approach
Stoichiometric producer-grazer models
Angela Peace
7/49
Ecological Stoichiometry
bringing food quality into the picture
study of the balance of energy and
elemental resources in ecological
interactions
constraints that provide mechanisms that
can be formulated into mathematical
models
example: producer-grazer model
assume that both producer and
grazer are composed of two
essential elements, carbon (C)
and phosphorus (P)
consider the P:C ratio of the
producer
brings ”food quality” into the
model
Stoichiometric producer-grazer models
Sterner and Elser 2002
Angela Peace
8/49
Stoichiometric producer-grazer LKE Model
I. Loladze, Y. Kuang, and J.J. Elser 2000.
dx
= bx
dt
1−
x
!
K
− f (x)y
dy
= ef (x) y − dy
dt
Stoichiometric producer-grazer models
Angela Peace
9/49
Stoichiometric producer-grazer LKE Model
I. Loladze, Y. Kuang, and J.J. Elser 2000.


dx
x
 − f (x)y
= bx 1 −
dt
min(K , (P − θy )/q)
dy
Q
= ê min(1, )f (x) y − dy
dt
θ
Where
P − θy
x
describes the variable P:C ratio of the producer (Quota).
Q=
b maximum growth rate of producer
K producer carrying capacity
P total phosphorus in the system
θ grazer’s constant P:C
Stoichiometric producer-grazer models
q producer minimal P:C
ê maximum production efficiency
d grazer loss rate.
f (x) grazer ingestion rate
Angela Peace
10/49
Stoichiometric algae-Daphnia model
I. Loladze, Y. Kuang, and J.J. Elser 2000.
0.5
1
2
1.75
Grazer
0.25
0.75
density mg C/L
density mg C/L
density mg C/L
Producer
0.5
0.25
1.5
1.25
1
0.75
0.5
0.25
0
0
10
20
30
40
50
60
70
80
t (days)
Low light
Stoichiometric producer-grazer models
0
0
10
20
30
40
50
t (days)
High light
60
70
80
0
0
10
20
30
40
50
60
70
80
t (days)
Very high light
Angela Peace
11/49
Stoichiometric algae-Daphnia model
I. Loladze, Y. Kuang, and J.J. Elser 2000.
Urabe, J. et al. 2002
Stoichiometric producer-grazer models
Angela Peace
12/49
Stoichiometric producer-grazer LKE Model
I. Loladze, Y. Kuang, and J.J. Elser 2000.
Stoichiometric producer-grazer models
Angela Peace
13/49
We’ve seen what happens when
there is too little phosphorus.
Stoichiometric producer-grazer models
Angela Peace
14/49
We’ve seen what happens when
there is too little phosphorus.
What happens when there is
too much phosphorus?
Stoichiometric producer-grazer models
Angela Peace
14/49
Excess phosphorus
Bennett et al. 200, Elser and Bennett 2011
P concentrations in freshwater systems worldwide are 75%
greater than preindustrial levels
We should try to understand how we are altering the global P
cycle
Stoichiometric producer-grazer models
Angela Peace
15/49
Stoichiometric Knife Edge
1400
Boersma and Elser 2006, Elser et al. 2006
1200
150
A
B
140
1000
WGR (%)
WGR (%)
130
800
120
110
100
600
0
0.5
1
1.5
2
90
80
2.5
2.8
Abalone
0
0.5
1
1.5
Production (mg)
120
2.4
2.2
80
40
Mexithauma
European Whitefish
empirical data reported by
Boersma and Elser 2006
0
0.5
1
1.5
GR (1/d)
0
2
0.5
0
0.2
0.4
0.6
0.8
1
0.1
E
F
0.4
0.08
0.3
0.06
0.2
0.1
0.04
0.02
Mayfly
Waterflea
0
2.5
D
C
2.6
2
2
160
C
GR (1/d)
Low P:C
High P:C
400
SGR (% body mass/d)
grazer dynamics are affected
by food nutrient content
(P:C)
grazer growth is reduced
Shrimp
0
1
2
3
4
5
0
0
0.5
1
1.5
2
2.5
Dietary P (% of dry mass)
Stoichiometric producer-grazer models
Angela Peace
16/49
Stoichiometric Knife Edge
Plath and Boersma 2001
Feeding-appendage beat rate of Daphnia Magna under different
phosphorus limitations.
Stoichiometric producer-grazer models
Angela Peace
17/49
Stoichiometric Knife Edge
Hypothesis suggested by Plath and Boersma 2011
High P content of food causes the animal to strongly decrease
their ingestion rate, perhaps leading to insufficient C intake
and thus decreased growth rate
Daphnia may follow a simple feeding rule: “eat until you get
enough P, then stop”.
Used this hypothesis to formulate our model
Stoichiometric producer-grazer models
Angela Peace
18/49
Stoichiometric knife-edge model
Peace et al. 2013
modify the grazer’s ingestion rate
dx
x
= bx(1 −
) − f (x) y
dt
min(K , (P − θy )/q)
dy
Q
= ê min(1, ) f (x) y − dy
dt
θ
Stoichiometric producer-grazer models
Angela Peace
19/49
Stoichiometric knife-edge model
Peace et al. 2013
modify the grazer’s ingestion rate
dx
x
= bx(1 −
) − f (x) y
dt
min(K , (P − θy )/q)
Q
dy
= ê min(1, ) f (x) y − dy
dt
θ
Assume the grazer ingests P up to the rate required for its
maximal growth but not more.
(
)
f (x) for f (x)Q < fˆθ
fˆθ
u(x, y ) =
}
=
min{f
(x),
ˆ
fθ
Q
for f (x)Q > fˆθ
Q
Stoichiometric producer-grazer models
Angela Peace
19/49
Stoichiometric knife-edge model
Peace et al. 2013
modify the grazers biomass growth rate
dx
fˆθ
x
= bx(1 −
) − min{f (x), } y
dt
min(K , (P − θy )/q)
Q
Q
dy
fˆθ
= ê min(1, ) min{f (x), } y − dy
dt
θ
Q
Stoichiometric producer-grazer models
Angela Peace
20/49
Stoichiometric knife-edge model
Peace et al. 2013
modify the grazers biomass growth rate
dx
x
fˆθ
= bx(1 −
) − min{f (x), }y
dt
min(K , (P − θy )/q)
Q
dy
Q
fˆθ
= ê min(1, ) min{f (x), } y − dy
dt
θ
Q
Stoichiometric producer-grazer models
Angela Peace
21/49
Stoichiometric knife-edge model
Peace et al. 2013
modify the grazers biomass growth rate
dx
x
fˆθ
= bx(1 −
) − min{f (x), }y
dt
min(K , (P − θy )/q)
Q
dy
Q
fˆθ
= ê min(1, ) min{f (x), } y − dy
dt
θ
Q
grazer growth rate may be limited by P; however, if P is in excess,
the growth rate may be limited by the amount of available C.
P
P
Q
e
Q
C
Stoichiometric producer-grazer models
C
Angela Peace
22/49
Stoichiometric knife-edge model
Peace et al. 2013
modify the grazers biomass growth rate
No excess P
Q
ê
<θ
Grazer’s growth rate is limited
by P
The grazer ingests u(x, y )Q
units of P
Grazer’s growth rate satisfies
g (x, y )θ = u(x, y )Q
Stoichiometric producer-grazer models
Excess P
Q
ê
>θ
Grazer’s growth rate is limited
by C
The grazer ingests u(x, y ) units
of C and u(x, y )ê units of C are
available for growth.
The growth rate satisfies
g (x, y ) = u(x, y )ê
Angela Peace
23/49
Stoichiometric knife-edge model
Peace et al. 2013
modify the grazers biomass growth rate
Q
θ u(x, y )
Q
ê
Q
ê
Q
<θ
g (x, y ) =
= min{ê, }u(x, y )
êu(x, y )
>θ
θ
Q
fˆθ
= min{ê, } min{f (x), }.
θ
Q
for
for
Since êf (x) < fˆ, we see that
g (x, y ) = min{êf (x),
Q
fˆθ
Q
fˆθ
f (x), ê , fˆ} = min{ f (x), ê , êf (x)}.
θ
Q
θ
Q
Biologically, this translates into three cases in which growth is
determined by, food quality due to P limitation ( Qθ f (x)), food
quality due to P excess (ê fˆQθ ), and food quantity (êf (x))
Stoichiometric producer-grazer models
Angela Peace
24/49
Stoichiometric knife-edge model
Peace et al. 2013
incorporating the dynamics of the stoichiometric knife edge
We arrive at the modified model:
dx
x
fˆθ
= bx(1 −
) − min{f (x), }y
dt
min(K , (P − θy )/q)
Q
dy
Q
θ
= min{êf (x), f (x), ê fˆ }y − dy
dt
θ
Q
Where Q =
P−θy
x .
Stoichiometric producer-grazer models
Angela Peace
25/49
Stoichiometric knife-edge model
Peace et al. 2013
1
0.9
0.8
y (mg C/L)
0.7
0.6
0.5
0.4
II
0.3
I
0.2
III
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x (mg C/L)
Stoichiometric producer-grazer models
Angela Peace
26/49
Stoichiometric knife-edge model
1.8
1
1.6
0.9
0.8
1.4
0.7
1.2
y (mg C/L)
y (mg C/L)
Peace et al. 2013
0.6
0.5
0.4
II
0.3
0
0
0.2
0.4
I
III
0.4
III
0.1
II
0.6
I
0.2
1
0.8
0.2
0.6
0.8
1
1.2
0
1.4
0
0.2
0.4
0.6
x (mg C/L)
0.8
1
1.2
1.4
x (mg C/L)
low P=0.03 mg P / L
P=0.05 mg P / L
2.5
6
2
I
y (mg C/L)
y (mg C/L)
1.5
II
I
5
II
III
1
III
4
3
2
0.5
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x (mg C/L)
P= 0.08 mg P / L
Stoichiometric producer-grazer models
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x (mg C/L)
excess P=0.2 mg P / L
Angela Peace
27/49
Stoichiometric knife-edge model
1.6
1.6
1.4
1.4
1.2
Density (mgC/L)
Density (mgC/L)
1.2
producer
1
0.8
grazer
0.6
0.4
1
0.8
0.6
0.4
0.2
0
0.2
0
10
20
30
40
50
60
70
0
80
0
10
20
30
day
50
60
70
80
P=0.05 mg P / L
1.6
1.6
1.4
1.4
1.2
1.2
Density (mgC/L)
Density (mgC/L)
40
day
low P=0.03 mg P / L
1
0.8
0.6
0.4
1
0.8
0.6
0.4
0.2
0
Peace et al. 2013
0.2
0
10
20
30
40
50
60
70
80
day
P= 0.08 mg P / L
Stoichiometric producer-grazer models
0
0
10
20
30
40
50
60
70
80
day
excess P=0.2 mg P / L
Angela Peace
28/49
Stoichiometric knife-edge model
Stoichiometric producer-grazer models
Peace et al. 2013
Angela Peace
29/49
Stoichiometric knife-edge model
0.4
0.35
f = 0.8
0.35
0.3
f = 0.9
0.3
Grazer Growth (1/day)
Grazer Growth (1/day)
0.4
Peace et al. 2013
f=1
0.25
0.2
0.15
0.1
0.05
0
θ=0.02
θ=0.03
θ=0.04
0.25
0.2
0.15
0.1
0.05
0
0.04
0.08
0.12
Producer P:C (mass)
Stoichiometric producer-grazer models
0.16
0.2
0
0
0.04
0.08
0.12
0.16
0.2
Producer P:C (mass)
Angela Peace
30/49
Stoichiometric knife-edge model
0.4
0.35
f = 0.8
0.35
0.3
f = 0.9
0.3
Grazer Growth (1/day)
Grazer Growth (1/day)
0.4
Peace et al. 2013
f=1
0.25
0.2
0.15
0.1
0.05
0
θ=0.02
θ=0.03
θ=0.04
0.25
0.2
0.15
0.1
0.05
0
0.04
0.08
0.12
Producer P:C (mass)
0.16
0.2
0
0
0.04
0.08
0.12
0.16
0.2
Producer P:C (mass)
Data of algae P:C ratios from aquatic habitats compiled by Sterner
et al. 2008 shows that up to 10% of aquatic habitats have algal
P:C near 0.05 (mass).
Stoichiometric producer-grazer models
Angela Peace
30/49
Stoichiometric knife-edge model
Peace et al. 2013
Assumptions of the knife-edge model:
1. The total mass of phosphorus in the entire system is fixed,
i.e., the system is closed for phosphorus with a total of P
(mgP/L).
2. P:C ratio in the producer varies, but it never falls below a
minimum q (mgP/mgC ); the grazer maintains a constant
P:C, θ (mgP/mgC ).
3. All phosphorus in the system is divided into two pools:
phosphorus in the grazer and phosphorus in the producer.
4. The grazer ingests P up to the rate required for its
maximal growth but not more.
Stoichiometric producer-grazer models
Angela Peace
31/49
Stoichiometric knife-edge model
Peace et al. 2013
Assumptions of the knife-edge model:
1. The total mass of phosphorus in the entire system is fixed,
i.e., the system is closed for phosphorus with a total of P
(mgP/L).
2. P:C ratio in the producer varies, but it never falls below a
minimum q (mgP/mgC ); the grazer maintains a constant
P:C, θ (mgP/mgC ).
3. All phosphorus in the system is divided into two pools:
phosphorus in the grazer and phosphorus in the producer.
4. The grazer ingests P up to the rate required for its
maximal growth but not more.
Assumption 3 may present a problem for the knife-edge model.
Stoichiometric producer-grazer models
Angela Peace
31/49
Expand the Model
and Track Free Nutrients
in the Environment
Stoichiometric producer-grazer models
Angela Peace
32/49
Full stoichiometric knife-edge model
Peace et al. 2014
Tracking Phosphorus
(
)
dx
x
q
fˆθ
= bx min 1 − , 1 −
− min f (x),
y
dt
k
Q
Q
dy
Q
θ
= min êf (x), f (x), ê fˆ
y − dy
dt
θ
Q
n
o
dQ
x
= v (Pf , Q) − b min Q(1 − ), (Q − q)
dt
k
(
) dPf
fˆθ
Q
= −v (Pf , Q)x + θdy + min f (x),
y Q − min ê,
θ
dt
Q
θ
Q = producer P:C ratio
Pf = Free phosphorus in the environment
Stoichiometric producer-grazer models
Angela Peace
33/49
Full stoichiometric knife-edge model
Peace et al. 2014
Tracking Phosphorus
Since total phosphorus is constant, Pf (t) = P − Q(t)x(t) − θy (t).
The model may be reduced down to three equations.
(
)
dx
x
q
fˆθ
= bx min 1 − , 1 −
− min f (x),
y
dt
k
Q
Q
dy
Q
θ
= min êf (x), f (x), ê fˆ
y − dy
dt
θ
Q
n
o
x
dQ
= v (P − Qx − θy , Q) − b min Q(1 − ), (Q − q)
dt
k
Stoichiometric producer-grazer models
Angela Peace
34/49
Full stoichiometric knife-edge model
Peace et al. 2014
Tracking Phosphorus
v (Pf , Q) is the P:C uptake rate of the producer
v (Pf , Q) =
ĉPf Q̂ − Q
â + Pf Q̂ − q
where Q̂, the maximum Quota, ĉ is the maximum
phosphorus:carbon uptake rate of the producer, and â is the
phosphorus half saturation constant of the producer.
Stoichiometric producer-grazer models
Angela Peace
35/49
Full stoichiometric knife-edge model
Peace et al. 2014
Tracking Phosphorus
0.16
1.6
0.16
1.4
0.14
1.4
0.14
0.12
1.2
0.12
producer (mgC/L)
0.1
1
0.08
grazer (mgC/L)
0.6
0.06
0.4
0.2
0
Producer (P:C)
0
10
20
30
40
50
60
70
0.1
1
0.8
0.08
0.6
0.06
0.04
0.4
0.04
0.02
0.2
0.02
0
80
0
10
20
30
day
50
60
70
80
day
low P=0.03 mg P / L
P=0.05 mg P / L
0.16
1.6
0.16
1.4
0.14
1.4
0.14
1.2
0.12
1.2
0.12
0.1
1
0.1
1
0.8
0.08
0.6
0.06
0.4
0.04
0.4
0.04
0.2
0.02
0.2
0.02
0
0
10
20
30
40
50
60
70
day
P= 0.08 mg P / L
Stoichiometric producer-grazer models
80
0.8
0.08
0.6
0.06
0
0
10
20
30
40
50
60
70
P:C
Density (mgC/L)
1.6
P:C
Density (mgC/L)
40
P:C
0.8
P:C
Density (mgC/L)
1.2
Density (mgC/L)
1.6
80
day
excess P=0.2 mg P / L
Angela Peace
36/49
Full stoichiometric knife-edge model
Peace et al. 2014
Interior equilibria
Investigate where the null-surfaces intersect
Varying P only changes the Quota null-surface
(
)
x
dx
q
fˆθ
− min f (x),
= bx min 1 − , 1 −
y
dt
k
Q
Q
dy
Q
θ
= min êf (x), f (x), ê fˆ
y − dy
dt
θ
Q
n
o
dQ
x
= v (P − Qx − θy , Q) − b min Q(1 − ), (Q − q)
dt
k
Stoichiometric producer-grazer models
Angela Peace
37/49
Full stoichiometric knife-edge model
Peace et al. 2014
Interior equilibria
Grazer (y) Null-surface: Yellow
Producer (x) Null-surface: Blue
Quota (Q) Null-surface: Red
P=0.01 mg P /L
Stoichiometric producer-grazer models
P=0.03 mg P /L
P=0.14 mg P /L
Angela Peace
38/49
Full stoichiometric knife-edge model
Peace et al. 2014
bifurcation diagram
1.8
Attracting equilibria or limit cycle
1.4
Grazer density (mg C/L)
1.6
1.2
1.2
1
0.8
0.6
Hopf bifurcation
0.4
0.4
0.2
0
0.02
0.04
0.06
0.08
0.1
Total phosphorus (mg P/L)
Stoichiometric producer-grazer models
0.12
0.081
0.082
0.083
0.084
0.085
0.086
0.087
0.088
Total phosphorus (mg P/L)
1.6
bistability
1.4
0.14
1.2
1
0.8
saddle node bifurcation
0.6
0
0.08
Grazer density (mg C/L)
0.8
0.2
saddle node bifurcation
1
saddle node
bifurcation
Grazer density (mg C/L)
1.4
0
bistability
1.6
Unstable equilibria
Hopf bifurcation
1.8
0.6
0.4
0.2
0
0.112
0.114
0.116
0.118
0.12
0.122
0.124
Total phosphorus (mg P/L)
Angela Peace
39/49
Comparing the models
1.8
1.8
Attracting equilibria or limit cycle
Unstable equilibria
1.6
saddle node
0.4
0.2
0
0.02
0.04
0.06
0.08
0.1
Total phosphorus (mg P/L)
knife-edge model
Stoichiometric producer-grazer models
0.12
1
0.8
0.6
0.4
0.2
0.14
0
saddle node bifurcation
0.6
saddle node bifurcation
0.8
1.2
Hopf bifurcation
1
saddle node
bifurcation
Grazer density (mg C/L)
1.4
1.2
Hopf bifurcation
Grazer density (mg C/L)
1.4
0
Attracting equilibria or limit cycle
Unstable equilibria
1.6
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Total phosphorus (mg P/L)
full knife-edge model
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Comparing the models
1.8
1.8
Attracting equilibria or limit cycle
Unstable equilibria
1.6
saddle node
0.4
0.2
0
0.02
0.04
0.06
0.08
0.1
Total phosphorus (mg P/L)
knife-edge model
0.12
1
0.8
0.6
0.4
0.2
0.14
0
saddle node bifurcation
0.6
saddle node bifurcation
0.8
1.2
Hopf bifurcation
1
saddle node
bifurcation
Grazer density (mg C/L)
1.4
1.2
Hopf bifurcation
Grazer density (mg C/L)
1.4
0
Attracting equilibria or limit cycle
Unstable equilibria
1.6
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Total phosphorus (mg P/L)
full knife-edge model
Excess P (mg P/L) in US lakes (Dodds et al. 2008):
Great Plains grassland: 0.05 central Great Plains: 0.08
northern Great Plains: 0.085 Mississippi Alluvial Plains: 0.061
Stoichiometric producer-grazer models
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Comparing the models
Analytical analysis shows the 2D knife-edge model is the
limiting case of the full knife-edge model when ĉ → ∞,
Q̂ → ∞.
1.8
1.8
Attracting equilibria or limit cycle
Unstable equilibria
1.6
0.4
0.2
0
0.02
0.04
0.06
0.08
0.1
Total phosphorus (mg P/L)
ĉ = 0.2 mgP/mgC/d
Stoichiometric producer-grazer models
0.12
1
0.8
0.6
0.4
0.2
0.14
0
0
0.02
saddle node bifurcation
0.6
1.2
Hopf bifurcation
0.8
saddle node bifurcation
Hopf bifurcation
1
saddle node
bifurcation
Grazer density (mg C/L)
1.4
1.2
saddle node
bifurcation
Grazer density (mg C/L)
1.4
0
Attracting equilibria or limit cycle
Unstable equilibria
1.6
0.04
0.06
0.08
0.1
0.12
0.14
Total phosphorus (mg P/L)
ĉ = 0.8 mgP/mgC/d
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Ecological Stoichiometry
has alot to offer
Stoichiometric producer-grazer models
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Producer-grazer models
How ecological stoichiometry has impacted their maturity
1963
2013
grazer
2000
grazer
grazer
grazer
1920
Region II
Region I
Region II
Region I
Region III
producer
Lotka Volterra
producer
Rosenzweig MacArthur
Stoichiometric producer-grazer models
producer
LKE
producer
Knife Edge
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Producer-grazer models
How ecological stoichiometry has lead to insight into paradoxes
Attracting equilibria
or limit cycle
Unstable equilibria
0.4
0
0
0.4
A
0.8
1.2
0.8
of
Paradox of Paradox
Energy
Enrichment Enrichment
Paradox of
Energy
Enrichment
Paradox of
Nutrient
Enrichment
Paradox of
Enrichment
1.6
Grazer density (mg C/L)
Grazer density (mg C/L)
0.8
Grazer density (mg C/L)
Paradox of
Enrichment
1.2
0.4
0
0
Producer carrying capacity K (mg C/L)
Rosenzweig-MacArthur
Stoichiometric producer-grazer models
0.4
0.8
1.2
1.6
Producer carrying capacity (mg C/L)
LKE
2
1.2
0.8
0.4
0
0
B
0.04
A
0.08
C
0.14
Total phosphorus (mg P/L)
Knife Edge
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Current research projects
Stoichiometric food web
How does food quality affect population growth and the flow of
energy and nutrients up the food chain?
Ecological Stoichiometry and Ecotoxicology
Can Ecological Stoichiometry help improve testing protocols for
assessing risk of exposures?
Stoichiometric producer-grazer models
Angela Peace
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Stoichiometric food web
How does food quality affect population growth and the flow
of energy and nutrients up the food chain and across trophic
levels?
How will the addition of a predation third trophic level change
the dynamics of species growth and population structure?
What roles do consumer and predator nutrient recycling rates
play to alter ecosystem level nutrient availability and how does
this affects population dynamics of the food web?
x
dx
= bx 1 −
− f (x)y
dt
min(K , (P − θy y − θz z)/q)
(P − θy y − θz z)/x
dy
= êy min 1,
f (x)y − g (y )z − dy y
dt
θy
θy
dz
= êz min 1,
g (y )z − dz z
dt
θz
Stoichiometric producer-grazer models
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Ecological Stoichiometry and Ecotoxicology
Co-occuring nutrient and toxic stressors
nutrient stressor
food quality
Stoichiometric producer-grazer models
life history traits
growth
reproduction
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Ecological Stoichiometry and Ecotoxicology
Co-occuring nutrient and toxic stressors
nutrient stressor
food quality
Stoichiometric producer-grazer models
life history traits
growth
reproduction
toxic stressor
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Ecological Stoichiometry and Ecotoxicology
Co-occuring nutrient and toxic stressors
nutrient stressor
food quality
life history traits
growth
reproduction
population
dynamics
toxic stressor
trophic transfer
of nutrients
and toxins
Hows the bioacculumation of metals is affected by stoichiometrically
relevant traits (growth, ingestion, assimilation efficiency )
trophic transfer of metals from photoplankton to zooplankton to fish
Models will help determine the importance of including food quality toxic
risk assessment and water quality policies
Stoichiometric producer-grazer models
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Conclusion
Theory of ecological stoichiometry is a powerful tool
Including food quality and multiple currencies can improve the
quantitative and qualitative predictions of ecological models
Development of stoichiometric food web models will further
our efforts in understanding food quality issues
This modeling framework has high potential in
ecotoxicological applications
Stoichiometric producer-grazer models
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Thank you
Stoichiometric producer-grazer models
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