Stoichiometric food web models
How light and nutrients affect trophic efficiencies
Angela Peace
Biomathematics Seminar
Department of Mathematics & Statistics
September 2, 2015
Stoichiometric food web models
Angela Peace
1/39
Outline
Intro: Population models and Trophic transfer efficiencies
Goal: Determine consequences of stoichiometric constraints
on Food Chain Efficiency (FCE)
Model: Mathematical models of ecological food chains
Analysis: Analyze the models, compare with data
Application: Use the models to achieve the goal
Stoichiometric food web models
Angela Peace
2/39
Trophic transfer efficiencies
Important gauges of ecosystem
function and trophic transfer of
nutrients and energy
predicting fish harvested as a
function of primary production
primary production
efficiency of which it is
converted to consumer
production at each trophic
coupling
Stoichiometric food web models
Angela Peace
3/39
Trophic transfer efficiencies
Consumer Efficiency
CE =
Stoichiometric food web models
consumer C production rate
producer C production rate
Angela Peace
4/39
Trophic transfer efficiencies
Consumer Efficiency
CE =
consumer C production rate
producer C production rate
Predator Efficiency
PE =
Stoichiometric food web models
predator C production rate
consumer C production rate
Angela Peace
4/39
Trophic transfer efficiencies
Consumer Efficiency
CE =
consumer C production rate
producer C production rate
Predator Efficiency
PE =
predator C production rate
consumer C production rate
Food Chain Efficiency
FCE =
Stoichiometric food web models
predator C production rate
producer C production rate
Angela Peace
4/39
Dickman et al. 2008 PNAS
Stoichiometric food web models
Angela Peace
5/39
Important roles of stoichiometric constraints
global environmental perturbations change nutrient availability
and light supply
Light and nutrient availability influence population dynamics,
community structure, and can constrain FCE
Fishery yields are constrained by FCE
Here, we try to understand how light and nutrient availability
mediate FCE
Stoichiometric food web models
Angela Peace
6/39
Modeling goals
Develop stoichiometric di- and tritrophic food chain models
Investigate how light, nutrients, and food chain length affects
trophic transfer efficiencies
Hypotheses
FCE is highest under light and nutrient conditions such that
the stoichiometric composition of the primary producer is near
that of the consumer
CE is lowered by predation constraints
Stoichiometric food web models
Angela Peace
7/39
Model development
Algae∗
Daphnia
Gizzard Shad∗∗
Ditrophic model
Use well known Stoichiometric LKE model; Loladze et al. 2000
primary producer, consumer
Tritrophic model
Expand models to include higher trophic level
primary producer, consumer, predator
*Image credit: http : //protist.i.hosei.ac.jp/pdb/images/chlorophyta/scenedesmus
**Image credit: http : //www .dcnature.com/photosfull/herring 3.jpg
Stoichiometric food web models
Angela Peace
8/39
Ditrophic model
Stoichiometric LKE model
Loladze et al. 2000
Stoichiometric food web models
Angela Peace
9/39
Modeling algae-Daphnia systems
Rosenzweig MacArthur variation of Lotka-Volterra Predator-Prey
dx
x
= bx 1 −
− f (x)y
dt
K
dy
= ef (x)y − δy
dt
x(t) algae density
y (t) Daphnia density
b max algae growth rate
K algae carrying capacity
Stoichiometric food web models
e production efficiency
δ Daphnia loss rate.
f (x) Daphnia ingestion rate
Angela Peace
10/39
Model simulations
Low light (K=0.25)
Stoichiometric food web models
High light (K=1)
Angela Peace
11/39
Investigate empirical data
Urabe, J. et al. 2002
organisms are composed of several chemical elements
single currency vs. multiple currency approach
Stoichiometric food web models
Angela Peace
12/39
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-consumer model
assume that both producer and
consumer 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 food web models
Sterner and Elser 2002
Angela Peace
13/39
Stoichiometric compositions
Stoichiometric food web models
Angela Peace
14/39
Incorporating Ecological Stoichiometry into the model
I. Loladze, Y. Kuang, and J.J. Elser 2000.
dx
= bx
dt
1−
x
K
!
− f (x)y
dy
= ef (x)y − δy
dt
Consider both Carbon (C) and Phosphorus (P)
Is algal growth limited by C or P?
Rethink the algae carrying capacity
Stoichiometric food web models
Angela Peace
15/39
Leibig’s Law of the Minimum
Justus von Liebig
(1803-1873)
An organism’s growth is
limited by whichever single
resource is in lowest
abundance relative to its
needs.
Algae is limited by C or P
Stoichiometric food web models
Angela Peace
16/39
Modifying the algae carrying capacity
dx
= bx
dt
1−
x
K
!
− f (x)y
dy
= ef (x)y − δy
dt
Stoichiometric food web models
Angela Peace
17/39
Modifying the algae carrying capacity


dx
= bx 
1 −
dt


x
n
o
 − f (x)y
P−θy
min K , q
dy
= ef (x)y − δy
dt
Stoichiometric food web models
Angela Peace
17/39
Modifying the algae carrying capacity


dx
x
n
o  − f (x)y
= bx 1 −
P−θy
dt
min K , q
dy
= ef (x) y − δy
dt
Stoichiometric food web models
Angela Peace
17/39
Modifying the Daphnia growth rate


dx
x
n
o  − f (x)y
= bx 1 −
P−θy
dt
min K , q
dy
Q
= e min 1,
f (x) y − δy
dt
θ
Stoichiometric food web models
Angela Peace
17/39
Ecological Stoichiometric model
I. Loladze, Y. Kuang, and J.J. Elser 2000.


dx
x
n
o  − f (x)y
= bx 1 −
dt
min K , P−θy
q
dy
Q
= e min 1,
f (x)y − δy
dt
θ
Where
P − θy
x
describes the variable P:C ratio of the producer (Quota).
Q=
Stoichiometric food web models
Angela Peace
18/39
Ecological Stoichiometric model
I. Loladze, Y. Kuang, and J.J. Elser 2000.


dx
x
o  − f (x)y
n
= bx 1 −
P−θy
dt
min K , q
dy
Q
= e min 1,
f (x)y − δy
dt
θ
b maximum growth rate of producer
θ consumer’s constant P:C
K producer carrying capacity
e maximum production efficiency
P total phosphorus in the system
δ consumer loss rate.
q producer minimal P:C
f (x) consumer ingestion rate
Q producer’s variable P:C
Stoichiometric food web models
Angela Peace
18/39
Ditrophic model
LKE with slight change of variables
dx
x
= bx 1 −
− f (x)y
dt
min{K , (P − θy y )/q}
dy
Q
= min ey ,
f (x)y − δy y
dt
θy
b maximum growth rate of producer
θy consumer’s constant P:C
K producer carrying capacity
ey maximum production efficiency
P total phosphorus in the system
δy consumer loss rate.
q producer minimal P:C
f (x) consumer ingestion rate
Q producer’s variable P:C
Stoichiometric food web models
Angela Peace
19/39
Ditrophic model: Phase portrait analysis
consumer density (mg C/L)
1
high nutrient
intermediate nutrient
low nutrient
0.8
0.6
0.4
0.2
0
0
0.4
0.8
1.2
1.5
producer density (mg C/L)
Stoichiometric food web models
Angela Peace
20/39
Ditrophic model: Phase portrait analysis
high nutrient
intermediate nutrient
low nutrient
0.6
0.4
0
0.4
0.8
producer density (mg C/L)
Stoichiometric food web models
1.2
1.5
0.5
0.25
0
0
ht
lig
0
0.75
ht
lig
0.2
1
low
consumer density (mg C/L)
0.8
gh
hi
consumer density (mg C/L)
1
0.25
0.5
0.75
1
1.25
1.5
producer density (mg C/L)
Angela Peace
20/39
Ditrophic Model: Bifurcation analysis
Stoichiometric food web models
Angela Peace
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Tritrophic model
Expand model to include predator
Stoichiometric food web models
Angela Peace
22/39
Tritrophic model
dx
x
= bx 1 −
− f (x)y
dt
min{K , (P − θy y − θz z)/q}
dy
Q
= min ey ,
f (x)y − g (y )z − δy y
dt
θy
θy
dz
= min ez ,
g (y )z − δz z
dt
θz
Stoichiometric food web models
Angela Peace
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Tritrophic model: Phase portrait analysis
Stoichiometric food web models
Angela Peace
24/39
Tritrophic model: Phase portrait analysis
Stoichiometric food web models
Angela Peace
24/39
Tritrophic model: Phase portrait analysis
Stoichiometric food web models
Angela Peace
24/39
Tritrophic model: Bifurcation analysis
Stoichiometric food web models
Angela Peace
25/39
Trophic transfer efficiencies
Efficiencies are defined at equilibrium conditions: (x̄, ȳ ), (x̄, ȳ , z̄) where
Q̄ =
P − θy ȳ
x̄
Stoichiometric food web models
and
Q̄ =
P − θy ȳ − θz z̄
x̄
Angela Peace
26/39
Trophic transfer efficiencies
Efficiencies are defined at equilibrium conditions: (x̄, ȳ ), (x̄, ȳ , z̄) where
Q̄ =
P − θy ȳ
x̄
and
Q̄ =
P − θy ȳ − θz z̄
x̄
Consumer Efficiency CE
CE =
consumer C production rate
producer C production rate
Stoichiometric food web models
Angela Peace
26/39
Trophic transfer efficiencies
Efficiencies are defined at equilibrium conditions: (x̄, ȳ ), (x̄, ȳ , z̄) where
Q̄ =
P − θy ȳ
x̄
and
Q̄ =
P − θy ȳ − θz z̄
x̄
Consumer Efficiency CE
n
o
min ey , θQ̄y f (x̄)ȳ
CE =
bx̄
Stoichiometric food web models
Angela Peace
26/39
Trophic transfer efficiencies
Efficiencies are defined at equilibrium conditions: (x̄, ȳ ), (x̄, ȳ , z̄) where
Q̄ =
P − θy ȳ
x̄
Consumer Efficiency CE
n
o
Q̄
min ey , θy f (x̄)ȳ
CE =
bx̄
Stoichiometric food web models
and
Q̄ =
P − θy ȳ − θz z̄
x̄
Predator Efficiency PE
PE =
predator C production rate
consumer C production rate
Angela Peace
26/39
Trophic transfer efficiencies
Efficiencies are defined at equilibrium conditions: (x̄, ȳ ), (x̄, ȳ , z̄) where
Q̄ =
P − θy ȳ
x̄
Consumer Efficiency CE
n
o
min ey , θQ̄y f (x̄)ȳ
CE =
bx̄
Stoichiometric food web models
and
Q̄ =
P − θy ȳ − θz z̄
x̄
Predator Efficiency PE
n
o
min ez , θθyz g (ȳ )z̄
n
o
PE =
min ey , θQ̄y f (x̄)ȳ
Angela Peace
26/39
Trophic transfer efficiencies
Efficiencies are defined at equilibrium conditions: (x̄, ȳ ), (x̄, ȳ , z̄) where
Q̄ =
P − θy ȳ
x̄
Consumer Efficiency CE
n
o
min ey , θQ̄y f (x̄)ȳ
CE =
bx̄
and
Q̄ =
P − θy ȳ − θz z̄
x̄
Predator Efficiency PE
n
o
θy
min ez , θz g (ȳ )z̄
n
o
PE =
Q̄
min ey , θy f (x̄)ȳ
Food Chain Efficiency CE
FCE =
Stoichiometric food web models
predator C production rate
producer C production rate
Angela Peace
26/39
Trophic transfer efficiencies
Efficiencies are defined at equilibrium conditions: (x̄, ȳ ), (x̄, ȳ , z̄) where
Q̄ =
P − θy ȳ
x̄
Consumer Efficiency CE
n
o
min ey , θQ̄y f (x̄)ȳ
CE =
bx̄
and
Q̄ =
P − θy ȳ − θz z̄
x̄
Predator Efficiency PE
o
n
min ez , θθyz g (ȳ )z̄
n
o
PE =
Q̄
min ey , θy f (x̄)ȳ
Food Chain Efficiency CE
n
o
θy
min ez , θz g (ȳ )z̄
FCE =
bx̄
Stoichiometric food web models
Angela Peace
26/39
Ditrophic model: Ecological efficiencies
0.6
Consumer Efficiency
0.5
0.4
0.3
0.2
0.1
0
0
0.02
0.04
0.06
0.08
P (Nutrient level) mg P/L
Stoichiometric food web models
Angela Peace
27/39
0.6
0.6
0.5
0.5
Consumer Efficiency
Consumer Efficiency
Ditrophic model: Ecological efficiencies
0.4
0.3
0.2
0.1
0
0
0.4
0.3
0.2
0.1
0.02
0.04
0.06
P (Nutrient level) mg P/L
Stoichiometric food web models
0.08
0
0
1
2
3
K (Light level) mg C/L
Angela Peace
27/39
Tritrophic model: Ecological efficiencies
Consumer Efficiency
0.4
0.3
0.2
0.1
0
0
0.02
0.04
0.06
0.08
P (Nutrient level) mg P/L
CE
Stoichiometric food web models
Angela Peace
28/39
0.4
0.4
0.3
0.3
Predator Efficiency
Consumer Efficiency
Tritrophic model: Ecological efficiencies
0.2
0.1
0
0
0.2
0.1
0.02
0.04
0.06
0.08
P (Nutrient level) mg P/L
CE
Stoichiometric food web models
0
0
0.02
0.04
0.06
0.08
P (Nutrient level) mg P/L
PE
Angela Peace
28/39
0.4
0.3
0.3
0.2
0.1
0
0
0.05
0.04
Food Chain Efficiency
0.4
Predator Efficiency
Consumer Efficiency
Tritrophic model: Ecological efficiencies
0.2
0.1
0.02
0.04
0.06
0.08
P (Nutrient level) mg P/L
CE
Stoichiometric food web models
0
0
0.03
0.02
0.01
0.02
0.04
0.06
P (Nutrient level) mg P/L
PE
0.08
0
0
0.02
0.04
0.06
0.08
P (Nutrient level) mg P/L
FCE
Angela Peace
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Tritrophic model: Ecological efficiencies
Consumer Efficiency
0.3
0.2
0.1
0
0
1
2
3
K (Light level) mg C/L
CE
Stoichiometric food web models
Angela Peace
29/39
Tritrophic model: Ecological efficiencies
0.3
Predator Efficiency
Consumer Efficiency
0.3
0.2
0.1
0
0
1
2
3
K (Light level) mg C/L
CE
Stoichiometric food web models
0.2
0.1
0
0
1
2
3
K (Light level) mg C/L
PE
Angela Peace
29/39
Tritrophic model: Ecological efficiencies
0.3
0.3
0.05
0.1
Food Chain Efficiency
Predator Efficiency
Consumer Efficiency
0.04
0.2
0.2
0.1
0.03
0.02
0.01
0
0
1
2
3
K (Light level) mg C/L
CE
Stoichiometric food web models
0
0
1
2
K (Light level) mg C/L
PE
3
0
0
1
2
3
K (Light level) mg C/L
FCE
Angela Peace
29/39
Compare model results with data
Dickman et al. 2008 PNAS
Stoichiometric food web models
Angela Peace
30/39
Dickman et al. 2008
PNAS
Stoichiometric food web models
Angela Peace
31/39
Production rates
Ditrophic
0.3
y Production (mg C/L)
x Production (mg C/L)
1.5
1
0.5
0
Low
High
Nutrient
Nutrient
Low Light
Low
High
Nutrient
Nutrient
High Light
0.2
0.1
0
Low
High
Nutrient
Nutrient
High Light
0.5
0
Low
High
Nutrient
Nutrient
High Light
Low
High
Nutrient
Nutrient
Low Light
Stoichiometric food web models
0.04
z Production (mg C/L)
y Production (mg C/L)
x Production (mg C/L)
0.3
1
0.2
0.1
0
Low
High
Nutrient
Nutrient
Low Light
Low
High
Nutrient
Nutrient
High Light
Low
High
Nutrient
Nutrient
Low Light
0.02
0
Low
High
Nutrient
Nutrient
High Light
High
Nutrient
Angela Peace
Low
Nutrient
Low Light
32/39
0.06
Ecological efficiencies
FCE
0.04
0.02
0
High
Nutrient
Low
Nutrient
High
Low
Nutrient
Nutrient
Low Light
0.4
2
0.3
1.5
CE
CE
High Light
0.2
0.1
0
1
0.5
High
Nutrient
Low
Nutrient
High
Nutrient
Low
Nutrient
Low Light
High Light
0
Low
High
Nutrient
Nutrient
High Light
Low
High
Nutrient
Nutrient
Low Light
1.5
PE
1
0.5
0
Stoichiometric food web models
Low
High
Nutrient
Nutrient
High Light
Low
High
Nutrient
Nutrient
Low Light
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Model application
Testing hypotheses
FCE is highest under light and nutrient conditions such that
the stoichiometric composition of the primary producer is near
that of the consumer
CE is lowered by predation constraints
Stoichiometric food web models
Angela Peace
34/39
Nutrient enrichment
Stoichiometric food web models
Angela Peace
35/39
Nutrient enrichment
Ditrophic
0.6
Tritrophic
0.06
Q=θy
0.05
Food Chain Efficiency
Low Light
Consumer Efficiency
0.5
0.4
0.3
0.2
0
0.04
0.03
0.02
0.01
0.1
0
0.02
0.04
0.06
0
0
0.08
P (Nutrient level) mg P/L
0.06
0.08
Q=θy
0.04
Food Chain Efficiency
High Light
Consumer Efficiency
0.04
0.05
Q=θy
0.5
0.4
0.3
0.2
0.03
0.02
0.01
0.1
0
0.02
P (Nutrient level) mg P/L
0.6
0
Q=θy
0.02
0.04
0.06
P (Nutrient level) mg P/L
Stoichiometric food web models
0.08
0
0
0.02
0.04
0.06
0.08
P (Nutrient level) mg P/L
Angela Peace
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Light enrichment
Stoichiometric food web models
Angela Peace
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Light enrichment
Ditrophic
Tritrophic
0.05
Q=θy
Food Chain Efficiency
0.4
0.3
0.2
0
0.03
0.02
0.01
0.1
0
1
2
0
0
3
K (Light level) mg C/L
3
Q=θy
0.04
Food Chain Efficiency
Consumer Efficiency
High Nutrient
2
0.05
Q=θy
0.5
0.4
0.3
0.2
0.03
0.02
0.01
0.1
0
1
K (Light level) mg C/L
0.6
0
Q=θy
0.04
0.5
Consumer Efficiency
Low Nutrient
0.6
1
2
K (Light level) mg C/L
Stoichiometric food web models
3
0
0
1
2
3
K (Light level) mg C/L
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Conclusions
Nutrient, light, and food chain length affect ecological
transfer efficiencies
FCE is highest under nutrient conditions such that Q̄ ≈ θy
In low nutrient conditions, FCE is highest under light
conditions such that Q̄ ≈ θy
In high nutrient conditions, FCE is highest under light
conditions such that Q̄ > θy
FCE is lower in 3 level food chains than 2 level food chains
Stoichiometric food web models
Angela Peace
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Model limitations
θy and θz are assumed constant
The models used do not track nutrients in the environment
The models used only consider P limitation and not P excess
These models consider only C and P
The minimum functions used are approximations
These models only model simple aquatic food chains
Stoichiometric food web models
Angela Peace
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Thank you
Angela Peace, PhD
a.peace@ttu.edu
Department of Mathematics & Statistics
Texas Tech University
Stoichiometric food web models
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