Online supplemental material:
Online Appendix A: Detailed conceptual diagram of our full ecosystem model
Online Appendix B: Model analytical results
Online Appendix C: Plant and herbivore C:N parameter sets used in our
simulations
Online Appendix D: Analytic formulas of ecosystem properties and functions
Online Appendix E: Figure of herbivore respiration flux
Online Appendix F: Sensitivity of model results to variation in plant and
herbivore C:N
Predator-driven elemental cycling: the impact of predation and risk effects on
ecosystem stoichiometry
Shawn J. Leroux1 and Oswald J. Schmitz2
1
Department of Biology, Memorial University of Newfoundland, St John’s, NL A1B 3X9
CANADA
2
School of Forestry and Environmental Studies, Yale University,
New Haven, Connecticut, 06511 USA
1
Online Appendix A: Detailed conceptual diagram of our full ecosystem model
τεβaDNHND
βrDND +
(1-τ)εβaDNHND
βNRδΦD,NR
D
CD
μρψαaHNPNH + WCH
βrHNH +
(1-μ)ρψαaHNPNH
βaDNHND
N
predators
aDNHND
ρ aHNPNH + rHNH + WNH
CH
ψαaHNPNH
ψαaPCSNSNP
εaDNHND +
rHNH
CP
ψαrPNP
CS
NH
herbivores
aHNPNH
rPNP
NP
plants
aPCSNSNP
NS
qCS
soil
nutrient
pools
I
kNS
Figure A1 Conceptual diagram of general ecosystem model where resources flowing to
herbivores are explicitly partitioned between active metabolism and growth and reproduction.
We remove the consumptive effects of predators by removing the top trophic level. For models
with predation risk WCH = 0 and for models without predation risk WNH = 0. See Table 1 for
variable and parameter definitions.
2
Online Appendix B: Model analytical results
Equilibrium solutions of stoichiometrically-explicit ecosystem model
Here, we provide the equilibrium solutions (where * denotes equilibrium stock) for our
stoichiometrically-explicit ecosystem models. We present all equilibrium solutions for four
model sets (i.e., treatments); 1) a herbivore-plant-soil model where predators heighten herbivore
metabolism due to perceived predation risk (i.e., “Risk”); 2) a predator-herbivore-plant-soil
model where predators cause direct biomass loss (i.e., “Predation”); 3) a predator-herbivoreplant-soil model with both predator-induced heightened herbivore metabolism and direct
predation (i.e., “Risk & Predation”); and 4) a herbivore-plant-soil model without predatorinduced heightened herbivore metabolism or direct predation (i.e., “Control”), i.e., a control for
predator affects. Because we assume that plants, herbivores, and predators maintain C:N
homeostasis (i.e., dCP/dt = (dNP/dt); dCH/dt = (dNH/dt); dCD/dt = (dND/dt)) our model can
be reduced to five independent variables and their corresponding dynamical equations; dNS/dt
(eq. 1a), dCS/dt (eq. 1b), dNP/dt (eq. 2a), dNH/dt (eq. 3a), dND/dt (eq. 4a). We set the time
derivatives of these five dynamical equations to zero and solve the system of equations for all
equilibria. For our analyses, we focus on the single feasible equilibrium where all ecosystem
compartments are positive (i.e., equilibrium Ci and Ni > 0). We present the feasibility conditions
for this equilibrium point for each model.
1) Solutions for “Risk” model:
In a model without direct predation and with predation risk, WCH = 0. There are 4 equilibrium
points in the “Risk” model. Note that equilibrium 1-ii) = 3-ii), 1-iii) = 3-iii), and 1-iv) = 3-v)
when ND = CD = 0.
Equilibrium i)
3
N S* = 0; CS* = 0;
N P* = 0; CP* = ya N P* ;
N H* = 0; CH* = b N H*
Equilibrium ii)
I
N S* = ; CS* = 0;
k
*
N P = 0; CP* = ya N P* ;
N H* = 0; CH* = b N H*
Equilibrium iii)
I
kr
N S* = ; CS* = P ;
k
aP I
N P* =
kq
; CP* = ya N P* ;
ya aP I
N H* = 0; CH* = b N H*
Equilibrium iv)
I * aH N H* + rP
N = ; CS =
;
k
aP N S*
*
S
N P* =
b rH
; CP* = ya N P* ;
ya aH - yar aH
qrP - ya aP rP N S* N P*
N =
; CH* = b N H* ;
*
* *
b aP rH N S + (1- m ) yar aP aH N S N P - aH q
*
H
Feasibility conditions of equilibrium 1) - iv)
For 1 > μ > 0, 1 > τ > 0, 1 > ψ > 0, 1 > ε > 0, 1 > ρ > 0 and positive values for all other
parameters, this equilibrium leads to positive stocks for all state variables if:
b aP rH I + ( r -1) aH kq > 0 and aP (- ( r -1) aH kq + (mr -1) b aP rH I ) > 0 and b aP rH I < aH kq . An
additional constraint imposed by the conditions for herbivores to maintain homeostasis under
predation risk by differential assimilation is that  > .
4
2) Solutions for “Predation” model:
In a model with direct predation and without predation risk, WNH = 0. There are 6 equilibrium
points in the “Predation” model.
Equilibrium i)
N S* = 0; CS* = 0;
N P* = 0; CP* = ya N P* ;
N H* = 0; CH* = b N H* ;
N D* = 0; CD* = b N D*
Equilibrium ii)
I
N S* = ; CS* = 0;
k
*
N P = 0; CP* = ya N P* ;
N H* = 0; CH* = b N H* ;
N D* = 0; CD* = b N D*
Equilibrium iii)
I
kr
N S* = ; CS* = P ;
k
aP I
kq
N P* =
; CP* = ya N P* ;
ya aP I
N H* = 0; CH* = b N H* ;
N D* = 0; CD* = b N D*
Equilibrium iv)
b rH rD ( b rH rD - e (t -1) aD )
I
N S* = ; CS* =
;
k
(e -1) aD2 q
N P* = 0; CP* = ya N P* ;
N H* =
rD
; CH* = b N H* ;
aD - e aD
N D* = -
rH
; CD* = b N D*
aD
5
Equilibrium v)
I
a N* + r
N S* = ; CS* = H H * P ;
k
aP N S
N P* =
rH
; CP* = ya N P* ;
aH - a r H
N H* =
qrP - ya aP rP N S* N P*
; CH* = b N H* ;
*
* *
b aP rH N S + (1- m ) yar aP aH N S N P - aH q
N D* = 0; CD* = b N D*
Equilibrium vi)
I
a N* + r
N S* = ; CS* = H H * P ;
k
aP N S
aH b ( r -1) ( rD - aDe N
N H* =
rD
; CH* = b N H* ;
aD - e aD
N D* =
(
-b rH rD + aD -qCS* + rH b N H* (1+ e (t -1))
N =
*
P
*
H
)
(t -1)) - aD rPay + aD aH ayr N (m -1)
*
H
; CP* = ya N P* ;
(1- r ) aH N P* - rH ; C* = b N *
aD
D
D
Feasibility conditions of equilibrium 2) - vi)
For 1 > μ > 0, 1 > τ > 0, 1 > ψ > 0, 1 > ε > 0, 1 > ρ > 0 and positive values for all other
parameters, this equilibrium leads to positive stocks for all state variables if:
aP <
( r -1) aH kq ( aH rD - aDrP (e -1)) . An additional constraint imposed by the conditions for
ya rH I ((e -1) aD rP + ( m -1) r aH rD )
herbivores to maintain homeostasis without predation risk by differential assimilation is that  <
.
3) Solutions for “Risk & Predation” model:
In a model with direct predation and predation risk, WCH = 0. There are 6 equilibrium points in
the “Risk & Predation” model.
6
Equilibrium i)
N S* = 0; CS* = 0;
N P* = 0; CP* = ya N P* ;
N H* = 0; CH* = b N H* ;
N D* = 0; CD* = b N D*
Equilibrium ii)
I
N S* = ; CS* = 0;
k
*
N P = 0; CP* = ya N P* ;
N H* = 0; CH* = b N H* ;
N D* = 0; CD* = b N D*
Equilibrium iii)
I
kr
N S* = ; CS* = P ;
k
aP I
kq
N P* =
; CP* = ya N P* ;
ya aP I
N H* = 0; CH* = b N H* ;
N D* = 0; CD* = b N D*
Equilibrium iv)
I
bet rH rD
N S* = ; CS* =
;
k
aD q - aD qe
N P* = 0; CP* = ya N P* ;
rD
N H* =
; CH* = b N H* ;
aD - aDe
r
N D* = - H ; CD* = b N D*
aD
Equilibrium v)
7
I * aH N H* + rP
N = ; CS =
;
k
aP N S*
*
S
N P* =
b rH
; CP* = ya N P* ;
ya aH - yar aH
N H* =
qrP - ya aP rP N S* N P*
; CH* = b N H* ;
*
* *
b aP rH N S + (1- m ) yar aP aH N S N P - aH q
N D* = 0; CD* = b N D*
Equilibrium vi)
I
a N* + r
N S* = ; CS* = H H * P ;
k
aP N S
N =*
P
N H* =
N
*
D
b aP rH rD + aD ( aH qN S* N H* + qrP N S* + b (e - et -1) aP rH N H* )
(
ya aP -aD rP + aH ( rD ( r -1) - aD N
*
H
( r - mr + e ( r -1) (t -1))))
; CP* = ya N P* ;
rD
; CH* = b N H* ;
aD - e aD
1- r ) aH N P* æ b - ya ö rH
(
=
1- ; C* = b N *
aD
ç
è
b
÷
ø aD
D
D
Feasibility conditions of equilibrium 3) - vi)
For 1 > μ > 0, 1 > τ > 0, 1 > ψ > 0, 1 > ε > 0, 1 > ρ > 0 and positive values for all other
parameters, this equilibrium leads to positive stocks for all state variables if:
aP <
( r -1) aH kq ( aH rD - aDrP (e -1)) . An additional constraint imposed by the conditions for
b rH I ( aD rP (e -1) + aH rD ( mr -1))
herbivores to maintain homeostasis under predation risk by differential assimilation is that  >
.
4) Solutions for “Control” model:
In a model without direct predation and without predation risk, WNH = 0. There are 4 equilibrium
points in the “Control” model. Note that equilibrium 2-ii) = 4-ii), 2-iii) = 4-iii), and 2-v) = 4-iv)
when ND = CD = 0.
8
Equilibrium i)
N S* = 0; CS* = 0;
N P* = 0; CP* = ya N P* ;
N H* = 0; CH* = b N H*
Equilibrium ii)
I
N S* = ; CS* = 0;
k
*
N P = 0; CP* = ya N P* ;
N H* = 0; CH* = b N H*
Equilibrium iii)
I
kr
N S* = ; CS* = P ;
k
aP I
N P* =
kq
; CP* = ya N P* ;
ya aP I
N H* = 0; CH* = b N H*
Equilibrium iv)
I
a N* + r
N S* = ; CS* = H H * P ;
k
aP N S
N P* =
rH
; CP* = ya N P* ;
aH - r a H
N H* =
qrP - ya aP rP N S* N P*
; CH* = b N H*
*
* *
b aP rH N S + (1- m ) yar aP aH N S N P - aH q
Feasibility conditions of equilibrium 4) - iv)
For 1 > μ > 0, 1 > τ > 0, 1 > ψ > 0, 1 > ε > 0, 1 > ρ > 0 and positive values for all other
parameters, this equilibrium leads to positive stocks for all state variables if:
- ( r -1) aH kq
ya rH I
< aP <
( r -1) aH kq
. An additional constraint imposed by the
( b ( r -1) + yar (m -1)) rH I
9
conditions for herbivores to maintain homeostasis without predation risk by differential
assimilation is that  < .
10
Online Appendix C: Plant and herbivore C:N parameter sets used in our simulations
Table C1 Empirically-derived plant (α) and herbivore (β) C:N parameter sets used in our
simulations. Elser et al. (2000) reported a mean terrestrial plant C:N = 36 (s.d. = 23) and a mean
terrestrial invertebrate herbivore C:N = 6.5 (s.d. = 1.9). Syntheses of C:N for risk vs no risk
conditions are unavailable, but Hawlena and Schmitz (2010b) reported mean terrestrial
invertebrate (i.e., grasshopper) C:N under no risk that is 0.93 x than with risk (C:N of 4.0 vs 4.3).
We set plant C:N to Elser et al. (2000) mean terrestrial plant C:N (i.e., 36) + 1 s.d. (i.e., 59) and –
1 s.d. (i.e., 13). We set herbivore C:N under risk to Elser et al. (2000) mean terrestrial
invertebrate herbivore C:N (i.e., 6.5) + 1 s.d. (i.e., 8.4) and – 1 s.d. (i.e., 4.6) and herbivore C:N
without risk to 0.93X herbivore C:N under risk (i.e., 6.05, 4.275, and 7.8). In-text results are
reported for the “mean” parameter set 1. Results for other parameters sets are summarized in
Appendix F.
Parameter set
1 ("mean”)
2
3
4
5
6
7
8
9
Models with Risk (i.e., “Risk” and
“Risk & Predation”)
Plant (α)
Herbivore (β)
36
6.5
36
4.6
36
8.4
13
6.5
13
4.6
13
8.4
59
6.5
59
4.6
59
8.4
11
Models without Risk (i.e.,
“Control” and “Predation”
Plant (α)
Herbivore (β)
36
6.05
36
4.275
36
7.8
13
6.05
13
4.275
13
7.8
59
6.05
59
4.275
59
7.8
Online Appendix D: Analytic formulas of ecosystem properties and functions
Table D1 Formulas used to calculate total flux of nitrogen and carbon recycled to the soil nutrient pools within the ecosystem, and
trophic-level specific contributions as well as production and ecological efficiency of each trophic level. Ni and Ci are equilibrium
stocks.
Trophic level
N Recycling Flux
C Recycling Flux
Production
Ecological
Whole ecosystem
Plants
Herbivores
(Secondary)
Risk
No Risk
Risk
No Risk
rPNP + ρaHNPNH +
rPNP +
ψαrPNP + βrHNH + (1
ψαrPNP + βrHNH + (1
rHNH + (1 - ρ)
ρaHNPNH +
- μ) ρψαaHNPNH +
- μ) ρψαaHNPNH +
aHNPNH ((β -ψα)/β)
rHNH + rDND
βrDND + (1 - τ)
βrDND + (1 - τ)
+ rDND + εaDNHND
+ εaDNHND
εβaDNHND
εβaDNHND
rPNP
rPNP
ψαrPNP
ψαrPNP
efficiency
aPCSNSNP
aPCSNSNP/I
ρaHNPNH + rHNH +
ρaHNPNH +
βrHNH + (1 - μ)
βrHNH + (1 - μ)
(1 - ρ)
(1 - ρ) aHNPNH ((β -
rHNH
ρψαaHNPNH
ρψαaHNPNH
aHNPNH
(1 - ρ) aHNPNH/
aPCSNSNP
ψα)/β)
Predators
(Tertiary)
rDND + εaDNHND
rDND +
βrDND + (1 -
βrDND + (1 -
εaDNHND
τ)εβaDNHND
τ)εβaDNHND
12
(1 - ε)aDNHND
(1 - ε) aDNHND /(1
- ρ) aHNPNH
log2(Herbivore respiration)
0
10
20
30
Online Appendix E: Figure of herbivore respiration flux
CONTROL
RISK
PREDATION
RISK & PREDATION
Figure E1 Log2 of herbivore respiration flux for four different models (“Control” = 3level no risk, “Risk” = 3-level with risk, “Predation” = 4-level no risk, “Risk &
Predation” = 4-level with risk). Results are for 1,000 random parameter (uniform
distribution) sets that meet feasibility conditions (i.e., equilibrium stocks of Ni and Ci > 0)
and risk models: α = 36, β = 6.5, no risk models: α = 36, β = 6.05.
13
Online Appendix F: Sensitivity of model results to variation in plant and herbivore
C:N
Here we show results for all 9 plant and herbivore C:N parameter sets (see Table C1 for
specific C:N parameter sets used and main text for a description of these parameters).
These figure depict how sensitive our results are to variations in plant and herbivore C:N.
We provide 4 treatment contrasts: i) Risk vs Control (Risk/Control), ii) Predation vs
Control (Predation/Control), iii) Risk & Predation vs Control (Risk & Predation/Control,
and iv) Risk & Predation vs Predation (Risk & Predation/Predation). Magnitudes greater
than 1 indicate a positive effect of a treatment on an ecosystem property relative to
another treatment and magnitudes less than 1 indicate a negative effect. This qualitative
breakpoint is depicted by a solid vertical line in all figures.
Risk/Control
Predation/Control
Risk & Predation/Control
Risk & Predation/Predation
Soil
Soil
Plant
Plant
Herbivore
Trophic level
Herbivore
Predator
0
1
2
3
4
5
6
0.0
1.0
2.0
3.0
0
2
4
6
8
10
0
5
10
15
20
25
30
0
5
10
15
20
25
30
Soil
Soil
Plant
Plant
Herbivore
Herbivore
Predator
0.0
0.5
1.0
1.5
2.0
0.0
1.0
2.0
Effect size
3.0
0
1
2
3
4
Effect size
Figure F1 Ratio of median log2 mass of soil, plant, herbivore and predator N (top panels)
and C (bottom panels) in Risk vs Control, Predation vs Control, Risk & Predation vs
Control, and Risk & Predation vs Predation treatments. Each circle represents 1 of 9 plant
and herbivore C:N parameter combinations (see Table C1). Results are for 1,000 random
14
parameter (uniform distribution) sets that meet feasibility conditions (i.e., equilibrium
stocks of Ni and Ci > 0).
Risk/Control
Predation/Control
Risk & Predation/Control
Risk & Predation/Predation
Primary
Primary
Secondary
Secondary
Trophic level
Tertiary
0
20 40 60 80
120
0
1
2
3
4
5
0
5
10
15
20
0
10
30
50
Primary
Primary
Secondary
Secondary
Tertiary
0
1
2
3
4
5
0.0
1.0
2.0
Effect size
3.0
0
5
10
15
20
0
2
4
6
Effect size
Figure F2 Ratio of median log10 primary, secondary and tertiary production (top panels)
and efficiency (bottom panels) in Risk vs Control, Predation vs Control, Risk &
Predation vs Control, and Risk & Predation vs Predation treatments. Each circle
represents 1 of 9 plant and herbivore C:N parameter combinations (see Table C1).
Results are for 1,000 random parameter (uniform distribution) sets that meet feasibility
conditions (i.e., equilibrium stocks of Ni and Ci > 0).
15
8
Risk/Control
Predation/Control
Risk & Predation/Control
Risk & Predation/Predation
Plant
Plant
Herbivore
Herbivore
Predator
Trophic level
Total
Total
0
5
10
15
20
25
0
1
2
3
4
0
10
20
30
40
50
60
0
5
0
5
10
15
20
25
10
15
20
Effect size
25
Plant
Plant
Herbivore
Herbivore
Predator
Total
Total
0
5
10
15
20
25
0
5
10
15
20
Effect size
25
0
5
10
15
20
25
Figure F3 Ratio of median log2 plant, herbivore, predator and total N flux (top panels)
and C flux (bottom panels) in Risk vs Control, Predation vs Control, Risk & Predation vs
Control, and Risk & Predation vs Predation treatments. Each circle represents 1 of 9 plant
and herbivore C:N parameter combinations (see Table C1). Results are for 1,000 random
parameter (uniform distribution) sets that meet feasibility conditions (i.e., equilibrium
stocks of Ni and Ci > 0).
Risk/Control
Predation/Control
Risk & Predation/Control
Risk & Predation/Predation
Respiration
0
1
2
3
4
0.0
1.0
2.0
3.0
0
Effect size
10
20
30
40
0
10
20
Figure F4 Ratio of median log2 herbivore respiration in Risk vs Control, Predation vs
Control, Risk & Predation vs Control, and Risk & Predation vs Predation treatments.
Each circle represents 1 of 9 plant and herbivore C:N parameter combinations (see Table
C1). Results are for 1,000 random parameter (uniform distribution) sets that meet
feasibility conditions (i.e., equilibrium stocks of Ni and Ci > 0).
16
30
40