Electronic Supplementary Material (ESM)
S1. Full model description and analysis ....................................................................................... 2
S1.1. Model equations ............................................................................................................... 2
S1.2. Stoichiometric constraints and composite parameter derivations .................................... 4
S1.2.1 Plants .......................................................................................................................... 4
S1.2.2 Detritus ....................................................................................................................... 4
S1.2.3 Decomposers .............................................................................................................. 5
S1.2.4 Herbivores .................................................................................................................. 6
S1.3. Equilibrium analysis ........................................................................................................ 8
S1.3.1 C-limited decomposer equilibrium: ........................................................................... 9
S1.3.2 X-limited decomposer equilibrium: ......................................................................... 10
S2. Effects of herbivore feeding behaviours and physiological characteristics ...................... 11
S2.1. Equilibrium stocks according to herbivory scenarios: ................................................... 11
S2.2. Signs of the effects of herbivory nutritional processes on equilibrium stocks: ............. 14
S3. Physiological alterations to plants by herbivores ................................................................ 15
S3.1. Increased root exudation following defoliation ............................................................. 15
S3.2. Alteration of plant biomass allocation to root tissues .................................................... 17
S3.3. Alteration of plant nutrient content ................................................................................ 18
S3.4. Alteration of plant secondary compound content .......................................................... 19
S3.5. Conclusion ..................................................................................................................... 19
S4. First-order mineralization ..................................................................................................... 20
S5. Functional responses .............................................................................................................. 21
S6. References in ESM ................................................................................................................. 26
1
S1. Full model description and analysis
S1.1. Model equations
The equations of the model are presented in Table S1.1.
Table S1.1: Model equations
Producers:
Decomposers:
Detritus:
Ingestion
Uptake
Senescence
ì
dX
P
ï
= uX I - hxH X P - lP X P
ï dt
í
Ingestion
Senescence
Fixation
ï dC
P
= ua X I - a hxH X P - lPa X P
ï
î dt
Decomposition
Mineralisation/Im mobilization
ì
Loss
ï dX D
C
d
1 m -d
= Min[m M ,
rX I ] - lD X D + Min[
mCM , rX I ]
ï
ï dt
m m -d
d m
í
Decomposition
ï
Loss
ï dCD = c Min[m CM , d rX ] - l b X
I
D
D
ïî dt
m m -d
Decomposition
Loss
ì
Defecation
Senescence
ï dX M
C
d
C
= lP X P + (1- aX )hxH X P - Min[m M ,
rX I ] - lM M
ï
ï dt
m m -d
m
í
Decomposition
ï
Defecation
Senescence
Loss
ï dCM = l a X + (1- a )a hx X - Min[mC , m d rX ] - l C
P
P
C
H P
M
I
M M
ïî dt
m -d
Mineralization/Im mobilization
ì
Input
Uptake
Excretion
Loss
ï dX I
1 m -d
= I X - lI X I - uX I + (1- nX )aX hxH X P - Min[
mCM , rX I ]
Inorganic resource: í
d m
ï dt
î
Table S1.2 shows all the variables and parameters used in the model, as well as the values
used to generate the results of Figs. 2 and 3. The parameters are matched to the case of forest and
shrubland insect herbivores.
2
Table S1.2: Symbol definitions
Class
Variables
Stoichiometric
parameters


Ecosystem
parameters
Symbol
XP
CP
XD
CD
XM
CM
XI
α
Definition


c
lP
lD
lM
lI
C:X ratio of detritus
C:X ratio of herbivores
C :X ratio of detritus from herbivores
Decomposer C:X TER
Herbivore C:X TER
Uptake rate of XI by plants
Uptake rate of XI by XI -limited
decomposers
Uptake rate of plant detritus by CM-limited
decomposers
Uptake rate of herbivore detritus by CMlimited decomposers
Uptake rate of total detritus by CM-limited
decomposers
Decomposer gross growth efficiency for C
Production rate of detritus by plants
Loss rate of decomposers from ecosystem
Loss rate of detritus from ecosystem
Loss rate of XI from ecosystem
IX
xH
h
Supply rate of XI
X stock in herbivores
Ingestion rate of producers by herbivores
aX
Herbivore assimilation efficiency for X
0.7
g. m-2.day-1
g.m-2
(g.m-2)1.day-1
dim.
aC
Herbivore assimilation efficiency for C
0.6
dim.
nX
nC
nXmax
Herbivore net growth efficiency for X
Herbivore net growth efficiency for C
Maximum nX
varies
varies
0.95
dim.
dim.
dim.
nCmax
Maximum nC
0.6
dim.

φ
δ

u
r
a
j
m
Herbivore
parameters
X stock in plants
Carbon stock in plants
X stock in decomposers
C stock in decomposers
X stock in detritus
C stock in detritus
Stock of inorganic X
C:X ratio of plants
C:X ratio of decomposers
Values
Units
varies
7.37
g.m-2
g.m-2
g.m-2
g.m-2
g.m-2
g.m-2
g.m-2
g.g-1
g.g-1
varies
5.49
varies
24.57
10.14
0.34
0.09
g.g-1
g.g-1
g.g-1
g.g-1
g.g-1
day-1
day-1
1.6 10-3
day-1
0.008
day-1
varies
day-1
0.3
4.8 10-6
3.3 10-3
8.4 10-4
3 10-4
dim.
day-1
day-1
day-1
day-1
0.03
0.3
3 10-5
Ref.
Driving factor
Cleveland
&
Liptzin 2007
Function of α
Elser et al 2000
Function of α
Calculated
Calculated
Barber 1995
Lovett & Ruesink
1995
Cebrián 1999
Lovett & Ruesink
1995
Calculated
Moore et al 2005
Cebrián 1999
Hunt et al 1987
Cebrián 1999
Christenson et al
2002
Chapin et al 2002
Cebrián 1999
Cebrián 1999
Carisey & Bauce
1997
Karasov
&
Martínez del Rio
2007
Calculated
Calculated
Carisey & Bauce
1997
Carisey & Bauce
1997
3
S1.2. Stoichiometric constraints and composite parameter derivations
Our model incorporates stoichiometric constraints on the elemental composition of the
compartments (homeostatic constraint) and on the fluxes of elements exchanged among them
(mass-balance constraint). These constraints are reflected in the parameters and functions of each
organic compartment:
S1.2.1 Plants
Their C:X ratio α is held homeostatically constant. As a result, all related fluxes of C and X in
and out of the compartment are in a ratio equal to α (see table 1.1).
S1.2.2 Detritus
The C:X ratio of detritus is m =
detritus by decomposers is m =
lP
hx H (1 - aX )
a+
j , and the uptake rate of
lP + hx H (1 - aX )
lP + hx H (1 - aX )
lP
(1 - aC )hx H
a+
j.
lP + (1 - aC )hx H
lP + (1 - aC )hx H
The implicit assumptions behind these equations are:
-
Plant- and herbivore-produced detritus are well mixed and are not discriminated by
decomposers, such that the C:X ratio and decomposition rate of detritus reflect their relative
proportions.
-
There are no losses of plant material from the ecosystem besides from herbivory (in
scenarios I, IE, IED and IEDA): losses of plant material in terrestrial ecosystem are mainly
through leaching, runoff, sorption and accumulation in refractory organic pools in soil [1, 2]. All
these processes occur when plant material is already part of the detritus compartment. In aquatic
ecosystems though, living primary producers can be lost through water convection. This case is
not covered by our model, but was included in a version of our model more tuned to aquatic
systems. The results were qualitatively similar.
4
-
C and X lost from decomposers ( l D X Dand lD bX D) do not enter the detritus pool and are lost
from the ecosystem entirely: Dead decomposer biomass is generally very labile and accessible to
the microbial community and so, cycles internally to the microbial community very fast [3]. This
is why we did not add it to the detritus pool. As we explained above, our definition of the
microbial decomposer pool is rather loose and includes all of bacteria, fungi protozoa, dead and
alive and their direct predators. However, there is a fraction of the dead decomposer biomass that
is not recycled internally and that contributes to the refractory soil organic pool. This is our loss
term from the decomposer pool. In any case, a flux of organic matter from decomposers to the
pool of detritus would likely not affect the relation between plant nutrient content and the impact
of herbivory on nutrient availability, as long as its elemental composition is independent from the
elemental composition of plants.
-
Likewise, there is no contribution from the carcasses of herbivores to the pool of detritus.
The importance of the contribution of carrion to soil organic matter is still disputed [4-6]. So we
ignored it to keep an already complex model as simple as possible. At any rate, addition of such a
flux would not affect the main conclusions of the model, as long as the elemental composition of
cadavers is independent from the elemental composition of plants.
S1.2.3 Decomposers
Decomposer C:X TER (TERD) is equal to d =
b
c
(~24.57 using parameter values in table S1.2),
where c is the net gross growth efficiency of decomposers and β their biomass C:X ratio.
When δ>μ, the mineralization/immobilization flux is negative, and hence decomposers mineralize
the inorganic nutrient XI. In contrast, when δ<μ, the mineralization/immobilization flux is
positive, and decomposers immobilize the inorganic nutrient XI (Figure 2, main text).
5
The decomposition rate depends on the availability of its two resources (detritus and inorganic
nutrients) according to Liebig’s law of the minimum, i.e., growth depends only on the availability
of detritus C when mCM < m d rX I and only on XI availability when mCM > m d rX I .
m -d
m -d
S1.2.4 Herbivores
aX n Xmax
g , where γ is the C:X
The C:X threshold elemental ratio (TER) for herbivores is h =
aC nCmax
ratio of herbivores, nCmax is their maximal net growth efficiency for C and n Xmax is their maximal
net growth efficiency for X. When the plant C:X ratio is smaller than the herbivore threshold
elemental ratio η, herbivore growth is limited by C availability. In contrast, when the plant C:X
ratio is larger than η, herbivore growth is limited by X availability. We assume that herbivores
use the limiting element most efficiently, which means that in the first case the net growth
efficiency for C nC = nCmax , while in the second the net growth efficiency for X nX = nXmax.
Herbivores need to keep their elemental composition constant, i.e.,
dCH
dX
= g H , and hence the
dt
dt
herbivore C:X ratio must be equal to the plant C:X ratio, corrected by C and X assimilation and
net growth efficiencies. Mathematically, one needs to set g =
aC nC
a . This equation is valid
aX n X
aC nCmax
max
max
when a = h (where η is TERH), yielding g =
max h (remember that nC = nC and n X = n X
aX n X
nC
nCmax
when a = h). Combining the two preceding equalities yields
a = max h. Therefore, when
nX
nX
6
herbivore growth is C-limited (α<η and nC = nCmax ), then n X =
herbivore growth is X-limited (α>η and nX = nXmax) then nC =
a max
n . In contrast, when
h X
h max
n .
a C
The C:X ratio of the herbivore-produced detritus φ is equal to the plant C:X ratio, corrected by
the assimilation efficiencies: j =
1 - aC
a ( 1 - aC and 1- aX are the fractions of ingested C and X
1 - aX
respectively that are not assimilated).
7
S1.3. Equilibrium analysis
The equilibrium analytical expressions for detritus C stock level (CM*) and X stock levels
of inorganic resources (XI*), producers (XP*) and decomposers (XD*), for the model are listed in
Table S1.3.
Because the stoichiometries of all organic compartments are fixed, any change in an
organic X pool, is matched by a similar change in the linked C pool, with a proportionality factor
equal to the C:X ratio of the affected compartment. E.g., a doubling of the XP pool corresponds to
a simultaneous doubling in the CP pool. Hence, the analysis of one of the 2 pools is sufficient for
each organic pool. We chose the pools XA, CM, XD and XI.
Table S1.3: Equilibrium analytical expressions
Equilibrium values
X I* =
C-limited
decomposers
X P* =
X I* =
X-limited
decomposers
X D* =
IX
,
æ lP + hx H (1 - aC ) m
m - d hx H (1 - n X )aX ö
lI + u + uç
a
÷
lP + hx H
lM + m
md
lP + hx H ø
è
u
1 lP + hx H (1 - aC ) *
1m *
CM , CM* = u
X I*, X D* =
aX I
lP + hx H
lM + m
lP + h
d lD
IX
u
, X P* =
X I*,
hx H (1 - n X )aX
lP + hx H
lI + u + r - u
lP + h
r m
X I*, CM* =
lD m - d
u
lP + hx H (1 - aC )
md
a -r
lP + hx H
m -d *
XI
lM
8
The local stability of these equilibriums and the persistence of the various ecosystem
components are analysed below. (The Jacobian matrix J has variables in the order XA, CM, XD and
XI in what follows)
S1.3.1 C-limited decomposer equilibrium:
é
-(lP + hx H )
ê
ê( lP + hx H (1 - aC ))a
J=ê
0
ê
ê
ê hx H (1 - n X )aX
ë
0
0
-(lM + m)
m
0
d
m -d
-m
md
-lD
0
ù
ú
0 ú
0 ú
ú
ú
-(lI + u) ú
û
u
Two eigenvalues are equal respectively to –(lM+m) and –lD. The two other eigenvalues are
solutions of the equation l2 + (lP + hx H + lI + u)l + (lP + hx H )(lI + u) - uhx H (1- nX )aX = 0.
We
can
calculate
the
determinant
of
2nd
this
degree
Rewriting the determinant as
equation:
shows that it
is always positive.
The first solution
The
second
solution
is thus always negative.
is
also
negative
because
.
All the eigenvalues of J are negative. Hence the equilibrium is always stable when
feasible.
This equilibrium is feasible if decomposers are limited by C, i.e. rX I* >
Table S1.3, the condition becomes r > u
m -d
*
. Using
mCM
md
lP + hx H (1 - aC ) m m - d
a.
lP + hx H
m + lM md
9
S1.3.2 X-limited decomposer equilibrium:
é
-(lP + hx H )
ê
ê[ lP + hx H (1 - aC )]a
J =ê
ê
0
ê
êë hx (1 - n )a
H
X
X
0
0
-lM
0
0
-lD
0
0
ù
ú
ú
ú
ú
ú
-(lI + u + r)úû
u
md
-r
m -d
m
r
m -d
Two eigenvalues are equal respectively to –lM and –lD. Two other eigenvalues are
negative solutions of the equation
l2 + (lP + hx H + lI + u + r)l + (lP + hx H )(lI + u + r) - uhx H (1- nX )aX = 0.
We can calculate the determinant of this 2nd degree equation:
Rewriting the determinant as
shows that
it is always positive.
The first solution
is thus always negative.
The second solution
is also negative because
.
All the eigenvalues of J are negative. Hence the equilibrium is always stable when
feasible.
This equilibrium is feasible if decomposers are limited by X, i.e. rX I* <
Table A3, the condition becomes r < u
m -d
*
. Using
mCM
md
lP + hx H (1 - aC ) m m - d
a . One can check easily that
lP + hx H
m + lM md
*
this condition is sufficient to guarantee that CM
> 0 in Table S1.3.
10
S2. Effects of herbivore feeding behaviours and physiological
characteristics
This appendix analyses the model as a function of the scenarios of herbivory, in order to
yield the signs of the effects of the herbivore nutritional processes on equilibrium stocks.
S2.1. Equilibrium stocks according to herbivory scenarios:
Table S2.1 presents the analytical expressions of inorganic X equilibrium levels under the
6 scenarios (0, I, IE, IED, IEDA and IEDAG) for both C-limited and X-limited decomposers:
Table S2.1: Equilibrium analytical expression of inorganic X stock level for the different scenarios
Scenario:
0
I
IE
IED
IEDA
IEDAG
CM-limited decomposers
XI-limited decomposers
IX
a a -d
l I + u + u(
a)
l M + a ad
IX
lP
a a -d
l I + u + u(
a)
l P + hx H l M + a ad
IX
æ l
a
a -d
hx (1- n X )a X h ö
P
l I + u + uç
a- H
÷
l P + hx H
è l P + hx H l M + a ad
ø
IX
æ l + hx (1- a ) a a - d
hx (1- n X )a X h ö
H
C
l I + u + uç P
a- H
÷
l P + hx H
l M + a ad
l P + hx H
è
ø
IX
æ l + hx (1- a ) a m - d
hx (1- n X )a X h ö
H
C
l I + u + uç P
a- H
÷
l P + hx H
l M + a md
l P + hx H
è
ø
IX
æ l + hx (1- a ) m m - d
hx (1- n X )a X h ö
H
C
l I + u + uç P
a- H
÷
l P + hx H
l M + m md
l P + hx H
è
ø
IX
lI + u + r
IX
lI + u + r
IX
hx (1- n X )a X
lI + u + r - u H
l P + hx H
IX
hx (1- n X )a X
lI + u + r - u H
l P + hx H
IX
hx (1- n X )a X
lI + u + r - u H
l P + hx H
IX
hx (1- n X )a X
lI + u + r - u H
l P + hx H
Equivalent tables can be generated for the equilibrium stocks of decomposer X (X*D),
plant X (X*P) and detritus C (C*M).
Fig. S2.1 presents inorganic X equilibrium levels as a function of plant C:X ratios, under
the 6 scenarios (0, I, IE, IED, IEDA and IEDAG) as calculated with the use of the parameters
11
from Table S1.2. The calculated values are similar to the values obtained through numerical
simulations (results not shown). We also checked that the levels of plant X and detritus C are of
the same order of magnitude as those reported for natural forests and shrublands in Cebrian [2].
12
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Figure S2.1: Equilibrium levels as a function of plant C:X ratios for inorganic nutrients, XI (A),
mineralization/immobilization rates (B), decomposer X, XD (C), plant X, XP (D), Detritus C, CM (E) and
the type of nutrient limiting decomposer growth – X or C (F), calculated with the parameter set from
Table S1.2. The six herbivory scenarios (I, IE, IED, IEDA and IEDAG are represented. In (B), negative
values correspond to mineralization, positive values to immobilization.
13
S2.2. Signs of the effects of herbivory nutritional processes on
equilibrium stocks:
The differences between scenarios can be used to isolate the effect on each process on the
variables of the model at equilibrium. To this purpose, the analytical expressions of two scenarios
(shown in Table S2.1) that differ only by one process can be compared. For example, the effect
of ingestion on X*I can be found by comparing X*I between the scenarios 0 and I: if the value in
the scenario I is larger, ingestion increases X*I; on the other hand; if it is smaller, ingestion
decreases the level of this stock.
Table S2.2 summarizes the signs of the effects of the five nutritional processes in isolation
on the equilibrium stocks.
Table S2.2: Signs of the effects of the nutritional processes on equilibrium inorganic X,
decomposer, and plant X stock levels and detritus C stock levels:
Ingestion
Variables:
C-limited
decomposers
X-limited
decomposers
Nutritional processes:
Excretion Defecation Differential
assimilation
X I*
- (>)
+ (<)
+
+ (>)
- (<)
X D*
-
+
+
XP*
-
+
CM*
-
+
+ (>)
- (<)
+
X I*
0
+
X D*
-
+
XP*
-
+
*
-
+
CM
Digestion
- (aX>aC)
+ (aX<aC)
-(aX>aC)
+ (aX<aC)
- (aX>aC)
+ (aX<aC)
- (aX>aC)
+ (aX<aC)
+ (>µ)
- (<µ)
0
0
0
+
-(aX>aC)
+ (aX<aC)
+
0
0
0
+
+ (aX>aC)
- (aX<aC)
0
+
+ (>µ)
- (<µ)
-
14
S3. Physiological alterations to plants by herbivores
This appendix discusses four mechanisms mediated by the physiological response of plants to
herbivory that are known to affect the effects of herbivores on nutrient availability but are not
included in our model.
S3.1. Increased root exudation following defoliation
The roots of some plants are known to increase their exudation of labile organic compounds in
response to aboveground herbivory [7]. This additional source of carbon then fuels the growth of
rhizospheric microorganisms. Eventually, mineral nitrogen availability is increased through a
complex chain of interactions that involves fine root tip elongation and protozoa [8, 9].
The universality of this mechanism, however, remains unclear. Herbivory-induced root carbon
exudation has been shown only in graminoid species so far [10, e.g., 11], though not in all of
them [12, 13]. Moreover, it is unclear whether the increase in mineral nutrient availability is
really due to the increase in carbon exudation or not. [Some, like in 14, invoke molecular
signaling by protozoans as an alternative] It is also possible that a major part of the nutrients
made available through this process comes from nitrogen exuded by the roots together with
carbon, resulting in little net gain for plants [15]. Finally, this mechanism seems to act mainly in
the short term. In the long term, herbivory generally induces a decrease in plant root biomass,
leading to lower amounts of root-derived C in the soil [7]. Given these restrictions, we decided to
defer the inclusion of this process in our model until robust generalizations are available
regarding the effects of herbivores and plant roots on rhizospheric nutrient cycling, in agreement
with Parkin et al [16].
15
Nevertheless, since this mechanism was shown to be important in several grassland ecosystems
[10, 11], we briefly introduce here a preliminary version of the model that includes additional
fluxes of organic C and X from plants to detritus. To mimic the induction of root exudation by
herbivory, we have set these two fluxes to 0 when h=0; and we set them equal to x CP (C flux)
and x XP (X flux) when h>0 (where x is the exudation rate).
Results from defoliation experiments suggest that herbivory can more than double the rate of C
root exudation [10]. Hence, for numerical calculations, we set x equal to lP (the production rate of
detritus by plants) and kept all the other parameters equal to the values used in our main study.
The simulation outputs for all the scenarios (Figure S3.1) show that this process does not
contradict the general predictions obtained from the original model (compare with figure 3), i.e.:
(i) excretion of excess X affects nutrient availability as postulated by Hobbs’ hypothesis only
when herbivores are C limited; (ii) the effects of herbivores mediated by the pool of detritus
depend on the mismatch between the nutrient content of this pool and the demand of microbial
decomposers; (iii) when both herbivores and microbial decomposers are limited by X, the effects
of herbivores do not depend on plant nutrient content.
16
" #$
%#$
! "#$%""
&#$
' #$
( #$
#$
!( #$
%$
( %$
' %$
&%$
%%$
" %$
) %$
&'( ) *"+,$"-( . / "
!' #$
!&#$
!%#$
*$
*+$
*+, $
*+, - $
*+, - , $
*$./ 0$1/ 23452$563278$
!" #$
Figure S3.1: Percent change in the equilibrium inorganic nutrient stock, XI, due to herbivory as
a function of plant C:X ratio for the various herbivory scenarios, with a version of the model
that includes a flux of C and X that emulates root exudation of organic compounds induced by
herbivory. For comparison, scenario I of the original model (no induced exud.) is also included.
This result is not surprising if one notices that the process of herbivore-induced root exudation is
very similar to the process of defecation in our model (they both lead to an increase in the
quantity of detritus). Their effects and response to plant nutrient content should thus be very
similar.
S3.2. Alteration of plant biomass allocation to root tissues
Some plants also react to herbivory by allocating more assimilates to roots [7]. However, this is a
short-term response. In the long term, herbivory shows both positive and negative effects on root
biomass [17]. Alteration of root biomass should lead to changes in the production of detritus from
dead roots, ultimately resulting in changes in nutrient availability.
17
As for the previous mechanism (herbivore-induced root exudation), the generality of this process
remains unknown [7, 17]. Moreover, empirical demonstrations of its effects on nutrient
availability are rare, if not nonexistent; consequently we decided not to include this process in our
model at this stage.
Nonetheless, it is possible to proceed by analogy and predict that effects of herbivore-induced
root-biomass alteration, if present, should be similar, or opposite, to the effects of defecation,
depending on whether herbivory increases or decreases allocation to root biomass respectively.
S3.3. Alteration of plant nutrient content
Herbivory is also known to enhance plant tissue nutrient concentrations [7, 18, but see, e.g., 19,
and the case of Abies alba in 20]. Higher nutrient contents in living tissues generally result in
more nutrient-rich litters and, eventually, higher mineralization rates [21].
Again, analogy is useful here. The effects of this process on nutrient availability should be
similar, but opposite in sign, to the effects of differential assimilation. Indeed, both processes
alter the detritus C:X ratio, but herbivore-induced plant nutrient content enhancement should lead
to a lower detritus C:X ratio, while differential assimilation of C and X by herbivores results in a
higher detritus C:X ratio.
In any case, since the effects of herbivore-induced enhancement of plant nutrient content on
nutrient availability are mediated by the pool of detritus, our predictions 2 and 3 should hold.
Hence, the effects of this process should depend on the mismatch between the nutrient content of
this pool and the demand of microbial decomposers and, when both herbivores and microbial
decomposers are limited by X, this process should not affect nutrient availability.
18
S3.4. Alteration of plant secondary compound content
Herbivory also alters the concentration of secondary chemical compounds in plants [22]. This
process seems to depend on the type of herbivore, since invertebrates generally lead to increased
concentrations, while vertebrate browsers typically decrease leaf secondary compound content
[7]. Plant litters richer in secondary compounds are harder to decompose [23].
The effects of this process should be opposite to those of the digestion process, which improves
litter quality and makes it easier to decompose. As such, it should depend in the same manner on
the mismatch between the nutrient content of detritus and the demand of microbial decomposers,
and on the type of element limiting decomposer growth (predictions 2 and 3).
S3.5. Conclusion
Although there are many examples of plant physiological responses to herbivory that alter
nutrient availability, little is known about the generality of these processes. Even less is known
about the factors that affect these processes and about the quantitative relationship between these
factors and the resulting changes in nutrient availability. This lack of knowledge makes
incorporation of these processes into mechanistic models difficult.
19
S4. First-order mineralization
This appendix presents the equations of a simplified version of our model where recycling of X
from detritus to inorganic X follows a first-order reaction. In such a model, the compartment of
decomposers is not explicitly modelled.
Table S4.1: Model equations
Producers:
Detritus:
Ingestion
Uptake
Senescence
ì
dX
ï P = uX I - hxH X P - lP X P
ï dt
í
Ingestion
Senescence
Fixation
ï dC
P
= ua X I - a hxH X P - lPa X P
ï
î dt
Loss
Mineralization
ì
Defecation
Senescence
ï dX M
C
C
= lP X P + (1- aX )hxH X P - m M - lM M
ï
m
m
í dt
Defecation
Decomposition
Senescence
Loss
ï
ï dCM = lPa X P + (1- aC )a hxH X P - mCM - lM CM
î dt
Input
Inorganic resource:
Loss
Uptake
Excretion
Mineralization
dX I
C
= I X - lI X I - uX I + (1- nX )aX hxH X P + m M
dt
m
These equations were used to generate Fig. 4C.
20
S5. Functional responses
Persistence of plant-decomposer systems, as well as nutrient levels at equilibrium, critically
depends on the shape of the nutrient uptake rates of plants and decomposers [24]. Uptake
functions generally fall within two categories: recipient-controlled functions that are proportional
to the density of consumers (examples are the law of mass action and the Michaelis-Menten
functions); and donor-controlled functions where the uptake rate is only marginally affected by
the density of consumers (e.g., first-order and ratio-dependent functions). Recipient control often
results in unstable or cyclic dynamics and competitive exclusion; while donor-controlled
functions generally result in more stable interactions. We chose donor-controlled functions for
both plants and decomposers. Donor control can result from a number of mechanisms, including
mutual interference among consumers, presence of external subsidies or spatial heterogeneity
[25, 26].
More precisely, our choice of a donor-controlled uptake function for decomposer is based on
empirical data showing a strong correlation between organic resource levels and microbial
biomasses, as predicted by donor control (but not recipient control) [27]. There is even more
justification for this choice if one considers the decomposer compartment to include both the
microbes that consume detritus (bacteria and fungi) and their predators (protists, microarthropods
and nematodes) [8]. In fact, theory shows that predators lumped with their preys behave similarly
to a compartment controlled by the resources of the prey [28]. Therefore, our decomposer
compartment should be understood to include both the saprotrophs and their predators.
As for plants, spatial heterogeneity in soils leads plants that control their local resource level to be
donor controlled at larger spatial scales [29]. In aquatic systems, however, particularly in pelagic
systems, the habitat is more homogeneous and plants are more likely to control inorganic nutrient
21
levels [30]. Therefore, we analyse here a version of the model with a recipient-controlled
Michaelis-Menten uptake function for plants (ESM, section S5).
In this version of the model, all the equations (see Table S1.1) are the same, apart from the plant
XI uptake rate, which is now set to be equal to:
u' X I
XP
Ku + XI
where u’ is the maximum uptake rate and Ku is the half-saturation constant.
At equilibrium, the inorganic nutrient XI is equal to X I* =
X I* =
lP
K in the scenario 0 and
u' -lP u
hxH + lP
Ku in the other scenarios.
u'- (hxH + lP )
These equilibrium expressions highlight the recipient-control of nutrient availability at
equilibrium, since, among all the nutritional processes of herbivory, only ingestion affects the
inorganic nutrient XI*.
In particular, XI is not affected by the plant C:X ratio, contrarily to the donor-controlled case. The
result is an effect of herbivory on nutrient availability at equilibrium that is invariant with respect
to the plant C:X ratio (Figure S5.1).
22
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' &! "
' %! "
! ""#$%""
' $! "
' #! "
' !!"
&! "
%! "
*"
*+"
*+, "
*+, - "
*+, - , "
$! "
#! "
&'( ) *"+,$"-( . / "
!"
$"
' $"
#$"
( $"
$$"
) $"
%$"
Figure S5.1: Effects of herbivory on equilibrium nutrient availability as a function of plant C:X
ratio (α) in a model with a Michaelis-Menten plant nutrient uptake function (u’=1.94 10-4 and
Ku=0.088).
Because of the recipient-control, the pattern of %ΔXI that is found in the donor-controlled
function (Fig. 3 A) is transferred to the plants (Fig. S5.2).
)$
,)$
+) $
*) $
))$
()$
' )$
! "#$%""
#$
%&' ( )"*+$",' - . "
!, #$
!+#$
!*#$
!) #$
!( #$
!' #$
!&#$
!%#$
-$
-. $
-. / $
-. / 0$
-. / 0/ $
!" #$
Figure S5.2: Effects of herbivory on equilibrium plant nutrient level (XP*) as a function of plant
C:X ratio (α) in a model with a Michaelis-Menten plant nutrient uptake function (u’=1.94 10-4
and Ku=0.088).
23
This is an indication the mechanisms that act in an ecosystem with a donor-controlled uptake of
nutrients by plants are also at work when plant uptake is recipient-controlled, but that their effects
are overridden at equilibrium by the plant control of nutrient availability. However, not all
ecosystems or experimental systems are at equilibrium. This is particularly true for many
exclosures and mesocosm experiments (the main experimental procedure used to test for the
effects of herbivory). We tested whether the donor- the recipient-controlled models would yield
similar predictions in the context of short-term exclosures and mesocosms by using transient
values instead of equilibrium values to calculate % ΔXI. Thus, we used the scenario 0 to conduct
our simulations, but starting with initial conditions equal to the equilibrium values of each of the
five other scenarios ( X I*(+herbivory)), in order to mimic the exclusion of herbivores from the
experimental settings. After a simulation time shorter than equilibrium time we recorded the level
of XI reached ( transient X I ("0")). The effect of herbivory was calculated as:
.
24
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' #! "
' !!"
&! "
%! "
$! "
*"
*+"
*+, "
*+, - "
*+, - , "
#! "
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!"
$"
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&' $
%' $
"' $
''$
) $"
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) #$
( #$
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&#$
#$
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$
(' $
)' $
&'( ) *"+,$"-( . / "
!%#$
!" #$
*$
*+$
*+, $
*+, - $
*+, - , $
Figure S5.3: Effects of herbivory on the transient nutrient level after the exclusion of herbivores
as a function of plant C:X ratio in (a) a model with a Michaelis-Menten plant nutrient uptake
function (u’=1.94 10-4 ,Ku=0.088 and stop time=2 105) and (b) the original model with a donorcontrolled plant nutrient uptake.
The profiles for % ΔXI as a function of the plant C:X ratio are qualitatively similar for the two
types of plant nutrient uptake, although the equalizing effect of plant control on nutrient levels is
already apparent in the Michaelis-Menten case. This suggests that the predictions derived from
the analysis of the donor-controlled model at equilibrium also apply to the donor- and recipientcontrolled models in a transient regime.
25
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