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
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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: /
Inorganic resource: /
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.
Table S1.2: Symbol definitions
Class / Symbol / Definition / Values / Units / Ref.Variables / XP / X stock in plants / g.m-2
CP / Carbon stock in plants / g.m-2
XD / X stock in decomposers / g.m-2
CD / C stock in decomposers / g.m-2
XM / X stock in detritus / g.m-2
CM / C stock in detritus / g.m-2
XI / Stock of inorganic X / g.m-2
Stoichiometric / α / C:X ratio of plants / varies / g.g-1 / Driving factor
parameters / b / C:X ratio of decomposers / 7.37 / g.g-1 / Cleveland & Liptzin 2007
m / C:X ratio of detritus / varies / g.g-1 / Function of α
g / C:X ratio of herbivores / 5.49 / g.g-1 / Elser et al 2000
φ / C:X ratio of detritus from herbivores / varies / g.g-1 / Function of α
δ / Decomposer C:X TER / 24.57 / g.g-1 / Calculated
h / Herbivore C:X TER / 10.14 / g.g-1 / Calculated
Ecosystem / u / Uptake rate of XI by plants / 0.34 / day-1 / Barber 1995
parameters / r / Uptake rate of XI by XI -limited decomposers / 0.09 / day-1 / Lovett & Ruesink 1995
a / Uptake rate of plant detritus by CM-limited decomposers / 1.6 10-3 / day-1 / Cebrián 1999
j / Uptake rate of herbivore detritus by CM-limited decomposers / 0.008 / day-1 / Lovett & Ruesink 1995
m / Uptake rate of total detritus by CM-limited decomposers / varies / day-1 / Calculated
c / Decomposer gross growth efficiency for C / 0.3 / dim. / Moore et al 2005
lP / Production rate of detritus by plants / 4.8 10-6 / day-1 / Cebrián 1999
lD / Loss rate of decomposers from ecosystem / 3.3 10-3 / day-1 / Hunt et al 1987
lM / Loss rate of detritus from ecosystem / 8.4 10-4 / day-1 / Cebrián 1999
lI / Loss rate of XI from ecosystem / 3 10-4 / day-1 / Christenson et al 2002
IX / Supply rate of XI / 0.03 / g. m-2.day-1 / Chapin et al 2002
Herbivore / xH / X stock in herbivores / 0.3 / g.m-2 / Cebrián 1999
parameters / h / Ingestion rate of producers by herbivores / 3 10-5 / (g.m-2)-1.day-1 / Cebrián 1999
aX / Herbivore assimilation efficiency for X / 0.7 / dim. / Carisey & Bauce 1997
aC / Herbivore assimilation efficiency for C / 0.6 / dim. / Karasov & Martínez del Rio 2007
nX / Herbivore net growth efficiency for X / varies / dim. / Calculated
nC / Herbivore net growth efficiency for C / varies / dim. / Calculated
nXmax / Maximum nX / 0.95 / dim. / Carisey & Bauce 1997
nCmax / Maximum nC / 0.6 / dim. / Carisey & Bauce 1997
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 , and the uptake rate of detritus by decomposers is .
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.
- C and X lost from decomposers (and) 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 (~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 δμ, the mineralization/immobilization flux is negative, and hence decomposers mineralize the inorganic nutrient XI. In contrast, when δμ, the mineralization/immobilization flux is positive, and decomposers immobilize the inorganic nutrient XI (Figure 2, main text).
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 and only on XI availability when .
S1.2.4 Herbivores
The C:X threshold elemental ratio (TER) for herbivores is , where γ is the C:X ratio of herbivores, is their maximal net growth efficiency for C and 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 , while in the second the net growth efficiency for X .
Herbivores need to keep their elemental composition constant, i.e., , and hence the 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 . This equation is valid when (where η is TERH), yielding (remember that and when ). Combining the two preceding equalities yields . Therefore, when herbivore growth is C-limited (αη and ), then . In contrast, when herbivore growth is X-limited (αη and ) then .
The C:X ratio of the herbivore-produced detritus φ is equal to the plant C:X ratio, corrected by the assimilation efficiencies: ( and are the fractions of ingested C and X respectively that are not assimilated).
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 valuesC-limited decomposers / , , ,
X-limited decomposers / , ,
,
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:
Two eigenvalues are equal respectively to –(lM+m) and –lD. The two other eigenvalues are solutions of the equation .
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 C, i.e. . Using Table S1.3, the condition becomes .
S1.3.2 X-limited decomposer equilibrium:
Two eigenvalues are equal respectively to –lM and –lD. Two other eigenvalues are negative solutions of the equation .
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. . Using Table A3, the condition becomes . One can check easily that this condition is sufficient to guarantee that in Table S1.3.
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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
CM-limited decomposers / XI-limited decomposersScenario:
0 / /
I / /
IE / /
IED / /
IEDA / /
IEDAG / /
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 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].
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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.
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.