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APPENDIX 1 - ROBUSTNESS OF THE RESULTS TO MODEL ASSUMPTIONS
2
In this appendix, we evaluate the robustness of the main patterns predicted by the model
3
(unimodal response of species richness to soil resource availability and the associated shift in
4
species composition from slow-growing and small species to fast-growing and larger species)
5
to various aspects of the model and the simulations.
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1.1 Species abundance
7
In the model, we assume that each species is represented in the community by a single
8
individual. Here we relax this assumption and model a community comprised of 5,000
9
individuals belonging to 100 species, with the abundance of each species being randomly
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drown from a log normal distribution (mean = 2 SD=0.2, for the normal distribution). The
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patterns obtained (Fig. S1) were very similar to those obtained in our original simulations.
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Fig. S1: Effect of soil resource availability on species richness (dotted lines) and species
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composition (grey marks) under different levels of asymmetry of light exploitation (θ). The
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upper panels are the results of the simulations in which each species is represented by one
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individual. The lower panels are the results of simulations with 5000 individuals belonging to
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100 species. Species are ranked from lowest (1) to highest (100) maximal growth rate on the
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y-axes. Grey colour indicates that the relevant species was capable of persisting (maintaining
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a positive growth) at the relevant level of soil resource, white area indicates species that were
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not able to persist. All parameters are the same as in Table 2 except for the supply rates (𝑅̅ ) of
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the soil resource and light which were adjusted (i.e. multiplied) to fit the 50 fold increase in
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community size in the 5000 individuals simulation.
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1.2 Maintenance cost of light
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In the original model we assumed that all species in the community have the same
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maintenance cost for light. Here we change this assumption and model a community where
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maintenance costs of light are given by equation (5) in the main text. This alternative
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assumption implies that the trade-off between maximum relative growth rate and
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maintenance cost applies for both soil resources and light. Although this change alternates the
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outcomes of asymmetric light exploitation from total dominance by fast-growing species to
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parameter-dependent dominance (see Table 1 in the main text), differences in species
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composition were apparent only under the intermediate levels of asymmetry (Fig. S2). The
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qualitative patterns of species richness and species composition were similar to the original
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simulations (Fig. S2), except less sensitivity to asymmetry (Fig. S2, compare the intermediate
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levels of θ in the upper and lower panels).
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Fig. S2: Effect of soil resource availability on species richness (dotted lines) and species
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composition (grey marks) under different levels of asymmetry of light exploitation (θ). The
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upper panels are the results of the simulations in which maintenance costs for light were
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equal for all species. The lower panels are results of simulations where light costs varied
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among species based on equation 5 (note the different levels of θ). Species are ranked from
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lowest (1) to highest (100) maximal growth rate on the y-axes. Grey colour indicates that the
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relevant species was capable of persisting (maintaining a positive growth) at the relevant
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level of soil resource, white area indicates species that were not able persist. All other
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parameters are the same as in Table 2.
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50
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1.3 Simulation length
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In our simulations we run the model for 1,000 time steps following preliminary analyses
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showing that this length is sufficient to reach asymptotic biomass of most species (see
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Appendix 2, Fig. 8 as an example). Still, since resource availability per biomass unit
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decreases as community biomass increases and the growth response to resource availability is
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modelled using a Michaelis-Menten function (e.g. Tilman 1988, Huston and DeAngelis
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1994), longer simulations can be expected to result in extinction of more species. To evaluate
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the sensitivity of the results to such differences, we repeated our simulations using both
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shorter and longer simulations. The results indicated that increasing the number of time steps
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reduces overall species richness but does not affect the qualitative patterns obtained for
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species richness and species composition (Fig. S3).
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Fig. S3. Effect of soil resource availability on species richness (dotted lines) and species
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composition (grey marks) under different levels of asymmetry of light exploitation (θ). The
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upper panels are the results of the simulations with 1,000 time steps. The two lower panels
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are the results of simulations based on 500 time steps (middle panels) and 2,000 time steps
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(bottom panels). Species are ranked from lowest (1) to highest (100) maximal growth rate on
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the y-axes. Grey colour indicates that the relevant species was capable of persisting
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(maintaining a positive growth) at the relevant level of soil resource, white area indicates
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species that were not able to persist. All other parameters are the same as in Table 2.
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1.4 Resource consumption by plants with negative balance between resource availability
and maintenance costs
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In the simulations we assume that species with a negative balance between resource supply
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and maintenance costs do not differ from species with a positive energetic balance in their
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consumption rates. To evaluate the sensitivity of the results to this assumption we model here
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a contrasting situation in which species with a negative balance do not consume any
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resources. The qualitative patterns obtained for both species richness and species composition
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are similar to those obtained in the original simulations besides some decrease in the
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sensitivity to asymmetry (Fig. S4).
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Fig. S4. Effect of soil resource availability on species richness (dotted lines) and species
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composition (grey marks) under different levels of asymmetry of light exploitation (θ). The
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upper panels are the results of the simulations in which plants with a negative balance
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between resource supply and maintenance cost did not differ from plants with a positive
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balance in resource consumption rates. The lower panels are the results of simulations in
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which species with negative energetic balance do not consume any resources (note the
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different levels of θ). Species are ranked from lowest (1) to highest (100) maximal growth
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rate on the y-axes. Grey colour indicates that the relevant species was capable of persisting
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(maintaining a positive growth) at the relevant level of soil resource, white area indicates
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species that were not able to persist. All other parameters are the same as in Table 2.
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1.5 Distribution of maximal growth rates
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In our simulations we introduced variation in growth strategies among species by drawing
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species-specific values for maximal relative growth rates from a normal distribution. To ask
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whether this affects the predictions of the model, we repeated our simulations using values of
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maximal relative growth rates drawn from a uniform distribution with the same mean (0.05)
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and standard deviation (0.01). The patterns obtained for both species richness and species
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composition were qualitatively similar to those obtained in our original simulations (Fig. S5).
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Fig. S5. Effect of soil resource availability on species richness (dotted lines) and species
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composition (grey marks) under different levels of asymmetry of light exploitation (θ). The
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upper panels are the results of the simulations in which species-specific maximal growth rates
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(µ) were drawn from a normal distribution. The lower panels are results from simulations
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where values of maximal growth rate were drawn from a uniform distribution. Species are
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ranked from lowest (1) to highest (100) maximal growth rate on the y-axes. Grey colour
110
indicates that the relevant species was capable of persisting (maintaining a positive growth) at
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the relevant level of soil resource, white area indicates species that were not able to persist.
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All other parameters are the same as in Table 2.
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1.6 Size constraints
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In the model we assumed that, in the absence of any limitation, plant growth is exponential.
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Here we make an alternative assumption and assume that plant growth at the absence of
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resource limitation is logistic:
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(S1)
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For simplicity, we assume that K is equal for all species (K =100). As could be expected, this
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alternative assumption limits the biomass of fast growing species under high levels of the soil
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resource (Fig. S6). Still, only the fastest-growing species were sensitive to this change (Fig.
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S6), and the qualitative patterns of both species richness and species composition were
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similar to those of the original simulations (Fig. S7).
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𝑆
𝑆𝑡+1 = 𝑆𝑡 + 𝑆𝑡 ∙ 𝜇 ∙ 𝑝 ∙ (1 − 𝐾𝑡 )
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Fig. S6. Individual biomass of 100 competing species (green bars) under different levels of
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the soil resource (𝑅̅ soil) and asymmetry of light exploitation (θ). Species are ranked by their
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maximal growth rate (µ) from lowest (1) to highest (100). Note the different scales of the y-
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axes under the different resource levels. Red and blue marks at the top of a panel represent
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limitations by soil vs. light, respectively. Grey background represents persistence
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(maintaining a positive growth rate throughout the simulation) while white background
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represents species that did not persist. All other parameters are the same as in Table 2.
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Fig. S7. Effect of soil resource availability on species richness (dotted lines) and species
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composition (grey marks) under different levels of asymmetry of light exploitation (θ). The
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upper panels are the results of the 'default' simulation where plant growth at the absence of
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resource limitation was assumed to be exponential. The lower panels are results of
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simulations in which plant growth was assumed to be logistic (Equation S1, K=100). Species
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are ranked from lowest (1) to highest (100) maximal growth rate on the y-axes. Gary colour
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indicates that the relevant species was capable of persisting (maintaining a positive growth) at
145
the relevant level of soil resource, white area indicates species that were not able to persist.
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All other parameters were the same as in table 2.
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APPENDIX 2 - COMPETITION BETWEEN TWO SPECIES FOR TWO
RESOURCES
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In this appendix we present a detailed analysis of competition between two species for two
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limiting resources (a soil resource and light) in order to better understand the patterns
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observed at the community level. In particular we were interested in asking (1) what growth
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strategy along the trade-off between fast growth and low maintenance cost enables a species
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to outcompete its competitor (i.e., to suppress its growth to lower level than that required for
156
maintaining a positive growth) under high vs. low levels of the soil resource gradient, and (2)
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how the intensity of competition experienced by each species, and the average intensity of
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competition experienced by the two species, depend on the level of the soil resource. Both
159
questions are major aspect of the Grime-Tilman debate (see main text).
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To this end, we simulated the growth of two species representing contrasting growth
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strategies under two scenarios: competition for two resources where both resources are
162
exploited symmetrically, and competition for two resources where one resource (representing
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a soil resource) is exploited symmetrically and the other (representing light) is exploited
164
asymmetrically. We then examined how the outcome of competition between the two species
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depends on the level of the soil resource. As a reference we also simulated the growth of each
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species under each level of the soil resource at the absence of competitive effects. This
167
procedure allowed us to investigate how the growth strategies of competing species interact
168
with the mode of resource exploitation and with the position of the site along the resource
169
gradient, in determining the outcome of competition and its 'intensity' as often measured in
170
experimental studies (relative reduction in biomass, (Smax-S)/Smax, where Smax is the biomass
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of an individual grown without competition and S is the corresponding biomass attained
172
under competition.
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The results show that the effect of asymmetric light exploitation on the outcome of
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competition is influenced by the level of the soil resource (Fig. S8). Under low level of the
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soil resource, the slow-growing species outcompetes the fast-growing species independently
176
of the mode of light exploitation. Although at early stage of the growth the fast-growing
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species is able to attain a larger size than the slow-growing one, when the plants get larger,
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the amount of soil resource available per unit biomass is reduced, and maintenance cost
179
becomes the main factor limiting the growth of the two species. At that stage, the growth rate
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of the fast-growing species is reduced below that of the slow-growing species (due to its
181
higher maintenance cost), the size difference is reversed, and at some stage (indicated by
182
asterisk) the amount of soil resource available for the fast-growing species drops below the
183
level required for its maintenance (Fig. 8).
184
Under a high level of the soil resource, the outcome of competition depends on the mode
185
of light exploitation. Under asymmetric light exploitation the fast-growing species is capable
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of outcompeting the slow-growing species by reducing light availability to levels that are
187
lower than those required for its maintenance (Fig. 8). Under symmetric exploitation the fast-
188
growing species is still able to attain a larger size due to its faster growth rate. However, since
189
(1) competition is predominantly for light, (2) both species get the same amount of light per
190
unit biomass, and (3) the maintenance cost of light is the same for both species, the two
191
species continue to grow in a diminishing rate and neither species is able to outcompete its
192
competitor (Fig. 8). This scenario is equivalent to competition between two species with the
193
same R* in Tilman's resource competition model (see also table 1 in the main text).
194
The results further show that, when competition is asymmetric and resource level is high, the
195
size difference between the two species increases dramatically: while biomass of the fast-
196
growing species is similar to its biomass without competition, the biomass of the slow-
197
growing species is much lower (compare upper and lower panels in Fig. S8).
198
Figure S9 shows the effect of soil resource availability on final biomass of individual plants
199
with contrasting growth strategies (a fast-growing and a slow-growing species) when grown
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with and without competition (upper panels), and the resulting variation in relative
201
competition intensity (bottom panels). Several important patterns emerge from this analysis.
202
First, increasing the level of the soil resource reverses the differences in biomass between
203
individuals of the two species (upper panels). Second, asymmetric competition increases the
204
magnitude of the differences in biomass between the two species but this effect is limited to
205
relatively high levels of the soil resource when the limiting factor is light (compare upper left
206
and upper right panels). Third, increasing the level of the soil resource increases the
207
competitive effect of the fast-growing species on the slow-growing species, but reduces the
208
competitive effect of the slow-growing species on the fast-growing species (bottom panels).
209
As a result, the average intensity of competition does not change along the gradient. While
210
limited to a particular measure of competition intensity, these overall results support Tilman's
211
view that increasing productivity causes a shift from belowground competition to
212
aboveground competition, but does not necessarily affect the average intensity of
213
214
competition.
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217
Fig. S8. Individual biomass of a fast-growing species (red, µ = 0.055) and a slow-growing
218
species (blue, µ = 0.045) competing for a soil resource and light, under different levels of the
219
soil resource (𝑅̅ soil) and asymmetry of light exploitation (θ). Continuous line – with
220
competition, dashed line - without competition. Stars represent the time at which a species
221
stops to grow due to negative balance between the amount of resource available per unit
222
biomass and the maintenance cost. All parameters are the same as in table 2 except for the
223
resource supply rates (𝑅̅ ) which were adjusted (i.e. divided) for the 50-fold decrease in
224
community size in the two individuals simulation compared with the 100 species simulation.
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229
Fig. S9. Effect of soil resource availability (𝑅̅ soil) on final plant biomass (top panels) and
230
competition intensity (bottom panels) under different levels of asymmetry of light
231
exploitation (θ). Competition intensity is the relative reduction in biomass under competition.
232
Continuous line – with competition, dashed line - without competition, red line - fast-growing
233
species (µ = 0.055), blue line - slow-growing species (µ = 0.045), black line - average
234
competition intensity. All parameters are the same as in table 2 except for the resource supply
235
rates (𝑅̅ ) which were adjusted (i.e. divided) for the 50 fold decrease in community size in this
236
two individuals simulation compared with the 100 species simulation.
237
238
239
REFRENCES
240
Huston, M.A. & De Angelis, D.L. (1994) Competition and coexistence - the effects of
241
resource transport and supply rates. American Naturalist, 144, 954-977.
242
Tilman, D. (1988) Plant strategies and the dynamics and structure of plant communities.
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Princeton University Press, Princeton, New Jersey, USA. 360.
244