1
Appendix S1: Model parameterization
2
In this Appendix we explain how the model parameter values were derived from
3
previous empirical research.
4
The maximum growth rate of Phalaris was derived from a study by Taub
5
(2002), in which a maximum growth rate of 0.25 day-1 was reported under a high
6
nitrogen treatment (daily application of a modified Johnson solution containing 5mM
7
NH4-NO3). This growth rate was the highest found among several studies that
8
analyzed Phalaris growth (Grime & Hunt 1975; Danais 1986; Brix & Sorell 1996;
9
Taub 2002). In this study, we assumed that the same maximum growth rate applies
10
for conditions of light limitation. Also, we assumed equal maximum growth rates for
11
Carex. The mortality rate of 0.01 day-1 was based upon an observed lifespan of
12
Phalaris leaves of 96 days (Ryser & Urbas 2000). We assumed a lower mortality rate
13
for Carex, in order to enable a situation of coexistence between the plant species.
14
Observations of the lifespan of other Carex species suggest that a leaf lifespan of
15
200 days is a reasonable assumption for Carex species (Aerts & De Caluwe 1995),
16
yielding a mortality rate of 0.005 day-1.
17
For the nitrogen content of Phalaris shoots, a range of 7-14 mg.g-1 has been
18
reported (McJannet, Keddy & Pick 1995; Taub 2002). A more detailed analysis over
19
a growing season, however, revealed that nitrogen content can be much higher
20
during certain periods (30-60 mg.g-1, Kätterer, Andrén & Pettersson 1998). Therefore,
21
we assumed a value slightly higher than the reported range of 15 mg.g-1. We
22
assumed the same nitrogen content of Carex shoots.
23
The rooting depth of Phalaris was derived from a study by Kätterer & Andrén
24
(1999), who measured nitrogen uptake in the upper 1 m of soils occupied by
25
Phalaris. They measured little nitrogen uptake deeper than 80 cm (Kätterer & Andrén
26
(1999). The density of the soil was taken from a study by Hansel et al. (2002), who
27
reported a dry bulk density of 530 kg.m-3 for Phalaris wetland soil. For decomposition
28
rates of litter in wetlands, a large range can be found in literature. The value taken in
29
this study, 0.003 day-1, is of the same order of magnitude as has been reported for
1
30
Phragmites and Typha communities and for riparian herbaceous vegetation (Boyd
31
1970; Mason & Bryant 1975; Brinson 1981; Hefting et al. 2005).
32
The recycling parameter in the model determines what fraction of nitrogen in
33
litter becomes available for plants during decomposition. For five sites with riparian
34
herbaceous vegetation, Hefting et al. (2005) reported a range of 0.37-1, with three of
35
the five sites within a range of 0.66-0.72. Therefore, we assumed a value of 0.7 for
36
the recycling parameter.
37
The values of the light interception parameters were based upon a study by
38
Herr-Turoff & Zedler (2007), who reported that Phalaris growing at a density of 500
39
g.m-2 reduced light availability by 90%. Parameters were chosen so that similar
40
reductions (~ 94-95% at 500 g m-2) were reached in our model. Further, the light
41
interception coefficient for Phalaris, was set slightly higher than for Carex, which
42
agrees with the observation that Phalaris has a larger Leaf Area Index (LAI) than
43
some Carex species (Danais 1986). The light interception coefficients for litter were
44
set lower to mimic different light absorbing characteristics of litter as compared to
45
living biomass (see Appendix S2 for details). In general, litter can be expected to limit
46
light availability for young plants, not adults (Violle, Richarte & Navas 2006).
47
The nutrient turnover parameter regulates the biomass of both plant species
48
in equilibrium. The value of this parameter was chosen in a way that the modelled
49
biomass of Phalaris and Carex for the intermediate nutrient-high light treatment was
50
within the range reported by Perry & Galatowitsch (2004) and Perry, Galatowitsch &
51
Rosen (2004). The values of the nutrient and light supply parameters correspond to
52
the treatment levels in the same studies. Finally, the saturation constants that
53
determine the growth curves of the plant species were calibrated as explained in the
54
main text.
55
56
References
57
58
59
Aerts, R. & De Caluwe, H. (1995) Interspecific and intraspecific differences in shoot
and lifespan of four Carex species which differ in maximum dry matter
production. Oecologia, 102, 467-477
2
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
Boyd, C.E. (1970) Chemical analyses of some vascular aquatic plants. Archiv für
Hydrobiologie, 67, 78-85.
Brinson, M.M. (1981) Primary productivity, decomposition and consumer activity in
freshwater wetlands. Annual Review of Ecology and Systematics, 12, 123–161.
Brix, H. & Sorrell, B. K. (1996) Oxygen stress in wetland plants: comparison of deoxygenated and reducing root environments. Functional Ecology, 10, 521–526.
Danais, M. (1986). The influence of some environmental factors on the production of
Carex vesicaria and Phalaris arundinacea. Vegetatio, 67, 45-56.
Daufresne, T. & Hedin, L.O. (2005) Plant coexistence depends on ecosystem
nutrient cycles: extension of the resource-ratio theory. Proceedings of the
National Academy of Sciences USA, 102, 9212–9217.
Grime, J.P. & Hunt, R. (1975) Relative growth rate: its range and adaptive
significance in a local flora. Journal of Ecology, 63, 393–422.
Hansel, C.M., La Force, M.J., Fendorf, S. & Sutton, S. (2002) Spatial and temporal
association of As and Fe species on aquatic plant roots. Environmental Science
and Technology, 36, 1988-1994.
Hefting, M.M, Clement, J.C., Bienkowski, P., Dowrick, D., Guenat, C., Butturini, A.,
Topa, S., Pinay, G., & Verhoeven, J.T.A. (2005) The role of vegetation and litter
in the nitrogen dynamics of riparian buffer zones in Europe, Ecological
Engineering, 24, 465–482.
Herr-Turoff, A., & Zedler, J.B. (2007) Does morphological plasticity of the Phalaris
arundinacea canopy increase invasiveness? Plant Ecology, 193, 265–277.
Kätterer, T. & Andrén, O. (1998) Growth dynamics of reed canarygrass ( Phalaris
arundinacea L.) and its allocation of biomass and nitrogen below ground in a
field receiving daily irrigation and fertilization. Nutrient Cycling in
Agroecosystems, 54, 21-29.
Kätterer, T., Andrén, O. & Pettersson, R. (1998) Growth and nitrogen dynamics of
reed canarygrass (Phalaris arundinacea L.) subjected to daily fertilisation and
irrigation in the field. Field Crops Research, 55, 153–164.
Mason, C.F. & Bryant, R.J. (1975) Production, nutrient content and decomposition of
Phragmites Communis Trin. and Typha Angustifolia L. Journal of Ecology, 63,
71-95.
McJannet, C. L., Keddy, P. A. & Pick, F.R. (1995) Nitrogen and phosphorus tissue
concentrations in 41 wetland plants: A comparison across habitats and
functional groups. Functional Ecology, 9, 231-238.
Perry, L.G. & Galatowitsch, S.M. (2004) The influence of light availability on
competition between Phalaris arundinacea and a native wetland sedge. Plant
Ecology, 170, 73-81.
Perry, L.G., Galatowitsch, S.M. & Rosen, C.J. (2004) Competitive control of invasive
vegetation: a native wetland sedge suppresses Phalaris arundinacea in carbonenriched soil. Journal of Applied Ecology, 41, 151-162.
Ryser, P. & Urbas, P. (2000) Ecological significance of leaf life span among Central
European grass species. Oikos, 91, 41-50.
Taub, D.R. (2002) Analysis of interspecific variation in plant growth responses to
nitrogen. Canadian Journal of Botany, 80, 34-41.
Violle, C., Richarte, J, & Navas, M.-L. (2006) Effects of litter and standing biomass on
growth and reproduction of two annual species in a Mediterranean old-field.
Journal of Ecology, 94, 196–205.
109
3
110
Appendix S2: Modelling light interception
111
112
In this Appendix, we explain the assumptions that were made to model light
113
interception by living biomass and litter.
114
115
The total amount of light that a monoculture of one plant species intercepts
can be modelled by Lambert-Beer’s law:
116
117
118
In which x1 is the height of the vegetation canopy relative to the soil surface, which is
119
depicted by x2. Lx1 and L0 the light availability at the top of the vegetation canopy, Lx2
120
is the light availability at the soil surface and LAD is the shoot density of the plant
121
species. Equation (1) shows that the amount of light intercepted is influenced by two
122
specific plant properties: the height of the vegetation (x1-x2) and the leaf density of
123
the plant species (LAD). To enable analytical tractability in ordinary differential
124
equation models, equation (1) can be approximated by a hyperbolic function of
125
biomass, summarizing the two plant properties into a single parameter (e.g.
126
Reynolds and Pacala 1993):
127
128
129
In which αB is a proportionality constant converting biomass to light-intercepting leaf
130
area. Differences in growth strategies between plant species can thus be included to
131
some extent by varying the values of the proportionality constants. In our example,
132
Phalaris generally grows higher than Carex, and has a larger leaf area index than
133
some Carex species (Danais 1986). As a result, a gram of Phalaris biomass will
134
generally intercept more light as compared to a gram of Carex biomass. This can be
135
reflected by a higher value of the proportionality constant of Phalaris.
136
The Lambert-Beer equation is generally applicable to any light absorbing
137
material (e.g. Grace and Woolhouse 1973), and can thus also be used to model light
138
interception by plant litter. Litter, however, will have a higher density and will have a
139
lower height to mass ratio than living aboveground biomass. In other words, litter is
4
140
relatively effective in intercepting light near the soil surface, but will not intercept
141
much light higher up in the canopy. The latter effect likely dominates because light
142
availability is rapidly decreasing within a canopy. Hence, the proportionality constants
143
for litter are set lower than those for the aboveground biomass of the plant species.
144
It should be noted however, that the above approach is a mean field
145
approximation: there is no explicit hierarchy in light interception between plant
146
species and litter. Also, it is assumed that the amount of light that is not intercepted
147
can equally stimulate biomass growth at any height in the vegetation canopy. On the
148
other hand, it also means that even a shallow litter layer will limit biomass growth,
149
because this litter will limit light availability for young and new stems. Note that our
150
approach considers biomass density as a whole, implicitly considering both older
151
(taller) stems as well as younger (shorter) stems.
152
Alternatively, Perry, Neuhauser and Galatowitsch (2003) developed an
153
analytically tractable model that included a hierarchy of plants within the canopy. In
154
their model, the biomass distribution of different plant species was simplified, in that
155
all the leaf biomass was assumed to grow at the top of the stems and stem height
156
was assumed to be proportional to biomass (Perry, Neuhauser and Galatowitsch,
157
2003). This created the possibility to model asymmetric competition for light, because
158
the species with highest biomass occupied the position highest in the canopy,
159
meaning that this species had the first access to light.
160
Similar to results from classical resource competition theory (Tilman 1982),
161
communities tended to develop toward species with lower light requirements (Perry,
162
Neuhauser and Galatowitsch, 2003). However, arrested succession could occur if
163
species with relatively high light requirements established that left little light available
164
for other plant species (Perry, Neuhauser and Galatowitsch, 2003). These model
165
results suggest that Phalaris, which generally grows taller than Carex as noted
166
above, may more quickly replace Carex under light-limited conditions than would be
167
predicted by a mean field approach. Thus, in comparison, our mean field approach
168
may be more conservative.
5
169
References
170
171
172
173
174
175
176
177
178
179
180
181
Danais, M. (1986). The influence of some environmental factors on the production of
Carex vesicaria and Phalaris arundinacea. Vegetatio, 67, 45-56.
Grace, J. & Woolhouse, H.W. (1973) A physiological and mathematical study of the
growth and productivity of a Calluna-sphagnum community. II. Light interception
and photosynthesis in Calluna. Journal of Applied Ecology, 10, 63-76.
Perry, L.G., Neuhauser, C. & Galatowitsch, S.M. (2003) Founder control and
coexistence in a simple model of asymmetric competition for light. Journal of
Theoretical Biology, 222, 425-436.
Tilman, D. (1982) Resource competition and community structure. Princeton
University Press, Princeton.
6
182
Appendix S3: Analytical justification of the results
183
In this Appendix, we provide an analytical analysis of the model, corresponding to the
184
graphical analyses presented in the main text.
185
The model comprises the following set of equations:
186
187
188
189
190
191
192
193
194
195
In the following, we will analytically analyze different topics that were analyzed
196
graphically in the main text.
197
198
Effect of increasing growth rate on the existence of a coexistence equilibrium
199
Effects of increasing growth rate in the model are similar as reported in classical
200
competition theory (Tilman 1982). Equation 1 shows that equilibrium of the plant
201
species (dBi/dt = 0) requires that plant losses through mortality are compensated by
202
growth of new plant material. If a plant species is limited by light, there is a specific
203
level of light where this requirement is met:
204
205
206
In case of limitation by soil nitrogen, the specific level of nutrients can be derived in
207
similar vein, yielding:
208
209
210
Note that equations 5 and 6 give the analytical expressions of the classical R* values
211
(Tilman 1982). Coexistence between plant species can occur if both species are
7
212
limited by a different resource at equilibrium. Assuming that Carex is limited by light,
213
Phalaris by soil nitrogen and using equations 5 and 6, the requirements become:
214
215
216
217
218
219
As shown in Fig. 3 of the main text, increasing the growth rates of Phalaris (gN,Ph or
220
gL,Ph) decreases the left hand side of equation 7 and the right hand side of equation
221
8. The requirements for coexistence may no longer hold if the increase in growth rate
222
is large enough (see below).
223
224
The effect of the C:N ratio of plant tissue on litter pools
225
The equations describing litter dynamics are relatively simple. At equilibrium, litter
226
density is given by:
227
228
229
Equation 9 shows that litter density at equilibrium is directly proportional to living
230
biomass. The litter:biomass ratio is determined by the first term on the right hand side
231
of equation 9. This term shows that litter density increases with decreasing
232
decomposability of litter. An increase in litter C:N ratio corresponds to a decrease in
233
the qN,i parameters in the model. Hence, an increase in C:N ratio decreases the
234
numerator of the first term on the right hand side of equation 9, and increases the
235
litter density at equilibrium.
236
237
The effects of litter feedbacks on biomass densities
238
The expressions of the biomass equilibria in terms of parameters leads to
239
expressions that are too cumbersome to display. However, the effect of litter
240
feedbacks on biomass equilibria can be shown relatively easily. First, consider the
241
model system (equations 1-4) without feedbacks being at equilibrium. Then, the
242
nutrient dynamics equation can be written as:
8
243
244
245
Where the hats indicate the equilibrium values of the state variables in the system
246
without litter-nutrient feedbacks. If at this point the nutrient-litter feedbacks are
247
switched on, equation 10 turns into:
248
249
250
Because all parameters are defined to be positive, the introduction of nutrient-litter
251
feedbacks decreases net uptake by plant species by a factor dependent on the
252
nutrient-litter feedback coefficients
253
Daufresne & Hedin (2005). As a result, there will be an increase in the nutrient
254
concentration in soil. From equation 1 it follows that higher nutrient availability will
255
increase growth of the Phalaris only:
and litter decomposability, see also
256
257
And from equation 9 follows:
258
259
Where the subscripts FB indicate equilibrium densities in the system with litter
260
feedbacks. Increased growth of Phalaris will also decrease light availability in the
261
system:
262
263
264
265
However, from equations 1,5 and 8, it follows that the coexistence equilibrium
266
requires:
267
268
Which can only be achieved if:
269
270
And hence:
271
9
272
From equation 8 it also follows that nutrient-litter feedbacks will lead to competitive
273
exclusion of Carex if:
274
275
Note that light-litter feedbacks have a similar effect, but start from a decrease in light
276
availability (due to the light absorption of litter). This requires Carex to decrease in
277
density, to restore required pre-feedback light levels for equilibrium (which follows
278
from equation 1 and 15), and this will leave more nutrients to consume for the
279
nutrient-limited species. In our model parameterization, the light interception
280
coefficients for litter were set lower than for living biomass (see Appendix S2). As a
281
result, the indirect positive effect on Phalaris due to light-litter feedbacks was smaller
282
than the direct positive effect of nutrient-litter feedbacks.
283
284
Stability of the coexistence equilibrium
285
Equations 7 and 8 already showed requirements for coexistence between plant
286
species, but these requirements are not sufficient (Tilman 1982). Another
287
requirement is that at equilibrium, the nutrient-light consumption ratio of the nutrient-
288
limited species is higher than the supply ratio, and the nutrient-light consumption ratio
289
of the light-limited species is lower than the supply ratio (Tilman 1982). In analytical
290
form, the requirements read:
291
292
293
294
295
Where the left hand sides indicate the consumption ratios of Carex and Phalaris and
296
the right hand sides the nutrient supply ratios. From equations 7,8,19 and 20 it can
297
be seen that an increased growth rate of Phalaris can violate both types of
298
requirements for coexistence. First, the coexistence equilibrium may disappear and
299
the system may develop toward an equilibrium with only Phalaris being present (as
300
shown in Fig. 3 of the main text). From equation 7 it follows that this occurs when:
10
301
302
303
The right hand side of equation 21 gives the critical growth rate above which Phalaris
304
excludes Carex within its climatic niche (see Scenario 2 in the main text).
305
Alternatively, Phalaris may invade regions outside its climatic niche when its R* value
306
for light is lower than that of the native species (see Scenario 2 in the main text).
307
From equation 8 it follows that this occurs when:
308
309
310
The coexistence equilibrium can also be destabilized by a change in C:N ratio of
311
plant tissue. The C:N ratio of plant tissue is determined by the parameters qN,i. As
312
can be seen in equations 7 and 8, changes in C:N ratio do not alter the resource
313
levels present in the coexistence equilibrium. However, equations 19 and 20 show
314
that they do alter the consumption vectors of the plant species. As shown in the
315
graphical analysis in the main text, the coexistence equilibrium destabilizes when the
316
nutrient:light consumption ratio of Phalaris becomes smaller than that of Carex.
317
Expressing the left hand sides of equations 19 and 20 in parameters only, this
318
situation occurs when:
319
320
321
In case of an increasing C:N ratio of Phalaris, equation 23 can be used to derive a
322
critical value of the nutrient content of Phalaris tissue:
323
324
325
326
327
328
329
330
(eqn 24)
Note that nutrient content of Phalaris tissue is inversely related to Phalaris’ C:N ratio.
Equation 24 includes the litter feedback coefficients. Hence the critical C:N
331
ratio is being affected by the presence or absence of litter feedbacks. Equation 24 is
332
used to calculate critical C:N ratios in scenario 3 in the main text, where litter
11
333
feedbacks were included. For scenario 2, in which litter feedbacks were not included,
334
equation 24 reduces to:
335
336
Finally, the main text considered a situation where the system shifted from a state of
337
Phalaris exclusion into a regime of alternative stable states. This shift requires the
338
slope of Phalaris’ nutrient-light consumption ratio to become smaller than the
339
nutrient-light supply ratio, meaning that equation 20 is no longer fulfilled. This occurs
340
when:
341
342
In which:
343
344
345
346
347
Again, equation 26 is an expression including the litter feedback coefficients. Hence
348
the critical C:N ratio is being affected by the presence or the absence of litter
349
feedbacks. Without litter feedbacks equation 26 reduces to:
350
351
352
The critical nutrient content of Phalaris tissue at which the competitive outcome of the
353
system changes depends on which of the equations (24 or 26 in the presence of litter
354
feedbacks, 25 or 27 in the absence of litter feedbacks) provides the weakest
355
constraint (i.e. smallest evolutionary change).
356
357
References
358
359
360
361
362
363
Daufresne, T. & Hedin, L.O. (2005) Plant coexistence depends on ecosystem
nutrient cycles: extension of the resource-ratio theory. Proceedings of the
National Academy of Sciences USA, 102, 9212–9217.
Tilman, D. (1982) Resource competition and community structure. Princeton
University Press, Princeton.
12
364
Appendix S4: Model sensitivity and robustness
365
In this Appendix we performed two types of sensitivity analysis. First, we analyzed
366
the sensitivity to small changes in model parameter values (local sensitivity) using an
367
elasticity analysis. Subsequently, we analyzed the robustness of the model results by
368
quantifying for each model parameter how much its value could be changed without
369
qualitatively altering the model result.
370
371
Methods to calculate local sensitivity
372
In the local sensitivity analysis we calculated for each parameter how a change in the
373
parameter value would affect the equilibrium values of the six state variables
374
(biomass density for two plant species, litter density for two plant species, nutrient
375
availability and light availability). We focused the analysis on an equilibrium with both
376
plant species having nonzero densities (referred to as the ‘coexistence equilibrium’).
377
The method is known as an elasticity analysis (e.g. Eppinga et al. 2009):
378
379
In which S denotes a state variable and p a parameter. The hat on S indicates that
380
this is the equilibrium value of the state variable as a function of parameters. The
381
second term on the right hand side of equation 1 indicates the change in the state
382
variable per unit change in the parameter. The first term on the right hand side of
383
equation (1) standardizes the outcome from an absolute into a relative sensitivity.
384
Relative sensitivities enabled comparisons between parameters, despite the large
385
variation in absolute parameter values (Eppinga et al. 2009). Further, for each
386
variable we rescaled sensitivities to the maximum sensitivity observed, meaning that
387
all sensitivities ranged between -1 and 1.
388
389
Results local sensitivity
390
We found that the plant biomass and litter variables were most sensitive to changes
391
in the litter feedback coefficients, especially the nutrient-litter feedback coefficients
13
392
Table D1). This result stressed the potential importance of litter feedbacks for plant
393
competition. These variables were also relatively sensitive to changes in soil nitrogen
394
and light supply (Table D1). Finally, these variables were more sensitive to changes
395
in Carex’ light uptake parameters rather than Phalaris’ nitrogen uptake parameters
396
(Table D1). This agreed with the notion that Phalaris’ competitiveness increases
397
because of nitrogen release through litter decomposition.
398
The equilibrium levels of nitrogen and light availability only depended on three
399
uptake parameters of the plant species that was limited by the particular resource
400
(see Appendix S3). Nitrogen availability was most sensitive to the uptake saturation
401
coefficient of Phalaris (Table D1). Light availability was found to be similarly sensitive
402
to all three parameters.
403
404
405
406
14
407
408
409
Table S4-1: Local sensitivity of the model obtained by an elasticity analysis, most
sensitive parameters are depicted in bold
Symbol Interpretation
Maximum growth rate of Carex under light
gL,C
limitation
Maximum growth rate of Phalaris under
gL,Ph
light limitation
Light availability at which Carex reaches
kL,C
half its maximal growth rate (if light limited)
Light availability at which Phalaris reaches
kL,Ph
half its maximal growth rate (if light limited)
Maximum growth rate Carex under nitrogen
gN,C
limitation
Maximum growth rate Phalaris under
gN,Ph
nitrogen limitation
Nitrogen availability at which Carex reaches
kN,C
half its maximal growth rate (if N limited)
Nitrogen availability at which Phalaris
kN,Ph
reaches half its maximal growth rate (if N
limited)
Mortality rate Carex
mC
Mortality rate Phalaris
mPh
Turnover rate of nutrient supply
a
Nitrogen availability in absence of plants
S
Nitrogen content of tissue of Carex
qN,C
Nitrogen content of tissue of Phalaris
qN,Ph
Soil bulk density
ρ
Rooting depth of plant species
lRoot
Nitrogen content of Carex litter at which it
QN,C
decomposes at rate dC
Nitrogen content of Phalaris litter at which it
QN,Ph
decomposes at rate dP
Nutrient-litter feedback coefficient Carex
αN,C
Nutrient-litter feedback coefficient Phalaris
αN,Ph
Carex litter decomposition rate
dC
Phalaris litter decomposition rate
dPh
Light supply rate
L0
Light interception coefficient Carex
γL,C
Light interception coefficient Phalaris
γL,Ph
Light-litter feedback coefficient Carex
αL,C
Light-litter feedback coefficient Phalaris
αL,Ph
B1
B2
D1
D2
N
L
0,89
-0,80
0,89
-0,80
0
-1
0
0
0
0
0
0
-0,87
0,79
-0,87
0,79
0
0,98
0
0
0
0
0
0
0
0
0
0
0
0
-0,13
0,13
-0,13
0,13
-0,10
0
0
0
0
0
0
0
0,13
-0,63
0,31
-0,81
-0,94
0,60
0,58
-0,81
-0,81
-0,13
0,53
-0,33
0,81
0,94
-0,58
-0,56
0,81
0,81
0,13
-0,58
0,31
-0,81
-0,94
0,55
0,58
-0,81
-0,81
-0,13
0,53
-0,30
0,81
0,94
-0,58
-0,60
-0,73
-0,73
1
0
0,10
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
-0,17
0,15
-0,12
0,15
0
0
-0,20
-1
-0,88
0,17
0,20
0,87
-0,30
-0,18
-0,17
-0,20
0,18
1
0,88
-0,15
-0,18
-0,79
0,28
0,16
0,15
0,18
-0,20
-1
-0,88
0,12
0,20
0,87
-0,30
-0,18
-0,17
-0,20
0,22
1
0,88
-0,15
-0,22
-0,79
0,28
0,16
0,15
0,18
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
410
411
412
413
15
414
Methods to calculate robustness of the model results
415
The elasticity analysis examined the effect of a change in a parameter value on the
416
coexistence equilibrium state of the model. To assess the robustness of the model
417
results, we also analyzed for each parameter the range in which this parameter could
418
be changed without leading to a qualitatively different model outcome (with a
419
maximum parameter variation of one order-of-magnitude smaller and larger, cf.
420
Eppinga et al. 2006). The default parameterization included all litter feedbacks and
421
yielded a range of nitrogen and light supplies under which coexistence between the
422
two plant species was possible. A qualitative change in model outcome occurred
423
when the parameter region of coexistence vanished entirely, and a parameter region
424
of so-called founder control (Bolker, Pacala & Neuhauser 2003) emerged instead.
425
426
Results model robustness
427
Almost all parameters could be varied over more than an order of magnitude without
428
qualitatively altering the model outcome (Table D2). This suggested that our findings
429
were not very dependent on specific parameter settings, and thus relatively robust.
430
However, the only exceptions were the plant mortality parameters (Table D2).
431
Relatively small changes in these parameters altered the model results (Table D 2).
432
However, this does not mean that coexistence between plant species was only
433
observed in this small range of plant mortality. It is important to note that the effect of
434
mortality rates was closely related to the plant growth rates (see Appendix S2 for
435
details). For mortality rates outside the parameter region indicated in Table D2, plant
436
coexistence still occurred, but for different growth rates. Thus, although the indicated
437
mortality ranges were relatively narrow, there was a broad range of growth-mortality
438
characteristics under which plant coexistence could occur.
439
16
440
441
442
Table S4-2: Robustness of the model results as indicated by the sensitivity range of
the model parameters
Symbol Interpretation
gL,C
gL,Ph
kL,C
kL,Ph
gN,C
gN,Ph
kN,C
kN,Ph
mC
mPh
a
qN,C
qN,Ph
ρ
lRoot
QN,C
QN,Ph
αN,C
αN,Ph
dC
dPh
γL,C
γL,Ph
αL,C
αL,Ph
443
444
445
446
447
448
449
450
451
452
Maximum growth rate of Carex under light
limitation
Maximum growth rate of Phalaris under
light limitation
Light availability at which Carex reaches
half its maximal growth rate (if light limited)
Light availability at which Phalaris reaches
half its maximal growth rate (if light limited)
Maximum growth rate Carex under nitrogen
limitation
Maximum growth rate Phalaris under
nitrogen limitation
Nitrogen availability at which Carex reaches
half its maximal growth rate (if N limited)
Nitrogen availability at which Phalaris
reaches half its maximal growth rate (if N
limited)
Mortality rate Carex
Mortality rate Phalaris
Turnover rate of nutrient supply
Nitrogen content of tissue of Carex
Nitrogen content of tissue of Phalaris
Soil bulk density
Rooting depth of plant species
Nitrogen content of Carex litter at which it
decomposes at rate dC
Nitrogen content of Phalaris litter at which it
decomposes at rate dP
Nutrient-litter feedback coefficient Carex
Nutrient-litter feedback coefficient Phalaris
Carex litter decomposition rate
Phalaris litter decomposition rate
Light interception coefficient Carex
Light interception coefficient Phalaris
Light-litter feedback coefficient Carex
Light-litter feedback coefficient Phalaris
Unit
Default
value
Lower
limit
Upper
limit
0.25
< 0.025
0.29
0.25
0.22
> 2.5
50
43
> 500
21
< 2.1
24
0.25
0.11
> 2.5
0.25
< 0.025
0.58
30
<3
71
35
0.005
0.01
0.005
15
15
530
1
15
0.0043
0.0082
< 0.0005
< 1.5
14
< 53
< 0.1
> 350
0.0058
0.0115
> 0.05
16
> 150
> 5300
> 10
15
< 1.5
16
15
14.1
> 150
0.7
0.7
0.003
0.003
0.03
0.04
0.01
0.013
0.67
< 0.07
< 0.0003
0.0026
0.026
< 0.004
0.008
< 0.0013
1 (=max)
0.72
0.004
> 0.03
> 0.3
0.048
> 0.1
0.015
day-1
day-1
mol m-2
mol m-2
day-1
day-1
mg kg
-1
mg kg-1
day-1
day-1
day-1
mg g-1
mg g-1
g.m-3
m
mg g-1
mg g-1
day-1
day-1
m2 g-1
m2 g-1
m2 g-1
m2 g-1
References
Bolker, B.M., Pacala, S.W. & Neuhauser, C. (2003) Spatial dynamics in model plant
communities: what do we really know? American Naturalist, 162, 135–148.
Eppinga, M.B., Rietkerk, M., Dekker, S.C., De Ruiter, P.C. & Van der Putten, W.H.
(2006) Accumulation of local pathogens: a new hypothesis to explain exotic
plant invasions. Oikos, 114, 168-176.
Eppinga, M.B., De Ruiter, P.C., Wassen, M.J. & Rietkerk, M. (2009) Nutrients and
hydrology indicate the driving mechanisms of peatland surface patterning.
American Naturalist, 173, 803-818.
17