title: emergent niche structuring leads to increased differences...
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Title: Emergent niche structuring leads to increased1
differences from neutrality in species abundance2
distributions3
Running Head: Emergent niches and neutral SADs4
Authors: Rosalyn C. Rael∗1,2, Rafael D’Andrea1, Gyorgy Barabas1,3, Annette Ostling15
∗Corresponding author, email: Rosalyn C. Rael, [email protected]
1Ecology and Evolutionary Biology7
University of Michigan8
830 North University9
Ann Arbor, MI 48109-104810
Present addresses: 2 Tulane University, Tulane-Xavier Center for BioenvironmentalResearch, 6823 St. Charles Ave., New Orleans, LA, 70118, USA; 3University of Chicago,Department of Ecology and Evolution, 1101 E 57th St, Chicago, IL 60637, USA
Rael, R. C. et al.
Abstract11
Both niche dynamics and neutral dynamics may contribute to the12
structural patterns of ecological communities. Understanding the relative13
contributions of each of these assembly processes to the maintenance of14
biodiversity remains an important and open question. Neutral dynamics15
involve equivalent competitors governed only by demographic stochasticity,16
and niche dynamics involve species trait differences enabling stable coexistence.17
Key recent studies of the community patterns produced by the complex,18
concerted action of niche dynamics and stochasticity employ simple models19
with enforced niche structure. These models suggest that species abundance20
distributions are insensitive to the relative contributions of niche and neutral21
mechanisms, especially when diversity is much higher than the number of22
niches. Here we analyze the outcomes of a stochastic competition model with23
species interactions driven by interspecific trait differences. With this model,24
niche structure emerges as clumps of species that persist along the trait axis,25
and leads to species abundance distributions that differ more substantially26
from neutral predictions than previously thought. We show that interactions27
across emergent niches are the dominant drivers of species abundances in this28
model, acting as a biotic filter favoring species at the centers of niches. These29
results suggest that mechanisms that produce niches also influence species30
abundance distributions and highlight between-niche interactions as a key31
element of emergent niche dynamics.32
Keywords : competition, coexistence, community assembly, trait axis, neutral theory,33
ecological equivalence, niche, species abundance distribution, self-organized similarity,34
emergent neutrality, stochastic niche theory35
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1 Introduction36
A question debated in community ecology is whether the pattern of species abundances37
in a given community reflects underlying mechanisms involved in assembling it, or if38
diverse mechanisms common not only across communities, but to a variety of complex39
systems, produce the same patterns of species abundance (Nekola and Brown 2007).40
Niche differentiation (Chesson 1991, Leibold 1995, Chase and Leibold 2003, Chesson 2000,41
Amarasekare 2003, Meszena et al. 2006) and neutral theory (Bell 2000, Hubbell 2001,42
Alonso et al. 2006, Rosindell et al. 2012) provide different hypotheses about mechanisms43
that drive the patterns of diversity we see in nature. The principle of competitive44
exclusion says that species must be sufficiently different from each other with regard45
to some trait relevant to competition in order to coexist (Hardin 1960). The resulting46
coexistence in this case is stable. Species can invade populations of other species from47
low abundance. Neutral theory suggests that coexistence can be explained by species’48
similarities rather than their trait differences, allowing species to persist together for long49
periods of time, although not in a way that would enable invasion from low abundance50
(Chesson 2000, Hubbell 2001, Siepielski and McPeek 2010). Despite these differences51
in the nature of coexistence proposed, recent studies have suggested that the species52
abundance distributions (SADs) of communities undergoing neutral and niche dynamics53
are too similar to be useful for inferring the presence or contribution of niche structure54
(Chave et al. 2002, Mouquet and Loreau 2003, Purves et al. 2005, Chisholm and Pacala55
2010, Haegeman and Loreau 2011, Pigolotti and Cencini 2013, Carroll and Nisbet 2015).56
Many of the recent studies considering the differences between niche and neutral57
SADs have mainly considered whether these community assembly modes produce the58
same range of SAD patterns with a broad range of model parameters (Chave et al.59
2002, Mouquet and Loreau 2003, Pigolotti and Cencini 2013). Speciation and dispersal60
rates in particular are difficult to measure and are therefore treated as free parameters61
of the neutral model. However, if these parameters could be estimated for particular62
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systems using data, then predicted neutral and niche-process SADs could be compared63
to gain insight into the degree to which the assembly process matters in those systems.64
In fact, information on dispersal rates is becoming increasingly available (Clark et al.65
1999, Muller-Landau 2001), as is data that could be used to approximate the abundance66
distribution of the regional pool in a neutral model and estimate speciation rate. For67
example, data becoming available on the abundances of tree species in a large region of68
the Panama basin surrounding Barro Colorado Island (Hubbell et al. 1995) might serve69
this purpose. Comparing SADs using the parameter estimates for a particular system may70
indicate that the shape of the SAD can reveal the process, and perhaps even stimulate71
further data collection required to constrain these parameters in other systems as well.72
Other recent studies have considered differences between niche and neutral SADs73
occurring for fixed speciation and immigration parameters. They conclude that a large74
amount of niche structuring is needed to create substantial differences between niche75
and neutral SADs. For example, Purves et al. (2005) and Chisholm and Pacala (2010)76
considered a simplified, extreme niche structure in which species fall into discrete,77
non-interacting guilds within which they interact neutrally. Chisholm and Pacala (2010)78
showed that this type of stochastic niche model produces SADs that are virtually79
indistinguishable from the neutral SAD when species richness is much higher than80
the number of niches, and that it takes a large number of niches to obtain substantial81
differences between niche and neutral cases. Haegeman and Loreau (2011) and Pigolotti82
and Cencini (2013) come to the same conclusion when considering another type of83
simplified niche structure in which intraspecific and interspecific competition are each84
respectively determined by a single parameter. They find that SADs change little as85
a small amount of niche structure is enforced by strengthening intraspecific relative to86
interspecific competition.87
However, it may be premature to draw conclusions about the community patterns88
typically expected in nature from these studies, as real interaction structures are expected89
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to be more complex than assumed by these models. Recent theoretical studies show90
that competitive interactions driven by species trait differences typically lead to niche91
structuring in the form of persistent clusters of similar species (Scheffer and van Nes 2006,92
Pigolotti et al. 2007, Segura et al. 2011, Vergnon et al. 2012), with interactions within93
clusters stronger than between. These clusters emerge from the dynamics themselves94
instead of being externally imposed. The niche dynamics studied by Purves et al. (2005)95
and Chisholm and Pacala (2010) could be viewed as a possible limiting case of this96
expected structure, with identical competitors within but no interaction at all between97
clusters.98
Here we evaluate the SADs from a stochastic competition model where structuring99
of species into niches emerges rather than being enforced. Specifically, we consider a100
stochastic version of the classic Lotka–Volterra competition model along a trait axis,101
where interaction strength declines with interspecific trait difference, a simple model that102
captures arguably the most salient feature of competition structuring many ecological103
communities. Often there is an array of resources for competing species to partition based104
on continuous trait values (e.g. water and nutrients available at different soil depths105
partitioned based on plant rooting depth). The Lotka–Volterra model of competition106
amongst species on a trait axis predicts system-specific limits to the similarity of107
coexisting species MacArthur and Levins (1967), May (1973), Abrams (1983), Szabo and108
Meszena (2006), Barabas and Meszena (2009), Barabas et al. (2012, 2013a). Perhaps109
counterintuitively, its transient state actually involves emergent clustering of species on110
the trait axis (the species nearest to those that coexist at equilibrium take the longest to111
be excluded). The addition of intraspecific negative density dependence, environmental112
fluctuation, mutation, or mass effects typically make this clustering persistent. This113
“self-organized similarity” or “emergent neutrality” was highlighted in a variety of recent114
studies (Bonsall et al. 2004, Scheffer and van Nes 2006, Holt 2006, Roelke and Eldridge115
2008, Vergnon et al. 2009, Segura et al. 2011, Barabas et al. 2013b). In the stochastic116
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version of the model with immigration we use here, the clusters persist through mass117
effects.118
We use this model to consider the potential for niche dynamics to produce differences119
from the neutral case when regional pool and dispersal parameters are fixed, and compare120
these differences to those produced by models with an extreme niche structure. By121
analyzing the model’s resulting variation in the strength of competition interactions122
across species, we also show that interactions across niches are of paramount importance123
in driving the observed species abundance patterns, even when they are much weaker124
than within-niche interactions. We previously noted that this stochastic niche model125
does not produce the multimodal average SADs predicted under unrealistic additional126
density-dependence (Vergnon et al. 2012, Barabas et al. 2013b, Vergnon et al. 2013).127
Here we demonstrate the differences this model produces from the neutral case in128
SAD predictions. This study lays the groundwork for further investigations on the129
distinguishability of assembly mode using SADs and other community patterns when130
niches emerge rather than being enforced. Furthermore, it highlights that continued131
development of a stochastic niche theory for SADs and other community properties132
may require a better understanding of the competitive interactions and emergent niche133
structuring that actually occurs in nature.134
2 Model and Simulation Methods135
We use a spatial structure often used in neutral models in ecology consisting of a136
“metacommunity” pool of species that can immigrate into a smaller local community137
(Hubbell 2001). We focus on the influence of niche differentiation on SADs only in the138
local community. We do not incorporate niche differentiation into the source pool, or139
model the dynamics in it explicitly. Instead we assume the source pool species abundances140
follow a Ewens sampling distribution, as would be expected for an infinite metacommunity141
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governed by the standard neutral model involving point speciation (Etienne et al. 2007).142
The dynamics of the local community include immigration from the source pool plus143
stochastic Lotka–Volterra competition, which in deterministic form is given by the model144
dxi
dt= βxi
(
1−1
K
S∑
j=1
α(wij)xj
)
. (1)
Here xi is the number of individuals of species i, S is the number of species (which145
changes over time in our model due to immigration and extinction), β is the intrinsic146
growth rate, and K is the carrying capacity of the species. We take the latter two to147
be species-independent for simplicity and to allow us to focus on the effects of niche148
differences rather than competitive asymmetries due to “fitness differences” which would149
be present if K varied across species (Chesson 2000, D’Andrea and Ostling 2015). The150
function α(wij) in Eq. 1 is the strength of competition between species i and j which151
are at distance wij from each other on the niche axis. Using finite circular boundary152
conditions, we define the distance to be153
wij := min{|ui − uj|, 1− |ui − uj|}, (2)
where ui ∈ [0, 1] is species i’s position on the axis. The form of the competition coefficients154
α(wij) determines the type of dynamics. For niche dynamics, we define155
α(wij) = exp[
−(wij
σ
)ρ ]
, (3)
and for neutral dynamics,156
α(wij) = 1. (4)
The model described by Eqs. 1 and 3 was first explored by MacArthur and Levins157
(1967). It involves niche dynamics in that a suite of species can stably coexist on the trait158
axis only if they are far enough apart in trait values, as long as ρ > 2 (Pigolotti et al.159
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2007, Hernandez-Garcıa et al. 2009, Pigolotti et al. 2010). The same holds for ρ = 2 except160
for certain special parameter combinations (Roughgarden 1979, Gyllenberg and Meszena161
2005, Meszena et al. 2006, Barabas et al. 2012). The ρ < 2 case is biologically unrealistic162
(Barabas et al. 2012, 2013a), so we do not consider it here.163
In our stochastic version of Eq. 1 with immigration, we assume that the community164
is under viability selection, so deaths are density-dependent, while birth and recruitment165
is density-independent. We assume the immigrant fraction m of recruits flow in at a166
rate mβJ , where J is the total community size. The probability that an immigrating167
individual belongs to a given species is equal to its relative abundance in the regional pool,168
which we take to follow the Ewens sampling distribution with parameter θ (Etienne et al.169
2007). Local birth and subsequent recruitment of individuals of species i occurs at a rate170
(1 − m)βxi, and deaths at a rate (β/K)xi
∑S
j=1α(wij)xj, which reduce to Eq. 1 in the171
deterministic limit for m = 0.172
To relate the predictions of the model to familiar neutral predictions, we set J to173
the size of the measured tree community on Barro Colorado Island (21,455 individuals >174
10 cm dbh in 1995), and m and θ to values under which neutral theory provides a good175
fit to the empirical species abundance distribution there (0.098, and 47.8 respectively;176
Etienne 2005, Ostling et al. 2015). Note that the total community size in our model is177
controlled by a combination of J and K. In the neutral case we can set both equal to the178
desired community size, but in the niche case we tune K to achieve a target stationary179
community of approximately 21,455 individuals.180
The model was simulated using Gillespie’s algorithm (Gillespie 1977). On every181
iteration, a single event (recruitment or death) happens with probability proportional to182
its relative rate (i.e. recruitment or death rate divided by the total rate of all possible183
events). Since all events occur at a rate proportional to β (set here to 1), its value does184
not affect the equilibrium. Simulations were initiated with 250 species at equal abundance185
with randomly assigned trait values between 0 and 1, and were run for a number of events186
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large enough that visual analysis suggested the average species abundance distribution187
across runs was near equilibrium. Simulations were performed using MATLAB and188
required over 20,000 hours of computation time, which was carried out on the Extreme189
Science and Engineering Discovery Environment (XSEDE). See the Appendix (A1) for190
further details of the model and simulation methods. The code we used for our simulations191
is available in the Supplementary Material.192
3 Results193
3.1 Emergent Niche Structuring194
Figure 1 shows the resulting abundances along a trait axis in an example neutral and195
niche simulation with ρ = 4 and σ = 0.15. The neutral case, as would be expected,196
shows no distinct pattern along the trait axis. Under niche dynamics however, the model197
produces clumps of densely packed and abundant species, separated by regions with198
fewer and less abundant species. The number of clumps is equal to the number of stably199
coexisting species that would be expected at equilibrium in the deterministic version of200
our model in Eq. 1. We term the clumping pattern in our model “emergent niches” to201
emphasize that groups form as a result of the dynamics rather than being prescribed.202
Each “niche” is a region approximately centered at the most abundant species in a clump203
and having a width roughly equal to σ. The number of niches can therefore be tuned204
by choosing σ appropriately. Due to the periodic boundary conditions and the fact that205
species interactions depend only on distance but not the absolute positions along the206
niche axis, only the relative positions of the clumps are determined by σ, with the exact207
locations varying through time and from one simulation to the next.208
All of our simulations with niche dynamics were conducted with ρ = 4, which is a209
conservative choice for discerning whether niche differentiation influences pattern. This210
value in the model leads to visible clumping of species along the trait axis that only211
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strengthens with higher ρ values. See the Appendix (A2) for a discussion of the model’s212
sensitivity to this parameter.213
3.2 Species abundance distribution patterns214
When niches are small in number relative to the number of species, the extreme niche215
model of Chisholm and Pacala (2010) produces SADs indistinguishable from the neutral216
case. To see if this is the case in our model, we first chose σ to allow for only five niches217
(as in Figure 1b), leading to an average of 233 species present in our local communities.218
The average richness in the corresponding neutral communities was 225 species. We then219
also considered abundance patterns with 20 and 50 niches. Figure 2 shows the species220
abundance distribution averaged over 1000 final community configurations for (a) 5, (b)221
20, and (c) 50 niches, along with the species abundance distributions analytically predicted222
by Chisholm and Pacala (2010) for the corresponding number of niches, and the average223
for our neutral runs.224
Our resulting niche communities have SADs that differ more substantially from225
the neutral SAD than those of the extreme niche model for the high diversity limit. In226
particular with even just five niches, differences between the niche and neutral SADs227
averaged over 1000 simulations are apparent in Figure 2. These communities exhibit a228
strong central peak in the mean SAD compared to the neutral mean SAD. This involves229
both a higher proportion of species of medium abundance (6th-8th abundance classes230
on the Preston-style SAD plot shown) than the neutral case, and lower proportions of231
intermediately rare and intermediately high abundance species (3rd-5th and 9th-10th232
abundance classes respectively). Our niche communities also exhibit large relative233
differences from the neutral case in the two highest abundance classes (i.e., relative to the234
number of species the neutral model predicts in those classes; see Appendix Figure 4). The235
Chisholm and Pacala prediction for the 5-niche case is virtually indistinguishable from the236
neutral case in Figure 2 and does not feature the strong central peak. It does, however,237
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exhibit small differences from the neutral SAD in the two highest abundance classes that238
are in the opposite direction from the SAD produced by our niche model. The relative239
differences are shown Appendix Figure 4.240
For a larger number of niches (20 and 50), the differences from the neutral case are241
still more substantial than predicted by Chisholm and Pacala. The predictions from242
our model and theirs are very close in the large abundance classes, with the directions243
of differences from neutrality in those classes being the same in both models. However,244
our resulting average SAD also differs strongly from the neutral case along the rest of245
the curve while their prediction does not (see both Figure 2 and Appendix Figure 4). In246
particular, it still generates a higher proportion of species of medium abundance (6th-8th247
abundance classes) and lower proportion of intermediately rare species (3rd-5th abundance248
classes) than seen in the neutral case.249
3.3 The importance of interactions across niches250
As mentioned above, in contrast to the extreme niche model of Purves et al. (2005) and251
Chisholm and Pacala (2010), niche structuring emerges in our model rather than being252
enforced. There are two key differences in how niche interactions are handled. First, the253
extreme niche model ignores interactions across niches, whereas in our model they are254
present and determined by the distances between species’ trait values. Second, we allow255
for heterogeneity in the strength of interactions between species in the same niche (since256
we also model those interaction strengths as depending on trait difference), while the257
extreme niche model assumes strict neutrality within each niche. Intuitively, the former258
must be playing the dominant role in shaping the patterning in species’ relative abundance259
on the trait axis in our model. As species closer to the dominant ones in the center of260
a niche generally reach higher abundance than those closer to the edge, there must be261
advantages related to their interactions with species in other clumps of abundant species262
on the axis that allow them to do so.263
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More rigorously, we can take the final configurations of our 5-niche simulations and264
measure the average strength of competition on a set of test focal species in one niche.265
We partition this total strength into contributions from a) species outside of that niche266
and b) species inside the niche (see Appendix (A3)). We then compare the patterns of267
total competition strength along the axis from each source (inside or outside the niche)268
to determine which of these two types of competition is dominant in shaping patterns of269
relative abundances within the niche.270
We find that, while competition from outside the niche is weaker than that from271
within the niche, the total competition strength reflects the shape of competition from272
outside the niche (Figure 3a). Also, the pattern of competition from outside the niche273
corresponds to the pattern of abundance within the niche: competition from outside is274
highest at the edges of the niche where abundance is lowest. The pattern of competition275
strength from inside the niche alone would lead one to predict the opposite pattern in276
abundance (see Figure 3b). These results indicate that competition from outside the277
niche is dominating over competition from within the niche in influencing the relative278
abundances of species within a niche.279
4 Discussion280
A key indicator in determining whether observed species abundance distributions can281
be used to infer community assembly processes is just how much they change with the282
presence of niche dynamics when other factors are held fixed. Purves et al. (2005) and283
Chisholm and Pacala (2010) recently argued that niche and neutral SADs are very similar284
when there are many species per niche, and in fact identical in the infinite diversity285
limit. They demonstrated this analytically for the case of discrete, non-interacting niches286
with neutral dynamics within each niche. Here we have shown that SADs show greater287
differences between niche and neutral communities if niche structuring is allowed to288
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emerge from the dynamics of a competition model instead of being modeled in a rigid289
manner. In particular, visually apparent differences arise in the SAD even with a small290
number of niches relative to the number of species. We have also identified between-niche291
interactions as being particularly important in shaping these differences in community292
structure pattern.293
It is clear from our study that the presence of niches in an interacting community can294
influence the shape of the SAD, and that while the extreme niche structuring of Purves295
et al. (2005) and Chisholm and Pacala (2010) allows for valuable analytical results to296
be derived, it is too extreme to reflect the processes that give rise to differences from297
a neutral SAD. This perhaps should not be surprising given that the extreme niche298
structure could more readily be interpreted as a group of phytoplankton put together299
with a group of trees in a rainforest, as well as a collection of island birds, etc., than300
niches in a community of interacting species. Indeed, Haegeman et al. (2011) point out301
that a model of independent, unregulated species gives the same SAD predictions as a302
zero-sum neutral model for all levels of diversity, and hence that it is not surprising that303
extreme niche structuring leads to the same distributions as a neutral model in the high304
diversity limit. When there are more species than niches, species in separate niches would305
likely instead retain some level of interaction, with heterogeneity in the intensity of those306
interactions due to variation in how far species are from the dominant species in a nearby307
niche. In our model where niche structure emerges from competition that depends on308
species trait differences, we find that species organize into niches in such a way that there309
are significant interactions across niches, and the heterogeneity in those interactions shapes310
species’ relative abundances.311
Understanding exactly how the effect of interactions across niches translates into the312
particular SAD differences from neutrality our model predicts is not intuitive, but there313
are some things we can highlight. First consider the 5-niche case, where one of the key314
SAD features of our model is a larger number of species in the highest abundance class315
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than expected in the neutral case. These highly dominant species always occur at or near316
the center of a niche. Hence it seems that the interactions across niches favor those species317
enough that they tend to achieve a high level of dominance more often than any species318
in the neutral model. This effect is not seen in the extreme niche models that assume319
neutrality within niches. With a higher number of niches, the community is essentially320
divided into more pieces, lowering the maximum abundance that can be achieved by321
any one species. This leads to fewer species in the highest abundance classes than in322
the neutral case in both models, but ours produces slightly more species in the highest323
abundance class that is not essentially empty of species (the 11th class in the 20-niche case324
and the 10th class in the 50-niche case), presumably because it incorporates the favoring325
of a particular species within the niche.326
In the 5-niche case, our model also produces fewer species of intermediately high327
abundance, and more species of moderate abundance than the neutral case. This328
suppression of species’ abundances could be due to some combination of the high329
competition from outside a niche on species near the niche edge, and the particularly high330
competition amongst species that happen to be in niches with very abundant species. We331
see evidence for both in Figure 1, where species at the outskirts of niches are rarely over332
about 250 individuals, and in the niche with the highly dominant species, the rest of the333
species are below 200 individuals. Our model also produces a strong central peak with334
more niches, presumably due to similar effects. We do not have an intuitive explanation to335
offer for the smaller number of intermediately rare species our model predicts (compared336
with the neutral case) for all the number of niches we examined, though varying the337
immigration rate can affect the number of rare species.338
Some of the changes in the SAD we find are reminiscent of differences in the SAD339
predicted by niche structure in some of the recent studies, although a detailed comparison340
is difficult because of the differences in spatial structure and parameter values used.341
Haegeman and Loreau (2011) and Pigolotti and Cencini (2013) obtain some substantial342
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differences in their resulting number of species with moderate abundance when they343
sufficiently lower the strength of competition between species. Although Chave et al.344
(2002) focused on the range of SAD patterns predicted as dispersal parameters varied,345
they also mentioned differences found for fixed parameter values (and significant346
niche structure), which involved an increase in the number of species of moderate347
abundance. However, we note that the niche structuring they assumed involved unrealistic348
assumptions, including discontinuities in the life history tradeoff they considered (Barabas349
et al. 2013a, D’Andrea et al. 2013), and perfect specialization and symmetric effects of350
enemies (Ødegaard et al. 2000).351
Further empirical inquiry into both observed patterns of niche structure and the352
processes behind it is needed to better guide theoretical studies of the influence of niche353
dynamics on trait patterns and ultimately on SADs. Some recent studies have highlighted354
observed clumped patterns of species on trait axes in support of those consistent with355
an emergent niche perspective (Vergnon et al. 2009, Segura et al. 2013, Yan et al.356
2012). With regard to modeling assumptions, the interaction symmetry used here in357
the Lotka-Volterra modeling construct is supported by Adler et al. (2010), who showed358
evidence for symmetry in the strength of intra- and interspecific interactions between four359
plant species. In contrast, some studies have found evidence of hierarchical interactions360
(e.g. Harpole and Tilman (2006) and Kunstler et al. (2012)), although in some of these361
cases it seems that a larger-scale perspective might reveal that competition depends on362
trait differences related to exploitation of patches with different characteristics, such as age363
since disturbance. Some studies have also directly highlighted evidence of niche differences364
and neutral processes in shaping community diversity (Levine and HilleRisLambers 2009,365
Rominger et al. 2009). While there are a number of empirical studies that lend support366
to the processes and resulting patterns given in this study, more work is needed to discern367
the links between traits, competitive interactions, niches, and resulting SADs in specific368
ecological systems to both guide and test theoretical predictions.369
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Gravel et al. (2006) also studied a stochastic model in which niche structure was370
allowed to emerge – one of competition along environmental gradients – and found371
patterns ranging from limiting similarity to random as the immigration rate and diversity372
increased, with no mention of clumping on the trait axis across that spectrum of373
parameter values. However, the model of Gravel et al. (2006) is, under global dispersal,374
equivalent to our model with ρ = 2 as the shape parameter of the competition kernel,375
which also did not exhibit visible clumping on the trait axis (Appendix Figure 2). Carroll376
and Nisbet (2015) presented the impacts on the SAD of stochastic niche dynamics that377
they argued approximated emergent neutrality in that it incorporated weak stabilization378
as described by Chesson (2000). However their model involved a hierarchical structure379
with a high degree of enforced symmetry and hence is quite different than the niche380
structuring we have found emerges on a trait axis under stochastic competitive dynamics.381
A couple of recent studies have considered slight relaxations of the extreme niche382
structure of Chisholm and Pacala (2010), and have also shown increased differences383
between SADs of neutral and niche structured communities, even for a small number384
of niches. Walker (2007) showed that when niches differed in their diversity, differences385
were produced in SADs, even in the high diversity limit. Bewick et al. (2015) recently386
considered a modification in which species can have membership in multiple niches,387
but interactions within niches are still neutral. Their model produced a plethora of388
rare species compared to the neutral case, even with a small number of niches. This389
effect was seemingly due to variation across species in niche breadth (i.e. the number of390
niches each can occupy) incorporated in their model, as the species with narrow niche391
breadths tended to be rare. Our model has no such differences in average diversity across392
niches or variation in resource or habitat-utilization widths. The results of Walker (2007)393
seem to point to the additional importance of the details of speciation and immigration394
across niche space, and both suggest that non-uniform niche structure can itself lead to395
differences in SADs, even for a small number of niches.396
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Our analysis has shown that more plausible, emergent niche structuring can in397
principle influence SAD patterning even with high diversity and a small number of398
niches. Future studies may determine whether it will typically have such an influence,399
and further, whether its influence is actually detectable in data (as Al Hammal et al.400
2015 have considered for the model of Pigolotti and Cencini 2013). Factors that should401
be considered include the shape of the competition kernel, the breakdown of population402
growth rates into birth and death rates, the immigration rate, and the metacommunity403
species abundance distribution. The inclusion of “fitness differences” (Chesson 2000)404
should also be considered, as recent studies in the context of enforced niche structure have405
found that they can counteract the effects of niche differences or “stabilization” on the406
SAD (Carroll and Nisbet 2015, Du et al. 2011). Ideally one would find a way to parse407
the effects of “fitness differences” and “stabilization” using SAD pattern. Evolutionary408
pressures could also play a role, potentially leading to greater similarity of species within409
niches (Scheffer and van Nes 2006, Vergnon et al. 2012). Coupling study of these factors in410
the context of a model like that studied here, with study of an array of more biologically411
detailed and empirically ground-truthed system-specific competition models may help412
place communities found in nature within the larger spectrum of models that can be413
mathematically constructed. Consideration of the impact of niche structure on community414
metrics containing more information than SADs may also prove worthwhile. Recent415
studies have suggested that the distribution of species lifetimes (Pigolotti and Cencini416
2013, Carroll and Nisbet 2015) and individual abundance distribution (Tang and Zhou417
2013) may be more revealing of assembly process than SADs. Another possible avenue is418
consideration of the distribution of abundances at various scales with regard to population419
size of the local community, and given the niche structuring on the trait axis exhibited in420
the model we study here, along various sizes of sampled trait range. Many details warrant421
further future consideration when determining the extent to which coarse metrics such as422
the SAD can reveal the underlying nature of community structure and interactions.423
17
Rael, R. C. et al.
Acknowledgments This material is based upon work supported by the National Science424
Foundation under grant no. 1038678, “Niche versus neutral structure in populations and425
communities,” funded by the Advancing Theory in Biology program. This work used the426
Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by427
National Science Foundation grant number OCI-1053575. We thank Veronica Vergara for428
assistance with running MATLAB on the Condor Pool, which was made possible through429
the XSEDE Extended Collaborative Support Service (ECSS) program.430
18
Rael, R. C. et al.
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List of Figures581
1 Example neutral (a) and niche (b) configurations at the end of the582
simulations. (a) No pattern is visible along the trait axis in the neutral583
case. K = 21, 455; total abundance for this example: 21, 235; run length:584
5 × 107 events. (b) Clumping of abundant species is visible along the trait585
axis. K = 5410, σ = 0.15, ρ = 4; total abundance for this example: 21, 930;586
run length: 5× 107 events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28587
2 Species abundance distributions (SADs), plotted as the proportion of588
species in logarithmically-scaled abundance classes delineated as in Volkov589
et al. (2003), averaged over 1000 simulations for neutral runs and each590
niche dynamics case. Also shown are the SAD predictions using Chisholm591
and Pacala’s analytical niche model (Chisholm and Pacala 2010). (a)592
5-niche simulation communities (K = 5410, σ = 0.15), (b) 20-niche593
simulation communities (K = 1310, σ = 0.037) and (c) 50-niche simulation594
communities (K = 519, σ = 0.015). Parameter value ρ = 4 was used for595
all niche runs. Note mean species richness was 225, 232, 236, and 247 in the596
neutral, 5, 20, and 50-niche simulation communities respectively. . . . . . . 29597
26
Rael, R. C. et al.
3 Average competition strength and abundances within a single niche in598
a 5-niche community. (a) Relative strength of outside and within-niche599
competition. A fixed set of equally spaced traits ui = 1...Sn, where Sn is600
the average number of species within a niche, is used as a test set on which601
to compute outside and within-niche competition from existing species.602
The test set is placed in the centermost niche of each final simulation603
configuration for consistency, and competition strength from existing604
outside and inside species is averaged over 1000 simulation configurations.605
(b) Average abundance across a single niche. Abundances from each of five606
niches over 1000 simulations are binned and averaged. . . . . . . . . . . . . . 30607
27
Rael, R. C. et al.
0 0.2 0.4 0.6 0.8 10
200
400
600
800
1000
1200
1400
1600
1800
Niche value ui
Abu
ndan
ce
Neutral
(a)
0 0.2 0.4 0.6 0.8 10
200
400
600
800
1000
1200
1400
1600
1800
Niche value ui
5−niche
(b)
Figure 1
28
Rael, R. C. et al.
1 4 16 64 256 1024 40960
0.05
0.1
0.15
0.2
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port
ion
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peci
es
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Neutral5−nicheC & P 5−niche
1 4 16 64 256 1024 40960
0.05
0.1
0.15
0.2
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port
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es
(b)
Neutral20−nicheC & P 20−niche
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0.05
0.1
0.15
0.2
Number of individuals (log2 scale)
Pro
port
ion
of s
peci
es
(c)
Neutral50−nicheC & P 50−niche
Figure 2
29
Rael, R. C. et al.
0 0.05 0.1 0.15 0.21000
2000
3000
4000
5000
6000
7000
Distance from left edge of niche
Com
petit
ion
stre
ngth
(a)
From outside of nicheWithin nicheTotal
0 0.05 0.1 0.15 0.240
60
80
100
120
140
160
180
Distance from left edge of niche
Mea
n ab
unda
nce
(bin
ned)
(b)
Figure 3
30
Supplementary Appendix and Figures
for
“Emergent niche structuring leads to increased differences from neutrality in species
abundance distributions”
Rosalyn C. Rael, Rafael D’Andrea, Gyorgy Barabas, Annette Ostling
Contents
A1 The stochastic Lotka–Volterra model with immigration . . . . . . . . . . . . 2A2 Model sensitivity to the shape parameter ρ . . . . . . . . . . . . . . . . . . . 4A3 Partitioning competition into within- and between-niche contributions . . . . 5
1
A1 The stochastic Lotka–Volterra model with immigration1
In our stochastic version of Eq. 1 from the main text with added immigration, we consider2
and write down formulae for the rates of the following set of possible events:3
• One of the species (say the ith) currently in the local community recruits a new indi-4
vidual – either an offspring of an individual in the local community or an immigrant5
offspring.6
• The ith species currently in the local community decreases in abundance by one due7
to death of an individual.8
• A species not currently in the local community has an immigrant offspring colonizing9
the local community with a single individual.10
In the simulation, exactly one of these events will happen on each iteration. To obtain the
event probabilities, we first write the rates at which these events would occur in a continuous-
time limit:
bi = βxi(1−m) + βJmpi (recruitment) (A1)
di =β
Kxi
S∑
j=1
α(wij)xj (death) (A2)
s = βJm
(
1−S∑
j=1
pj
)
(immigration) (A3)
In Eqs. A1 - A3, xi is the abundance of species i, β is the intrinsic growth rate, m is11
the probability that a newly recruited individual is an immigrant from the metacommunity12
(as opposed to the offspring of an individual already in the local community), J is the13
local community size, α(wij) the competition kernel as a function of the trait difference wij14
between species i and j, K is the carrying capacity, and pj is the relative abundance of15
2
species j in the metacommunity, assumed constant in time and given by the Ewens sampling16
formula (Etienne et al., 2007).17
The recruitment rate bi is simply the density-independent part of Eq. 1 in the main18
text, modified to include immigration. The death rate di is the density-dependent part of19
Eq. 1.The immigration rate s assumes that the likelihood for a species k not yet present20
in the local community to contribute an immigrant is equal to its relative abundance pk in21
the metacommunity. Hence the chance that a successfully immigrating individual is from a22
species not currently present in the local community is(
1−∑S
j=1pj
)
. Once a species goes23
extinct in the local community, we no longer keep track of its identity.24
To simulate the continuous-time stochastic dynamics efficiently, we use a Gillespie al-25
gorithm (Gillespie, 1977). With this algorithm, one event occurs on each iteration with26
probability determined by the rates above. For example, the probability of recruitment into27
species i currently in the local community is28
bi
s+∑S
i=1(bi + di)
. (A4)
Since we are interested in the stationary distribution, we do not keep track of how much29
time has passed between events. To determine the number of iterations for which to run the30
simulations, we plot the average species abundance distribution across simulations at inter-31
mediate time points and look for there to be little change in the average species abundance32
distribution towards the end of the simulation. We deemed the appropriate number of events33
for this to be 5 × 107 for the niche case and 1 × 107 for the neutral case. We note however34
that the slight changes in the distribution still occurring in the niche case are a bit more35
substantial than the neutral case, but that the niche case is moving towards even greater36
differences from the neutral case as the number of events increases.37
3
A2 Model sensitivity to the shape parameter ρ38
In the main text we used ρ = 4 to define the shape of the competition kernel. This parameter39
controls how “box-like” the competition kernel is: larger values of ρ correspond to more box-40
like kernels, with a perfect rectangular function recovered for ρ → ∞ (Appendix Figure 1).41
Here we comment on the sensitivity of our results to the choice of this parameter.42
In the deterministic Lotka–Volterra model (Eq. 1 in main text), though values of ρ less43
than two are mathematically meaningful, they have been shown to be biologically unrealistic44
(Adler and Mosquera, 2000; Barabas et al., 2012, 2013), requiring species to utilize their45
environments in discontinuous ways. Therefore, only the ρ ≥ 2 case is of interest.46
Our parameterization of the Lotka–Volterra model involves constant carrying capacities47
and periodic boundary conditions. A variable carrying capacity function would inflate the48
abundances of some species relative to others, potentially leaving a large footprint in the49
species abundance distribution (SAD). Similarly, nonperiodic boundary conditions would50
lead to edge effects, with species at the edges experiencing less competition than the ones in51
the middle, also distorting the SAD. Our parameterization steers safe of such confounding52
factors.53
However, for this parameterization, it has been shown that the ρ = 2 case leaves the54
system dangerously close to the boundary of structural stability and instability (Pigolotti55
et al., 2007; Hernandez-Garcıa et al., 2009; Pigolotti et al., 2010; Barabas et al., 2012),56
potentially creating artifacts in model predictions. Therefore, ρ = 2 should also be avoided.57
Indeed, our stochastic model with ρ = 2 produces no clear clustering of species into distinct58
“niches” (Appendix Figure 3a), even though that is what happens in the deterministic case59
(Scheffer and van Nes, 2006; Szabo and Meszena, 2006; Barabas and Meszena, 2009). In60
any case, the lack of patterning we see from our model suggests that niche dynamics for this61
value of ρ may have little impact on diversity maintenance and hence do not provide a good62
example for exploration of the impacts of niche differentiation on SAD pattern.63
4
We therefore examined values of ρ greater than two. Appendix Figure 3 shows some of64
the simulation results for other ρ values we explored. As mentioned before, the ρ = 2 case is65
non-representative, but out of all the other results, the ρ = 4 case has the least well-defined66
clustering of species into distinct clumps. Since the stronger the clumping, the greater its67
signature will be in the SAD, we used a very conservative choice in terms of potential pattern68
detection by opting for ρ = 4, reasoning that if one can detect SAD differences here, it is69
certainly true that differences will be detectable for other values of ρ as well.70
A3 Partitioning competition into within- and between-niche con-71
tributions72
In calculating the strength of competition on species from within and outside the niche as a73
function of their position within their niche, we must partition the trait axis into niches. We74
first assume the number of niches is equal to the number of species that would stably coexist75
in the deterministic formulation of the model (approximately 1/σ, rounded to an integer).76
To place the first niche, we designate the trait of the most abundant species in the final77
configuration of a simulation to be at the center of that niche. We then place the remaining78
niche centers equally spaced across the niche axis with the first.79
5
−0.5 0 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ui − u
j
α (
u i−u
j)
ρ = 2ρ = 4ρ = 6ρ = 10ρ = 100
Appendix Figure 1: Competition coefficients α(ui−uj) = exp(
−∣
∣
∣
ui−uj
σα
∣
∣
∣
ρ)
with σα = 0.15
and various shape parameters ρ.
6
0 0.2 0.4 0.6 0.8 10
200
400
600
800
1000
1200
1400
1600
1800
2000
Niche value ui
Abu
ndan
ce
Appendix Figure 2: Stem plots for examples of five-niche communities using the compe-tition coefficients shown in Appendix Figure 1, with shape parameter ρ = 2.
7
0 0.2 0.4 0.6 0.8 10
500
1000
1500
2000
Abu
ndan
ce
(a)
0 0.2 0.4 0.6 0.8 10
500
1000
1500
2000
(b)
0 0.2 0.4 0.6 0.8 10
500
1000
1500
2000
Niche value ui
Abu
ndan
ce
(c)
0 0.2 0.4 0.6 0.8 10
500
1000
1500
2000
Niche value ui
(d)
Appendix Figure 3: Stem plots for examples of five-niche communities using the compe-tition coefficients shown in Appendix Figure 1, with shape parameters (a) ρ = 4, (b) ρ = 6,(c) ρ = 10, and (d) ρ = 100.
8
1 4 16 64 256 1024 4096−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
Number of individuals (log2 scale)
Diff
eren
ce in
pro
port
ion
of s
peci
es
(a)
5−niche20−niche50−nicheC & P 5−nicheC & P 20−nicheC & P 50−niche
1 4 16 64 256 1024 4096−1
−0.5
0
0.5
1
1.5
2
2.5
3
Number of individuals (log2 scale)
Rel
ativ
e di
ffere
nce
in p
ropo
rtio
n of
spe
cies
(b)
5−niche20−niche50−nicheC & P 5−nicheC & P 20−nicheC & P 50−niche
Appendix Figure 4: (a) Absolute differences (mean niche −mean neutral) and (b) relativedifferences ((mean niche −mean neutral)/mean neutral) between the mean niche and neutralspecies abundance distributions (SADs) for SADs resulting from simulations of our nichemodel (solid lines) and for the distributions predicted by the model of Chisholm and Pacala(dashed lines). Shown are the differences in the mean proportion of species in the communityin logarithmically scaled abundance classes, with those classes delineated as in Volkov et al.(2003) and as in the main text.
9
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