Biodiversity Maintenance May Be Lower Under Partial Niche Differentiation Than Under Neutrality

Notes

1,3 1,2 RAFAEL D’ANDREA AND ANNETTE OSTLING 1Department of Ecology and Evolutionary Biology, University of Michigan, 2004 Kraus Natural Sciences, 830 North University Avenue, Ann Arbor, Michigan 48109 USA 2Centre for Macroecology, Evolution and Climate, Building 3, 2nd floor, Universitetsparken 15, 2100 København Ø, Denmark

Abstract. Niche differentiation is normally regarded as a key promoter of species coexistence in competitive systems. One might therefore expect that relative to neutral assemblages, niche-differentiated communities should support more species with longer persistence and lower probability of extinction. Here we compare stochastic niche and neutral dynamics in simulated assemblages, and find that when local dynamics combine with immigration from a regional pool, the effect of niches can be more complex. Trait variation that lessens competition between species will not necessarily give all immigrating species their own niche to occupy. Such partial niche differentiation protects certain species from local extinction, but precipitates exclusion of others. Differences in regional abundances and intrinsic growth rates have similar impacts on persistence times as niche differentiation, and therefore blur the distinction between niche and neutral dynamical patterns—although niche dynamics will influence which species persist longer. Ultimately, unless the number of niches available to species is sufficiently high, niches may actually heighten extinction rates and lower species richness and local persistence times. Our results help make sense of recent observations of community dynamics, and point to the dynamical observations needed to discern the influence of niche differentiation. Key words: community assembly; community dynamics; competitive coexistence; immigration; local persistence time; neutral dynamics; Partial niche differentiation; stochastic niche dynamics.

Furthermore, many if not most communities in nature are INTRODUCTION subject to propagule pressure from regional pools. In such Niche differentiation is widely considered a prime open-community scenarios, species may persist for a sub- force behind species coexistence, and thus instrumental stantial period of time due to mass effects, and extinction in maintaining biodiversity. Niche differentiation stabi- is not an absorbing state since later re-colonization is lizes communities by guaranteeing positive growth rates always possible. One measure of diversity maintenance in of rare species (Chesson 1991, 2000). Without such sta- this context is species mean persistence time – the average bilization, coexistence is only temporary, and biodiver- time between introduction through immigration and extir- sity can only be maintained if gains from immigration or pation through drift or competitive exclusion. speciation compensate losses to competitive exclusion or Persistence times have been studied as indicators of eco- drift as posited in neutral theory (Hubbell 2001). logical processes (Leigh et al. 1993, Magurran and Hen- Views on the implications of niche differentiation for derson 2003), and have been explored theoretically in the biodiversity maintenance mostly come from deterministic context of neutral dynamics (Leigh et al. 1993, McGill community models with no immigration. However, et al. 2005, Nee 2005, Pigolotti et al. 2005, Bertuzzo et al. stochastic formulations, which model the influence of vari- 2011, Condit et al. 2012, Segura et al. 2017). Recent stud- ability in finite populations whose deterministic details are ies found that stabilizing niche differentiation considerably unknown or unrelated to species differences, are more rele- prolongs persistence times (Pigolotti and Cencini 2013, vant for natural systems. In such settings, coexistence is Carroll and Nisbet 2015). However, the niche scenarios defined in a probabilistic sense (Schreiber 2015). proposed in these studies tend to be extreme. For example, Pigolotti and Cencini (2013) used a stochastic version of a simplified Lotka-Volterra model where all species interac- Manuscript received 11 March 2017; revised 22 August 2017; tions are identical except that competition is stronger accepted 29 August 2017. Corresponding Editor: Sebastian J. Schreiber. within than across species (Haegeman and Loreau 2011). 3 E-mail: [email protected] This corresponds to a biological scenario where all species

3211 3212 NOTES Ecology, Vol. 98, No. 12 stably coexist and in this sense occupy their own niche. METHODS The authors found that niche stabilization increased species persistence times compared with neutral assemblages. Simulated dynamics Given this extreme degree of stabilization, the higher persistence is not surprising. We use a lottery model to implement stochastic A niche for every species is unlikely in highly diverse dynamics in a local community subject to immigration systems such as tropical forests, where it has been argued from a regional pool. The local community has a fixed that the niches-to-species ratio can be low (Hunter and number of individuals, and in each time step a single Foster 1986, Holt 2006). Even in lower diversity systems, individual dies and is replaced by a new individual. there is no clear evidence that all species can mutually When a death occurs, the probability that it befalls spe- invade from low abundance. Empirical demonstration of cies i is ∑ jAijNiNj/ ∑ klAklNkNl, where Ni is species i’s frequency dependence is rare (Siepielski and McPeek abundance and the coefficient Aij > 0 reflects the degree 2010, but see Adler et al. 2010, Chu and Adler 2015). On of competition between species i and j. The new recruit the other hand, there is evidence suggesting competition can be either a local offspring with probability 1 m or increases with higher phenotypic similarity (Johansson an immigrant from the regional pool with probability m. and Keddy 1991, Jiang et al. 2010, Burns and Strauss If it is a local offspring, the probability that species i is 2011). Models with this type of competitive structure pre- chosen at this stage is riNi/ ∑ jrjNj, where ri is the intrin- dict limits to similarity between species that can coexist sic growth rate of species i. If the recruit is an immigrant, in stable equilibrium; those outside these limits are even- species i is chosen with probability givenP by its relative ¼ tually excluded by stronger competitors (MacArthur and abundance in the regional pool pi, where i pi 1. Levins 1967). However, some species are excluded faster If we set all ri = constant and Aij = 1, then per capita than others, resulting in the spontaneous formation of birth and death probabilities are the same across species transient clusters of similar species (Scheffer and van Nes and this model becomes identical to Stephen Hubbell’s 2006, Fort et al. 2009). In particular, those with the most (2001) neutral community model. Our model is an exten- unfavorable traits might be excluded faster than under sion of Hubbell’s model where we allow for species pure drift. Given a continuous supply of immigrants, a differences, which appear as differences in the ri and Aij. neutral scenario may even sustain more species. The niche mechanism consists of the map between spe- In addition to niche relations, other types of species dif- cies traits and the competition coefficients Aij. We note ferences also affect local dynamics. Environmental filters that we are placing density-dependent competitive effects may cause non-stabilizing fixed differences in species on deaths, but preliminary tests revealed that none of our growth and death rates, which can dramatically change results change qualitatively by placing them on births richness, persistence times, and abundance distributions instead. We also note that our model is similar to classi- relative to neutrality (Zhou and Zhang 2008, He et al. cal Lotka-Volterra dynamics, though in stochastic form. 2012). Furthermore, immigration will promote local per- In fact, we numerically verified that it yields the same sistence of regionally common species over regionally rare qualitative behavior as a Gillespie implementation of ones. Although immigration can stabilize communities as stochastic Lotka-Volterra dynamics (see Appendix S1). it pushes local relative abundances towards regional rela- By fixing community size, we can focus on the impact tive abundances, it is conceptually different from stabiliza- of competition; otherwise, fluctuations in community size tion caused by ecological advantages to rare species, i.e., would also affect dynamics regardless of the competitive niches. One wonders whether the effects of niche differ- structure. To facilitate comparison we use the same com- ences on persistence are similar or fundamentally differ- munity size in all our scenarios. We set the local commu- ent from those of species differences unrelated to niches. nity size at 21,000 individuals, which mirrors a typical Here we hypothesize that a local species assemblage real-world scenario of trees exceeding 10 cm diameter at where the number of niches available is less than the num- breast height in a 50-ha plot of tropical forest (e.g., Barro ber of species in the regional pool may have lower diversity Colorado Island, Pasoh Forest Dynamics Plot). We also and shorter persistence times on average than a completely verified that using a different community size had no neutral community of the same size and subject to the qualitative impact on our results. Abundances in the same propagule pressure. To test that hypothesis, we per- regional pool, which contains 150,000 individuals and form a simulation-based study of an open local commu- 400 species, are either identical or follow a neutral loga- nity under stochastic niche dynamics with a adjustable rithmic (log-series) distribution (Volkov et al. 2003). number of niches. We compare species richness, persis- Additional simulations verified that a larger regional tence times, and extinction rates against a fully neutral pool had no impact on quantitative results. assemblage. We then see how results change when we add We start the local community with a random draw of differences in regional abundances, immigration, and offspring from the regional pool, and then simulate intrinsic growth rates. Finally, we test how the number of dynamics for 50 million death and replacement events, niches affects comparison with neutrality. at which point species abundance distributions appear December 2017 NOTES 3213 stationary. We then run for another 50 million death 2010). Another common choice is Gaussian competition, and replacement events while keeping track of all intro- where the power 4 in the exponential function is replaced duction and extirpation events, for a total of 100 million with 2, but that has been shown to be mathematically steps, corresponding to roughly 5,000 community turn- idiosyncratic (Hernandez-Garc ıa et al. 2009), and leads overs. At the end of a run, we measure species richness, to very slow competitive sorting. The scale constant w mean extinction rate, and species persistence times. We determines how quickly competition decreases with trait calculate the latter as the average number of steps that difference. When w 0, each species competes only with each species remained in the local community between itself, corresponding to a case where each species has its being introduced through immigration and being extir- own niche (complete niche differentiation). In the oppo- pated due to competitive exclusion or drift. Given the site extreme w ? ∞, all species compete equally in the low variation in these indices across replicates of the same single niche (neutral competition). In between is same scenario (see Results), we decided that 10 replicates partial niche differentiation, where there are fewer niches of each is sufficient. than species: competition is stronger between more simi- Note that although we are measuring time in terms of lar species, and only those sufficiently niche-differentiated events (death immediately followed by replacement), we would be able to coexist in the absence of immigration. are not necessarily assuming constant intervals. If time is By analogy with common statistical terminology for measured in terms of days or years etc., there will be an null and alternative hypotheses, we will refer to commu- exponential distribution of intervals between consecutive nities lacking niche stabilization as H0 assemblages, and events (assuming deaths follow a Poisson process). those with niche stabilization as H1 assemblages. In Approximating the rate of deaths per unit time as fixed, terms of our parameters, H0 assemblages have w ? ∞ the expected value of the relative persistence times and hence Aij = 1 between all species pairs (neutral com- measured in physical time units will be very close to the petition), whereas in our H1 assemblages w is finite and relative persistence times measured in events. Aij vary based on trait differences. All simulations were coded in the R computer Table 1 describes all our scenarios and our corre- language (version 3.4.1, with code available on GitHub sponding hypotheses as to the impact caused by the fac- (D’Andrea 2017; version R Core Team 2017). tor being varied. Our first scenario, termed the Baseline, isolates the impact of partial niche differentiation by con- Scenarios sidering H0 and H1 assemblages where all species have identical regional abundances and intrinsic growth rates, We pin competitionhi to phenotypic (trait) similarity by so that the only differences between species are niche dif- ¼ xi xj 4 ¼ð Þ= setting Aij exp w ,wherexi i 1 400 is ferences. Next we check how niche differences affect per- the trait value of species i (traits range between 0 and 1). sistence when acting in concert with other species This is a commonly used decreasing function of trait dif- differences that do not contribute to stable coexistence. ference (Hernandez-Garc ıa et al. 2009, Pigolotti et al. In Variants 1 and 2, some species immigrate more

TABLE 1. Variants 1 through 4 are identical to the baseline scenario except where indicated in the Description and Parameters columns. Settings are applied to both the neutral (H0) and niche (H1) simulations where applicable.

Scenario Description Parameters Hypothesis

Baseline Isolate differences between neutral competition pi = const. Some species in H1 will have enhanced (H0) and partial niche differentiation (H1) ri = const. persistence and others lessened persistence m = 0.01 relative to H0 w = 0.063

Variant 1 Some species are more abundant than others in pi log - series Even in H0, some species will be more prone regional pool ? Inhomogeneous immigration to extinction than others due to lower rescue effect. Outcomes of H0 and H1 will be more similar than in Baseline

Variant 2 Same as Variant 1 but species have higher pi log - series Higher immigration increases persistence dispersal ability ? Higher immigration m = 0.1 times and magnifies the effects of regional abundances relative to internal dynamics. Outcomes of H0 and H1 even more similar than in Variant 1 Variant 3 Metabolic costs or other physiological limitations ri ¼ xið1 xiÞ Species with higher intrinsic growth will have cause lower intrinsic growth of species with higher persistence and lower extinction rates. extreme trait values Effect is stronger on H0 assemblage, bringing H0 outcomes closer to H1 than in Baseline Variant 4 Intrinsic growth rates are determined by factors ri U(0, 1) Qualitatively similar impact as Variant 3 unrelated to the niche trait, such as environmental filtering for unrelated traits 3214 NOTES Ecology, Vol. 98, No. 12 frequently than others because of differences in regional 2003). Results for Variant 1 (where m = 0.01) are shown abundances. In Variants 3 and 4, some species have in Fig. 3, and results for Variant 2 (where m = 0.1) are higher intrinsic growth rates than others because of phys- shown in the supporting information. iological constraints or environmental filters. In Variant 2, we additionally test for the impact of immigration by RESULTS giving all species ten times the dispersal ability used in the rest of the study (m = 0.1 vs. m = 0.01. As a reference When regional abundances and intrinsic growth are point, immigration rates at Barro Colorado Island are identical across all species (Baseline scenario), the estimated at m = 0.08, Chisholm and Lichstein 2009). assemblage with partial niche differentiation (H1) stands Finally, we look at the impact of varying the number of in high contrast with the neutral assemblage (H0), as niches available to our 400 species. We define the number shown in Fig. 1. In H0, persistence times show no of niches as the number of species that can coexist with- relation to species trait values (Fig. 1A) and do not out immigration in the deterministic formulation of our correlate with species abundances observed at the end of q ¼ : model. In all scenarios shown in Figs. 1 and 2 we set the simulation (correlation index logðNÞ;logðTÞ 0 008). w = 0.063, which leads to 12 niches and thus about 33 By contrast, in H1 persistence times and species abun- species per niche. By lowering parameter w we raise the dances are highly non-random (Fig. 1B), and correlate q ¼ : number of niches from 12 to 400 (1 species per niche, i.e., strongly with each other ( logðNÞ;logðTÞ 0 86, see also full niche differentiation). For this part, we use Variants 1 Appendix S2: Fig. S1). and 2, where regional abundances are logarithmically dis- A distinctive pattern is seen along the trait axis: species tributed as in a neutral metacommunity (Volkov et al. lump together, and the lumps or clusters are separated by

A B

1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 Relative abundance (log) abundance Relative 0.0 (log) abundance Relative 0.0 Mean relative persistence (log) Mean relative Mean relative persistence (log) Mean relative 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Trait value Trait value C D 80 4 All species Transient species 60 3 0

40 / H 1 2 H

20 1 Number of species

0 0 0.5 0.6 0.7 0.8 0.9 1 Richness Persistence Extinction Mean relative persistence (log)

FIG. 1. (A, B) Species abundances and persistence times plotted against relative trait values in an example H0 assemblage (A) and H1 assemblage (B). Stems show log N/log Nmax,whereN are species abundances and Nmax is the highest abundance observed. Red lines show log T/logTmax,whereT are species mean persistence times and Tmax ≃ 24,000 community turnovers is the maximum possible persistence time, corresponding to our entire tracking period (1 community turnover = 21,000 simulation steps). (C) Number of species by mean persistence time, log T/log Tmax, in the baseline assemblages with no stabilization (H0, blue bars) and partial stabilization (H1,red bars). Shown are averages across ten replicates. Error bars show standard error of the mean. (D) Comparison of species richness, community-averaged persistence time, and community-averaged extinction rates between baseline H1 and H0 assemblages. Bars show average ratio H1/H0 across ten replicates, error bars show standard error of the mean. [Color figure can be viewed at wileyonlinelibrary.com] December 2017 NOTES 3215

FIG. 2. Comparison of persistence time distribution, log T/log Tmax, between baseline (pale colors) and variant scenarios (dark colors) as described in Table 1. Top row shows H0 assemblages, bottom row shows H1 assemblages. Results shown are averaged across ten replicates of each scenario, with error bars showing the standard error of the mean. (A, E) species have random, logarithmically distributed regional abundances, independent of trait values; (B, F): logarithmically distributed regional abundances and higher immigration rate (m = 0.1); (C, G) intrinsic growth rates ri peak at the center of the axis and drop to zero at the edges; (D, H) species have random, uniformly distributed ri values, independent of trait values. [Color figure can be viewed at wileyonlinelibrary.com] gaps of low abundance (and corresponding low persis- than under neutrality, whether or not residents are tence times). This clustered pattern occurs in all our simu- accounted for (Fig. 1D). lations (see Appendix S2: Figs. S4–S6, S8), and indeed When some species are more abundant than others in has received attention in the recent literature on niche the regional pool causing differences in immigration dynamics (Holt 2006, Scheffer and van Nes 2006, Fort rates (Variant 1), the persistence time distribution shifts et al. 2009, 2010, Pigolotti and Cencini 2013, D’Andrea towards the extremes (Fig. 2A,E): on one hand, tran- and Ostling 2016). We verified in our simulations that if sients have shorter persistence times compared to when immigration is turned off, the clusters eventually disap- the pool is homogeneous; on the other hand, the neu- pear, and one species remains per each cluster. Each clus- tral community now also has residents, and in H1 the ter therefore represents a niche, and under immigration number of residents is higher than before. These new each niche is occupied by more than one species. The resi- residents arise not from stabilization but because of dents are the species best adapted to their niche, and are their comparatively high regional pool abundance the ones that remain in the absence of immigration. (Appendix S2: Fig. S2). As a result, H0 and H1 now There is much higher variation in mean persistence have more similar distribution of persistence times than time across species in the partially stabilized assemblage in the Baseline scenario (compare Fig. 2A,E). than the neutral counterpart (Fig. 1C; also compare red Keeping the differences in regional abundances and curves in Fig. 1A,B). The histogram of persistence times increasing immigration tenfold (Variant 2), we see most (Fig. 1C) reveals that (1) a few species in H1 but none in persistence times increase and many species become resi- H0 are residents, i.e. were present throughout the track- dents in both H0 and H1 assemblages (Fig. 2B,F, compare ing period of the simulation, never going extinct; (2) the with Fig. 2A,E). A few species now have actually lower vast majority of species in H1 not only are transient but persistence times due to regional rarity. Overall, regional in fact have lower persistence times than in H0. Residents abundances become stronger determinants of persistence benefit from stabilization and can in principle remain times compared with the lower immigration scenario. indefinitely (barred stochastic fluctuations), thus heavily Again, H0 and H1 communities are more similar to each driving mean persistence times in H1. Indeed if residents other than in the Baseline scenario (compare Fig. 2B,E). are discounted, the average persistence time in H1 drops It should be noted that although high immigration may from higher to lower than the neutral assemblage make the persistence time distributions essentially indistin- (Fig. 1D). Importantly, the outsize influence of residents guishable, niche differentiation still influences which spe- on community-averaged persistence times does not cies are residents (Appendix S2: Fig. S3). We also note translate to higher diversity or lower extinction. In fact, that a comparable increase in immigration when the pool both species richness and community-averaged extinc- is homogeneous trivially makes all species residents in tion rates are lower in the partially stabilized assemblage both the H0 and H1 assemblages (results not shown). 3216 NOTES Ecology, Vol. 98, No. 12

(Fig. 2D,H). As with previous variants, niche and neu- Richness tral communities are more similar than in the Baseline. Persistence all 3 In summary, differences in regional abundances and Persistence transients intrinsic growth have a qualitatively similar effect on persis- Extinction tence times as niche differentiation: persistence times of most species is very short, while those of a few species is

0 2 indefinitely long. As a result, the persistence time distribu-

/H tion is qualitatively similar whether or not niche differentia- 1

H tion is present. On the other hand, relative to the Baseline scenario where all regional abundances and intrinsic growth 1 rates are identical, the effect is much stronger on H0 communities than H1 communities (compare the dark- and light-shaded bars in the top and bottom rows of Fig. 2). As we increase the number of niches while keeping regio- 0 nal diversity fixed, all our biodiversity metrics improve 1 5 10 15 20 25 30 35 from worse to better than in the neutral community (Fig. 3). In particular, as fewer species compete for the Species per niche same niche tending towards full niche differentiation (one FIG. 3. Comparison of richness, community-averaged per- species to one niche, on the left end of the graph), extinc- sistence times, and community-averaged extinction rates tion tends to zero and richness escalates. Persistence times between neutral assemblages (H0) and partially differentiated increase dramatically. Most of the improvement stems assemblages (H ) with increasing number of niches and fixed 1 from the larger number of residents; persistence across regional diversity. Data points show the ratio between the H1 and H0 assemblages, error bars show one standard error of the transients also improves, but more slowly (compare blue mean, calculated across 10 replicates. Persistence times are and black curves in Fig. 3, see also Appendix S2: Fig. S5). shown averaged across all species in the community (black) and We note that the specific degree of stabilization across transients (blue). As the number of niches increases from required for higher persistence in the H assemblage 12 to 400, the number of species per niche decreases from 33 to 1 1 species per niche. All measured quantities cross the neutral depends on the particulars of the community, such as value (dashed gray line) at different points. In the limit of full immigration rates and niche axis geometry (Appendix S2: niche differentiation (one species per niche), extinction is zero Figs. S6, S7), but the qualitative results are the same: bio- and local persistence and richness are maximal. Regional diversity maintenance can be actually lower in a commu- abundances are logarithmically distributed, while all other parameters are set as in the Baseline scenario (Variant 1, see nity with niches compared with a neutral community, Table 1. Compare similar plot for Variant 2 in Appendix S2: unless the number of niches is sufficiently high. Fig. S7). [Color figure can be viewed at wileyonlinelibrary.com]

DISCUSSION The introduction of intrinsic growth advantages to species with intermediate trait values (Variant 3) dramat- By building on the framework of neutral models ically lowers persistence times for most species in H0 (Alonso et al. 2006), our study sheds light on biodiversity compared to the Baseline, while promoting those few maintenance in an open community where species similar- species with the highest intrinsic growth to resident sta- ity begets competition and the number of species exceeds tus (compare pale- and dark-colored bars in Fig. 2C). the number of niches. Partial niche differentiation stabi- The effects on the H1 assemblage are more subtle, with lizes species abundances and may indefinitely prolong persistence times actually increasing for several species local persistence times of certain species, but at the com- (Fig. 2G), and correlating negatively with intrinsic munity level it does not guarantee higher richness, longer q ¼: growth ( r;log T 0 2, see also Appendix S2: Fig. S4). persistence times, or lower extinction rates. Differences in However, the qualitative shape of the H1 persistence time regional abundances and intrinsic growth rates also affect distribution is unchanged, indicating that intrinsic dynamics, having a bigger impact on the assemblages growth rates have a small effect on H1 exclusion dynam- lacking stabilization entirely. This showcases the robust- ics relative to frequency-dependent competition. Again, ness of stabilized assemblages against other dynamical the H0 and H1 persistence time distributions become less forces, in contrast with the susceptibility of neutral distinguishable compared with the baseline where all dynamics. On the other hand, those species differences intrinsic growth rates are identical (compare Fig. 2C,G). have a qualitatively similar impact on persistence times as Differences in intrinsic growth rates that are uncon- niche differences: most species go out fast while a few last nected to the niche-related trait (Variant 4) shorten the indefinitely. Moreover, high immigration blurs the dynam- persistence times of almost all species in both the H0 ical differences between stabilized and non-stabilized com- and H1 assemblages, but once again the effect is much munities. We conclude that niche differentiation can raise stronger on the assemblage lacking stabilization or reduce richness, persistence, and extinction, depending December 2017 NOTES 3217 on a series of factors examined in this study, and the effect stochasticity in neutral models has been recently shown to is qualitatively similar to other types of species differences. be an important driver of species abundances (Chisholm It follows that these indices alone cannot distinguish et al. 2014, Kalyuzhny et al. 2015), and therefore presum- between neutral and partially stabilized dynamics, unless ably also of persistence times. It would be an interesting one knows sufficient details about the community of inter- next step to test its effects on stabilized assemblages. est to parametrize the niche model. Our results are compatible with real-life observations We note that the niche axis in our model is finite and of “core” species that persist for long times and “occa- carries edge effects. Although some modelers use circular sional” populations that regularly undergo local extinc- axes to avoid this, a finite linear axis is arguably more tion and recolonization events. Magurran and Henderson realistic. In a highly symmetric case like our Baseline sce- (2003) found that occasional species in their study typi- nario where the only differences between species are niche cally had different habitat requirements than core species, differences, circularity drastically dampens the influence which suggest their differences in persistence arise from of niches on persistence times (Appendix S2: Fig. S6). filtering effects analogous to Variants 3 and 4 in our This is because the positions of the niches on a finite axis model (Fig. 2C). However, the division between core and are set, whereas on a circular axis they constantly shift occasional species they find does not line up exactly with through time, thus mitigating the effects of stabilization. habitat requirements. This could occur under either neu- However if other asymmetries occur, such as differences tral or niche dynamics. Core species may last longer sim- in regional abundances, the effect of circularity disappears ply by virtue of having higher regional and local (Appendix S2: Fig. S7), making the choice of a finite or abundance, but our results show how they could also be circular axis inconsequential. Also, for the reader inter- the beneficiaries of niche stabilization. ested in the common choice of exponent 2 as opposed to Upon finding that the estimated levels of immigration 4 in the competition coefficients Aij, we show in the sup- on Barro Colorado Island are compatible with neutrality porting information (Appendix S2: Fig. S8) that the per- given its observed richness, Condit et al. (2012) wrote that sistence time distribution in that model is much closer to “species interactions, niche partitioning, or density-depen- neutrality (even on a finite axis). This is not surprising dence, while they may be present, do not appear to enhance given the slower and weaker exclusion dynamics. tree species richness at Barro Colorado.” Our findings con- Our niche model shows a distinctive pattern of abun- firm that the connection between immigration and result- dances by traits, mirrored by a similar pattern of persis- ing richness is not unique to process, implying that Condit tence times: species are organized in clusters and et al.’s observations are potentially also compatible with separated by gaps; those at the center of the clusters are niche differentiation, and indeed that it is possible for niche the residents and those at the gaps are the shortest-living partitioning to be present while not enhancing richness. transients. Each cluster is a group of species competing In general, insight into whether persistence dynamics for the same niche, and in the absence of immigration are shaped by niches in nature will require more exten- only one species remains in each niche. Clusters are a sive data than collected by Magurran and Henderson recent extension of classical ideas of limiting similarity (2003) or Condit et al. (2012). Inferring niches from spe- (MacArthur and Levins 1967, Holt 2006, Scheffer and cies persistence is only rigorous when one can tease apart van Nes 2006, Pigolotti and Cencini 2013). Fort et al. the effect of trait differences from the effect of regional (2009) showed mathematically that they arise as a tran- and local abundances. sient state on any deterministic niche model with a circu- ACKNOWLEDGMENTS lar axis and competition tied to species similarity. Our model is similar but adds demographic stochasticity, We thank Sebastian Schreiber, Gyuri Barabas, and the immigration, and a finite axis. Clusters have been seen anonymous reviewers for helpful suggestions to improve the under these circumstances before (Barabas et al. 2013), manuscript. This work was supported by the National Science Foundation under grant 1038678, “Niche vs. neutral structure and we recently proposed that they are a general outcome ” ’ in populations and communities funded by the Advancing of niche-axis models (D Andrea and Ostling 2016). Theory in Biology program, as well as by the Danish National Note that while not all species have enhanced persis- Research Foundation through support of the Center of Macroe- tence under partial stabilization, the number of species that cology, Evolution and Climate (grant DNRF96), and by the do is higher than the number of niches. In other words, it Miller Institute for Basic Research in Science at the University is not the case that only those species with a niche have of California, Berkeley. enhanced persistence relative to neutrality. As we empha- sized, the particular number will depend on immigration, LITERATURE CITED regional abundances, environmental filters, and the map Adler, P. B., S. P. Ellner, and J. M. Levine. 2010. Coexistence of between species traits and degree of competition. perennial plants: an embarrass ment of niches. Ecology In our study we did not account for temporal changes Letters 13:1019–1029. in environmental conditions which may affect competitive Alonso, D., R. S. Etienne, and A. J. McKane. 2006. The merits interactions and intrinsic growth rates. Environmental of neutral theory. Trends in Ecology & Evolution 21:451–457. 3218 NOTES Ecology, Vol. 98, No. 12

Barabas, G., R. D’Andrea, R. Rael, G. Meszena, and A. M. Jiang, L., J. Tan, and Z. Pu. 2010. An experimental test of Ostling. 2013. Emergent neutrality or hidden niches? Oikos Darwin’s naturalization hypothesis. American Naturalist 122:1565–1572. 175:415–423. Bertuzzo, E., S. Suweis, L. Mari, A. Maritan, I. Rodriguez- Johansson, M. E., and P. A. Keddy. 1991. Intensity and asym- Iturbe, and A. Rinaldo. 2011. Spatial effects on species persis- metry of competition between plant pairs of different degrees tence and implications for biodiversity. Proceedings of the of similarity: an experimental study on two guilds of wetland National Academy of Sciences of the United States of Amer- plants. Oikos 60:27–34. ica 108:4346–4351. Kalyuzhny, M., R. Kadmon, and N. M. Shnerb. 2015. A neutral Burns, J., and S. Strauss. 2011. More closely related species are theory with environmental stochasticity explains static and more ecologically similar in an experimental test. Proceedings dynamic properties of ecological communities. Ecology of the National Academy of Sciences of the United States of Letters 18:572–580. America 108:5302–5307. Leigh, E. G., S. J. Wright, E. A. Herre, and F. E. Putz. 1993. Carroll, I. T., and R. M. Nisbet. 2015. Departures from neutral- The decline of tree diversity on newly isolated tropical ity induced by niche and relative fitness differences. Theoreti- islands: a test of a null hypothesis and some implications. cal Ecology 8:449–465. Evolutionary Ecology 7:76–102. Chesson, P. 1991. A need for niches? Trends in Ecology & MacArthur, R. H., and R. Levins. 1967. The limiting similarity, Evolution 6:26–28. convergence, and divergence of coexisting species. American Chesson, P. 2000. Mechanisms of maitenance of species diversity. Naturalist 101:377–385. Annual Review of Ecology and Systematics 31:343–358. Magurran, A. E., and P. A. Henderson. 2003. Explaining the Chisholm, R. A., and J. W. Lichstein. 2009. Linking dispersal, excess of rare species in natural species abundance distribu- immigration and scale in the neutral theory of biodiversity. tions. Nature 422(April):714–716. Ecology Letters 12:1385–1393. McGill, B. J., E. A. Hadly, and B. A. Maurer. 2005. Community Chisholm, R. A., et al. 2014. Temporal variability of forest inertia of Quaternary small mammal assemblages in North communities: empirical estimates of population change in America. Proceedings of the National Academy of Sciences 4000 tree species. Ecology Letters 17:855–865. of the United States of America 102:16701–16706. Chu, C., and P. B. Adler. 2015. Large niche differences emerge Nee, S. 2005. The neutral theory of biodiversity: do the numbers at the recruitment stage to stabilize grassland coexistence. add up? Functional Ecology 19:173–176. Ecological Monographs 85:373–392. Pigolotti, S., and M. Cencini. 2013. Species abundances and Condit, R., R. A. Chisholm, and M. O. Hunter. 2012. Thirty lifetimes: from neutral to niche stabilized communities. Jour- years of forest census at Barro Colorado and the importance nal of Theoretical Biology 338:1–8. of immigration in maintaining diversity. PLoS ONE 7:1–6. Pigolotti, S., A. Flammini, M. Marsili, and A. Maritan. 2005. D’Andre, R 2017. rafaeldandrea/Persistence-Times-Code: First Species lifetime distribution for simple models of ecologies. release. Zenodo. http://doi.org/10.5281/zenodo.883450 Proceedings of the National Academy of Sciences of the D’Andrea, R., and A. M. Ostling. 2016. Challenges in linking United States of America 102:15747–15751. trait patterns to niche differentiation. Oikos 125:1369–1385. Pigolotti, S., C. Lopez, E. Hernandez-Garc ıa, and K. H. Fort, H., M. Scheffer, and E. H. van Nes. 2009. The paradox of Andersen. 2010. How Gaussian competition leads to lumpy or the clumps mathematically explained. Theoretical Ecology uniform species distributions. Theoretical Ecology, 3:89–96. 2:171–176. R Core Team. 2017. R: a language and environment for statisti- Fort, H., M. Scheffer, and E. van Nes. 2010. The clumping transi- cal computing. R Foundation for Statistical Computing, tion in niche competition: a robust critical phenomenon. Journal Vienna, Austria. of Statistical Mechanics: Theory and Experiment 2010:P05005. Scheffer, M., and E. H. van Nes. 2006. Self-organized similarity, Haegeman, B., and M. Loreau. 2011. A mathematical synthesis the evolutionary emergence of groups of similar species. Pro- of niche and neutral theories in community ecology. Journal ceedings of the National Academy of Sciences of the United of Theoretical Biology 269:150–165. States of America 103:6230–6235. He, F., D. Y. Zhang, and K. Lin. 2012. Coexistence of nearly Schreiber, S. J. (2015). Coexistence in the face of uncertainty. neutral species. Journal of Plant Ecology 5:72–81. ArXiv http://arxiv.org/abs/1512.03970, pages 1–25. Hernandez-Garc ıa, E., C. Lopez, S. Pigolotti, and K. H. Ander- Segura, A. M., D. Calliari, B. L. Lan, H. Fort, C. E. Widdi- sen. 2009. Species competi tion: coexistence, exclusion and combe, R. Harmer, and M. Arim. 2017. Community fluctua- clustering. Philosophical transactions. Series A, Mathemati- tions and local extinction in a planktonic food web. Ecology cal, Physical, and Engineering Sciences, 367:3183–3195. Letters 20:471–476. Holt, R. D. 2006. Emergent neutrality. Trends in Ecology & Siepielski, A. M., and M. A. McPeek. 2010. On the evidence for Evolution 21:531–533. species coexistence: a critique of the coexistence program. Hubbell, S. P. 2001. The unified neutral theory of biodiversity Ecology 91:3153–3164. and biogeography. Princeton University Press, Princeton, Volkov, I., J. R. Banavar, M. O. Hunter, and A. Maritan. 2003. New Jersey, USA. Neutral theory and relative species abundance in ecology. Hunter, M. O., and R. B. Foster. 1986. Biology, chance, and his- Nature 424:1035–1037. tory and the structure of tropical rain forest tree communities. Zhou, S.-R., and D.-Y. Zhang. 2008. A nearly neutral model of Community Ecology 19:314–329. biodiversity. Ecology 89:248–258.

SUPPORTING INFORMATION Additional supporting information may be found in the online version of this article at http://onlinelibrary.wiley.com/doi/ 10.1002/ecy.2020/suppinfo