Untangling the complexity of priority effects in multispecies communities - Stanford University
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Received: 30 March 2021 | Revised: 23 May 2021 | Accepted: 9 August 2021 DOI: 10.1111/ele.13870 PERSPECTIVE Untangling the complexity of priority effects in multispecies communities Chuliang Song1,2,3 | Tadashi Fukami4 | Serguei Saavedra1 1 Department of Civil and Environmental Abstract Engineering, MIT, Cambridge, MA, USA 2 Department of Biology, McGill The history of species immigration can dictate how species interact in local com- University, Montreal, Canada munities, thereby causing historical contingency in community assembly. Since 3 Department of Ecology and Evolutionary Biology, University of Toronto, Toronto, immigration history is rarely known, these historical influences, or priority ef- ON, Canada fects, pose a major challenge in predicting community assembly. Here, we provide 4 Department of Biology, Stanford University, Stanford, CA, USA a graph-based, non-parametric, theoretical framework for understanding the pre- dictability of community assembly as affected by priority effects. To develop this Correspondence Chuliang Song, Department of Biology, framework, we first show that the diversity of possible priority effects increases McGill University, 1205 Doctor Penfield, Montreal QC, Canada super-exponentially with the number of species. We then point out that, despite H3A 1B1. this diversity, the consequences of priority effects for multispecies communities Email: clsong.ecology@gmail.com can be classified into four basic types, each of which reduces community predict- Funding information National Science Foundation, Grant/ ability: alternative stable states, alternative transient paths, compositional cycles Award Number: DEB-1737758 DEB- and the lack of escapes from compositional cycles to stable states. Using a neural 2024349 network, we show that this classification of priority effects enables accurate ex- Editor: Tim Wootton planation of community predictability, particularly when each species immigrates repeatedly. We also demonstrate the empirical utility of our theoretical framework by applying it to two experimentally derived assembly graphs of algal and ciliate communities. Based on these analyses, we discuss how the framework proposed here can help guide experimental investigation of the predictability of history- dependent community assembly. K EY WOR DS alternative stable states, alternative transient paths, compositional cycle, predictability, priority effects I N T RODUC T ION history affects which species survive because species can interact in local communities differently depending on An ecological community's assembly history is defined arrival order and timing (Fukami, 2015), the phenom- by the order and timing of species arrival (Fukami, enon known as priority effects. Much of our present 2015). Arrival timing and order have a strong random understanding of priority effects, however, is based on component (Gould, 1990; Sprockett et al., 2018), making the simple two-species Lotka–Volterra model and its it difficult to predict which assembly history will take variants (Fukami et al., 2016; Gilpin & Case, 1976; Ke place. The large number of conceivable assembly histo- & Letten, 2018; Ke & Wan, 2020; May, 1977; Mordecai, ries has highlighted the necessity of understanding how 2013; Song et al., 2020; Wittmann & Fukami, 2018). It history affects the diversity, composition and function- remains unclear how the diversity and complexity of ing of an ecological community (Carlström et al., 2019; priority effects expand beyond these models and when Clay et al., 2020; Fukami & Morin, 2003; Toju et al., considering multispecies communities (Fukami, 2015; 2018). In this vein, mounting data suggest that assembly Lawton, 1999). Ecology Letters. 2021;00:1–13. wileyonlinelibrary.com/journal/ele © 2021 John Wiley & Sons Ltd. | 1
2 | UNTANGLING THE COMPLEXITY OF PRIORITY EFFECTS IN MULTISPECIES COMMUNITIES (a) Model-dependent assembly dynamics (c) Super-exponential increase of assembly dynamics ● topologically different assembly dynamics II. A,B 102 Deterministic 10150 exclusion 101 A B Estimatd number of Fitness ratio Deterministic Priority 100 10100 coexistence effects # of all atoms in the universe III. A,B 10−1 I. A,B Deterministic ● exclusion 1050 10 −2 A B A B ● 10−2 10−1 100 101 102 0 ● Niche overlap 10 ● 2 3 4 5 6 Community size (b) Graph-based assembly dynamics Deterministic Deterministic Historically contingent coexistence exclusion exclusion* I. A,B II. A,B III. A,B Community A B A B A B composition Invasion by species A Deterministic Historically contingent Compositional cycle* Invasion by species B coexistence coexistence and exclusion* V. A,B VI. IV. A,B A,B A B A B A B F I G U R E 1 Non-parametric graph-based approach. Panel (A) presents the traditional model-based phase diagram of assembly dynamics in two-species communities using coexistence theory. The balance between niche overlap (x axis) and fitness ratio (y axis) determines the types of assembly dynamics: (i) deterministic existence, (ii) deterministic exclusion or (iii) priority effects. Panel (B) presents all six types of possible assembly dynamics (model-free) using the graph-based approach. In an assembly graph, the nodes represent the species combinations, while the links represent how the species combination changes into another combination after a non-resident species invades. Take Case (ii) for example: if species B (blue line) invades the community with species A, species A is excluded and the community composition changes into species B (hence, the arrow goes from circle A to circle B); and if species A (red line) invades the community with species B, species A cannot establish and the community composition remains the same (hence, the arrow turns back on itself). Cases (iii), (v) and (vi) are all priority effects (marked with an asterisk) as at least two different assembly histories can lead to different compositions. Panel (C) plots how the diversity of priority effects increases with community size. We find a super-exponential increase of diversity of topologically different assembly dynamics (Equation 1) For example, arrival order dictates one of the three and (vi) all represent priority effects. The common omis- well-studied outcomes of two-species competition in the sion of cases (v) and (vi) in theoretical studies is a conse- Lotka–Volterra model (Figure 1A; Case, 2000): (i) coexis- quence of assuming that model parameters are fixed and tence (two species coexist regardless of who arrives first); history independent (Rudolf, 2019; Zou & Rudolf, 2020). (ii) deterministic exclusion (the same species always ex- In reality, interaction strengths can be history dependent cludes the other regardless of who arrives first); and (iii) (Carter & Rudolf, 2019; Poulos & McCormick, 2014; history-dependent exclusion (the species that arrives first Rasmussen et al., 2014; Sniegula et al., 2019; Vannette excludes the other). Although not as widely recognised, & Fukami, 2017). Thus, the scope of priority effects is three other outcomes are also possible in two-species largely underestimated by traditional parametric models competition: (iv) two species eventually coexist despite assuming history-independent interaction strengths. requiring a specific arrival order; (v) only one species Moreover, the scope of priority effects quickly in- survives or two species coexist depending on who ar- creases as more species are considered. For example, rives first; and (vi) the species that arrives late always even with only three species, 41,979 outcomes are the- replaces the species that arrives early. The presence of oretically possible, the bulk of which constitute priority these other possibilities has been supported by empirical effects (Figure 1D). In general, as the number of species evidence (Amor et al., 2020; Angulo et al., 2021; Drake, increases, the number of alternative outcomes increases 1991; Warren et al., 2003). Of these six scenarios, (iii), (v) super-exponentially (Figure 1D). The core reason behind
SONG et al. | 3
this super-exponential increase is that, while all history- approach maps any assembly dynamics uniquely onto
independent dynamics are alike, priority effects can be an assembly graph, where nodes represent combina-
unique in their own way (Fukami, 2015). Only when all tions of coexisting species and directed links represent
conceivable assembly histories result in the same com- how species combinations change when a new species in-
munity composition does history-independent dynamics vades. This form of graph presentation is not new (Hang-
arise. The extremely large number of assembly dynamics Kwang & Pimm, 1993), but has been largely underused.
in multispecies communities makes the prospect of pre- We illustrate this approach with two hypothetical spe-
dicting community assembly daunting (Lawton, 1999; cies, where species are denoted as A and B (Figure 1B).
Srivastava, 2018). This illustrative assembly graph has 22‒1 = 3 nodes:
While knowing the exact assembly dynamics operat- {A}, {B} and {A, B}, which represent all possible species
ing in a community may be an impossible task indeed, combinations. The link starting from community (node)
we may still be able to understand the predictability {A} is generated by the invasion of species B, which can
of community assembly affected by priority effects, lead to all three possible species combinations (similarly
or extent to which we can predict community compo- for community {B}). In turn, community {A, B} does
sition given the stochasticity of assembly history. For not have any directed link since all species are present
instance, if in one case all different assembly histories in this community. For example, case (II) in Figure 1B
lead to different outcomes, and if in another case only shows the graph representation of deterministic exclu-
one assembly history leads to a different outcome, the sion: invasion from species B into community {A} leads
latter case has a much more predictable outcome than to community {B}, whereas invasion from species A into
the former (Margalef, 1958). Understanding the pre- community {B} also leads to community {B}. Note that
dictability of community assembly is not only of basic there are in total 3 × 3 = 9 assembly dynamics without
interest to ecologists but can also aid applications of considering which species is named as {A} or {B} (which
community ecology for ecosystem management, in- is arbitrary). Formally, this only considers topologically
cluding ecological restoration, biological control and unique assembly dynamics (graphs), where an assem-
the medical treatment of the gut microbiome (Deng bly graph is unique up to the ordering of species labels.
et al., 2021; Rohr et al., 2020; Song & Saavedra, 2018; Thus, with two species, there are only six topologically
Sprockett et al., 2018). Yet, a theoretical framework unique assembly dynamics (Figure 1B).
is largely lacking for quantifying the predictability of We now generalise our approach to multispecies com-
community assembly in this context. munities. For simplicity, we present this extension with
Here, we introduce a non- parametric graph- based three species with species denoted as A, B and C. The
approach to studying the diversity of priority effects in assembly graph has 23 ‒1 = 8 nodes, representing all pos-
multispecies communities. Taking this approach, we sible species combinations. In a community with a single
propose an information-based metric to quantify com- species ({A}, {B} or {C}), there are two outgoing links rep-
munity predictability, with a focus on the influence of resenting invasions by the other two species respectively.
priority effects. We show that the predictability of com- For instance, in community {A}, one of the two outgo-
munity assembly has two types of regularities: (a) The ing links represents the invasion by species B, which can
higher the invasion times, the higher the predictabil- lead to all three possible combinations of species A and
ity; (b) Predictability of final composition and of tem- B ({A}, {B} or {A,B}). Then, in a community with two
porary changes in community compositions is strongly species ({A,B}, {A,C} or {B,C}), there is one outgoing link
correlated. We then show that the consequences of pri- representing the invasion by the only species not present
ority effects for community predictability can be clas- in the community. That is, in community {A,B}, the only
sified into four basic sources: the number of alternative outgoing link represents the invasion by species C, which
stable states, the number of alternative transient paths, can lead to all eight possible combinations of species A,
the length of compositional cycles and the presence or B and C. Note that community size may decrease after
absence of an escape from cycles to stable states. We an invasion. Finally, in the community with all three spe-
demonstrate both theoretically and empirically that this cies ({A,B,C}), there is no outgoing link since all species
classification allows for an accurate explanation of com- are present. In total, there are 41,979 topologically differ-
munity predictability. Finally, we discuss how our results ent assembly dynamics for just three species.
can be used to guide experimental studies for commu- The same procedure above applies to an arbitrary
nity assembly. number
( of)species. The assembly graph for S species
has 2S − 1 nodes. The nodes representing communities
with n species have (S‒n) outgoing links. Each of the as-
A NON - PA R A M ET R IC GR A PH - sociated outgoing ( links from
) the node with n species
( (n+1)
can)
BA SE D A PPROAC H possibly lead to 2(n+1) − 1 nodes. Then, of the (
2
)
−1
nodes, only one node contains (n + 1) species, Sn+ 1 nodes
To capture the full diversity of priority effects, we in- contain n species, while all the other nodes contain
troduce a non-parametric graph-based approach. This less than n species. As a first order of approximation,
4 | UNTANGLING THE COMPLEXITY OF PRIORITY EFFECTS IN MULTISPECIES COMMUNITIES
the diversity (number) of topologically different assem- infinite. While this assumption is typically applied to
bly dynamics can be calculated as (see Appendix A for allow statistical convergence, it is a rather strong assump-
derivation): tion that is often not met (Capitán et al., 2009; Hubbell,
1997; Serván & Allesina, 2021). As the assembly graph
⎛S ⎞ fully determines the trajectory of community composi-
(S−n)⎜ ⎟
∏S ⎜n⎟
⎝ ⎠
tion given any assembly history, we can study any arbi-
(2n+1 − 1) 22S−2S eS
2 (1)
n=1
∼√ , trary pool of assembly histories (Appendix B). Here, for
S! 1
2 S S+ 2 simplicity, we assume that each species invades m times.
With two species, if species can only invade m =1 time,
Figure 1d shows how the diversity of priority effects the pool of possible assembly histories consists of only
scales with the number of species. Note that already with two assembly histories: AB → (meaning species A invades
six species, the diversity is significantly greater than the first and then species B invades) and BA →. But if we have
total number of atoms in the entire universe (Kragh, three species that each invade m = 2 times, the pool of
2003). possible assembly histories consists of 2 ! 26 !! 2 ! = 90 assem-
bly histories. For example, ACBCAB
→ is a possible assembly
history, meaning that species A invades first, then spe-
PR IOR I T Y E F F E C T S A N D cies C invades, and so on until species B invades last. By
T H E PR E DIC TA BI L I T Y OF changing the number of invasion attempts (m), it is also
COM M U N I T Y A S SE M BLY possible to answer how large m has to be to effectively
generate the same effects as infinite invasions. Similarly,
Priority effects resulting from stochastic dispersal give our framework can be easily adapted to incorporate
rise to assembly dynamics that are not fully predictable. more ecological complexity, such as that some species
To quantify this lack of predictability owing to priority arrive with higher frequency than others (see Appendix
effects, it is necessary to define both the pool of possible B for details).
assembly histories for a given community and the type Shifting our focus to the type of uncertainty to anal-
of uncertainty to analyse (Figure 2). Focusing on the yse, we suggest that, given a pool of assembly histories,
pool of assembly histories, if an infinite number of inva- it is possible to measure two types of uncertainties re-
sions are possible, the pool of assembly histories is also lated to community composition. The first type is the
(a) General procedure Predictability on
Assembly
Potential dynamics Community
assembly histories trajectory
Predictability on
temporal composition
(b) Example (each species invading once)
A,B,C
A,B B,C A,C
A B C
F I G U R E 2 Quantifying the predictability of community assembly. Panel (A) presents the general procedure used for the calculation: First,
based on a given uncertainty (either final composition or temporary compositions), we define a pool of potential assembly histories. Second,
we need to know the assembly dynamics of the community. Third, each assembly history produces its community trajectory (how community
composition changes with an invading species) based on the assembly dynamics. Last, we compute predictability on the final composition
across the community trajectory (boxed with dashed lines) and on the temporary compositions along the community trajectories (boxed with
solid lines). The definition of predictability is based on normalised entropy (see the mathematical definition in Equation 2 or in the figure).
Panel (B) illustrates this procedure with an example of a three-species community. First, we define the pool of potential assembly histories
ABC ACB
when each species invades only once. This gives us six potential assembly histories: → , → , BAC BCA CAB CBA
→ , → , → , → . Second, we choose a three-species
assembly dynamics for illustration. Third, we show the community trajectories defined by assembly history and assembly dynamics. For
ABC
example, the community trajectory is {A} → {A, B} → {A, B} for the assembly history → . Last, we calculate the community predictability:
The predictability on final composition is the predictability of the states ({C} , 4 {A, B} , {A, B, C}), that is, the final compositions in each
community trajectory (boxed in dashed lines), which is 0.55; and the predictability on temporal composition is the average predictability of six
trajectories corresponding to six assembly histories (boxed in solid lines), which is 0.51.
SONG et al. | 5
uncertainty associated with the final community com- length of compositional cycles and the presence or ab-
position across all assembly histories. The second type sence of an escape from cycles to stable states.
is the uncertainty associated with transient community Alternative stable states occur when a community
compositions along all assembly histories. Here, to quan- has more than one stable composition (left column of
tify the predictability of assembly dynamics, we adopt Figure 4A) (Gilpin & Case, 1976; Schooler et al., 2011;
a normalised information entropy metric (Rohr et al., Schröder et al., 2005). Alternative transient states occur
2016). Although there are many alternative uncertainty when there are more than one assembly histories (or tra-
metrics (Vellend, 2016), information entropy has been jectories) from the founding species to the stable states
useful to quantify and explain different ecological pro- (middle column of Figure 4A) (Fukami & Nakajima,
cesses (Margalef, 1958; Marleau et al., 2020; O’Connor 2011, 2013; Sarneel et al., 2019). Compositional cycles
et al., 2019; Zu et al., 2020). We define the predictability occur when the assembly histories involve cyclic se-
of an assembly dynamics as quences of community composition (right column of
∑ Figure 4A) (Fox, 2008; Schreiber & Rittenhouse, 2004).
predictability: = 1 − entropy(uncertainties) = 1 + i P(xi )log(P(xi ))
, Our graph-based approach can identify these dynamical
log(2S − 1) sources as topological features. That is, alternative sta-
(2)
( ) ble states arise when the assembly graph has more than
where xi is a species combination and P xi is the proba- one sink (nodes that have incoming links but no outgoing
bility that combination xi occurs (Figure 2A). The entropy link); alternative transient states arise when more than
is normalised to [0, 1] to ensure interpretability across dif- one directed path exist from single species to a sink; and
ferent community sizes. Thus, a predictability of one (resp. compositional cycles occur when directed cycles exist in
zero) implies that that there is no (resp. full) uncertainty the assembly graph. These three dynamical sources of
about the assembly dynamics. unpredictability have already been hypothesised to be
To illustrate this measure, let us consider the assembly consequences of priority effects (Fukami, 2015).
dynamics defined by case (IV) shown in Figure 1B, where The three dynamical sources are not mutually exclu-
each species invades only once. In this example, if assem- sive, implying that assembly dynamics can potentially
bly history AB → takes place, the trajectory of the community exhibit all of these sources. For example, the assem-
is from {A} → {B}. Instead, if assembly history BA → takes bly graph for the three species shown in Figure 4B has
place, the trajectory is from {B} → {A, B}. Therefore, two alternative stable states ({A, C} and {C}), one
the predictability of the final community composition transient path for each stable state ({A} → {A, C} and
is given by predictability ({B} , {A, B}) = 1 + .5log(.5)log(3)
+ .5log(.5)
= 0.37. {A} → {B} → {A, C}) and one compositional cycle of
Similarly, the predictability of the assembly trajec- length 3 ({B} → {A, B} → {B, C} → {B}). However, the
tory (the temporary community compositions) is picture is not complete with only these three sources.
predictability ({A} , {B}) =. 37 and predictability ({B} , {A, B}) =. 37. Figure 4C shows another example of an assembly graph
Figure 2B illustrates the process of quantifying this pre- for three species with the same types and number of dy-
dictability for a set of three species. The agreement be- namical sources as shown in Figure 4B. Nevertheless, the
tween the two types of predictability is not a coincidence. assembly dynamics in Figure 4C has lower predictability
Figure 3A shows the strong correlation between the two than the one shown in Figure 4B. The difference between
types of predictability across all possible assembly dy- these two cases is that only in Figure 4B, the assembly
namics for three species. The level of correlation is invari- dynamics exhibit the possibility to escape from a cycle
ant across different pools of potential assembly histories. to a stable state (i.e. the trajectory can escape the com-
Thus, without loss of generality, hereafter our results are positional cycle ({B} → {A, B} → {B, C} → {B}) into
based on the first type of predictability (i.e. on the final a stable state ({C}). Because cycles are less predictable
community composition). Communities become more than stable states in general, this possibility to escape
predictable when species are allowed to invade multi- can increase the predictability of assembly dynamics.
ple times. Figure 3B shows how the distribution of pre- According to some original definitions of compositional
dictability across all possible assembly dynamics with cycles (Fukami, 2015; Morton & Law, 1997; Steiner &
three species increases as a function of the number of in- Leibold, 2004), compositional cycles are strictly perma-
vasion attempts. nent such that, if a ‘cycle’ has an escape to a stable state,
it actually does not represent a compositional cycle, but
rather constitutes alternative transient paths. In this
C L A S SI F Y I NG PR IOR I T Y E F F EC T S paper, we adopt a broader definition of compositional
cycles to refer to those with an escape to a stable state as
As shown above, there are significant differences in the well as those without.
predictability of community assembly. Here, we show Therefore, to classify priority effects by their predict-
that we can decompose community predictability into ability, we propose to use the four dynamical sources (to-
four basic sources: the number of alternative stable pological features): alternative stable states, alternative
states, the number of alternative transient paths, the transient states, compositional cycles and the presence
6 | UNTANGLING THE COMPLEXITY OF PRIORITY EFFECTS IN MULTISPECIES COMMUNITIES (a) (b) 1.00 Predictability on final compositions 15 0.75 10 Invasion times 8 0.50 5 4 3 0.25 200 150 2 100 50 1 0.25 0.50 0.75 0.00 0.25 0.50 0.75 1.00 Predictability on temporary compositions Predictability F I G U R E 3 Regularities in the predictability of community assembly. Panel (A) shows the correlation (Pearson correlation = .72 with 95% confidence interval [.68,.75]) between the predictability of community assembly on the temporary compositions (x axis) and on the final compositions (y axis). Recall that all assembly dynamics with predictability less than 1 correspond to priority effects. The colour map represents the number of data points. Panel (B) plots the predictability distribution of assembly dynamics as a function of invasion times. The predictability (x axis) corresponds to the predictability on the final composition. The distribution converges to a stationary distribution with increasing invasion times, that is, the multiple humps in the distributions would be fixed or absence of an escape from cycles to stable states. We importance of the number of stable states remains con- used a neural network to carry out this classification. In stant across invasion times, while the relative importance short, the architecture of the neural network is as follows: increases for both the number of transient paths and the the input layer is a four-dimensional vector, which en- presence of an escape from cycles to stable states, but it codes the four topological features of an assembly graph; decreases for the length of cycles. Moreover, this result the five hidden layers all have ReLU activation; and the shows that the relative importance of the number of sta- output layer is the explained predictability. Appendix C ble states can be smaller than the combined importance provides a more detailed description of the neural net- of the other three sources (similar results are found for work. For simplicity, we measure the explanatory (clas- larger communities, see Appendix D), revealing the need sification) power using the correlation between observed to account for these other topological features to bet- and classified predictability in the out-of-sample test set. ter understand the role of priority effects in ecological Focusing on the case of three species, we found that the communities. classification works better for assembly dynamics with multiple invasions. Specifically, the classification dis- played an explanatory power of 0.57 and 0.98 for single F ROM T H E ORY TO (m = 1; Figure 5A) and multiple (m = 15; Figure 5B) inva- T E STA BL E H Y P OT H E SE S sions, respectively. The explanatory power increases with the number of invasions and reaches a plateau around Our theoretical framework illustrates that priority ef- m = 8 invasions (Figure 5C). These results are qualita- fects can take a large set of possibilities, most of which tively the same for larger communities (Appendix D). are outside the realm of traditional theoretical predic- Finally, we used a regression- based scheme tions. However, only empirical evidence can discern (Grömping et al., 2006) to quantify the relative impor- which assembly dynamics are possible and which are tance (contribution) of each topological feature (dynam- not, given the internal and external constraints acting ical source) to the classification power of priority effects. on ecological communities (Medeiros et al., 2021). For Figure 5D shows that the relative importance of topo- example, Drake (1991) recorded all the necessary data to logical features changes with the number of species in- empirically map the assembly graph for three species of vasions. Additionally, this figure shows that the relative algae (see also Zimmermann et al. (2004)), while Warren
SONG et al. | 7
(a)
Alternative stable states Alternative transient paths Compositional cycle
species species
arrival arrival
Composition Composition Composition
(b) Example 1 (c) Example 2
# alternative stable states =2 # alternative stable states =2
A,B,C # alternative transient paths = 1 # alternative transient paths = 1
A,B,C
length of compositional cycle = 3 length of compositional cycle = 3
Escape from cycle to stable states NO escape from cycle to stable states
A,B B,C A,C A,B B,C A,C
A B C A B C
Predictability under single invasions = .55 Predictability under single invasions = .32
Predictability under multiple invasions = .78 Predictability under multiple invasions = .18
Invasion by species A Invasion by species B Invasion by species C
F I G U R E 4 Dynamical sources of priority effects. Panel (A) presents the three known dynamical sources of priority effects (Fukami,
2015): alternative stable states, alternative transient paths and compositional cycles. We illustrate these sources with two plants (cartoons).
Panels (B) and (C) illustrate how these dynamical sources can be represented as topological features of assembly graphs. Panel (B)
shows an assembly graph for three hypothetical species. It has two alternative stable states ({A, C} and {C}; boxed in purple lines), one
transient path for each stable state ({A} → {A, C} and {A} → {B} → {A, C}; boxed in green lines) and one compositional cycle of length 3
({B} → {A, B} → {B, C} → {B}; boxed in purple lies). Panel (C) shows an assembly graph for another set of three hypothetical species (B).
It also has two alternative stable states ({A} and {A, B, C}; boxed in purple lines), one transient path for each stable state ({A} → {A} and
{C} → {A, C} → {A, B, C}; boxed in green lines) and one compositional cycle of length 3 ({B} → {A, B} → {B, C} → {B}; boxed in purple lies).
However, the assembly graphs in Panels (B) and (C) have drastically different predictability. The cause of this difference is a missing fourth
topological feature: escape from compositional cycle to stable states. In Panel (B), the system can escape from the compositional cycles to stable
states (i.e. from {B} to {C}). In Panel (C), the system cannot escape
et al. (2003) empirically mapped the assembly graph for determining which species combinations can persist and
six species of ciliates. In Figure 6, we analysed these then mapping how these combinations change after a
two assembly graphs following our methodology. The new species is introduced. Although this direct approach
predictions of our framework apply to these two com- may seem to require an exhaustive combinatorial design
munities. First, they exhibit combinations of dynamical that is too labour-intensive to be feasible, the actual ex-
sources (topological features). Second, their predictabil- perimental workload can be much lower.
ity increases with more invasions and then saturates to a To explain the work required, we first focus on iden-
fixed value. Third, their predictability can be explained tifying the persistent species combinations (i.e. knowing
from the generic importance (obtained from the trained the nodes in assembly graphs). While a community with
neural network) of each dynamical source. S species has 2S ‒1 potential species combinations, it
Unfortunately, we were unable to locate any other has been shown that only a small fraction of such com-
empirical studies that mapped assembly graphs. The binations can persist (Angulo et al., 2021). Because of the
dearth of empirical studies regarding these dynamics sparsity of these persistent combinations, recently devel-
is not surprising given both the challenges involved in oped computational tools based on Bayesian inference
performing detailed experiments and also their under- (Maynard et al., 2020) and deep learning (Michel-Mata
appreciated potential to answer central questions in et al., 2021) can facilitate the inference of all persistent
community ecology (Fukami, 2015; Vellend, 2016). In combinations using a small number of experiments in a
this regard, we discuss some viable empirical designs for community with S species. In brief, the method devel-
inferring assembly graphs in ecological communities. A oped by Maynard et al. (2020) can infer the coexistence
direct approach to infer assembly graphs should include of all 2S ‒1 species combinations from a minimum of8 | UNTANGLING THE COMPLEXITY OF PRIORITY EFFECTS IN MULTISPECIES COMMUNITIES (a) Invasion time = 1 (b) Invasion times = 15 1.00 1.00 Observed predictability Observed predictability 0.75 0.75 0.50 0.50 0.25 5000 0.25 2000 4000 3000 1500 2000 1000 0.00 1000 0.00 500 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Classified predictability Classified predictability (c) Explanatory power (d) Feature importance 1.0 1.00 # stable states observed and classified Correlation between Relative importance 0.9 # transient paths 0.75 to classification 0.8 0.50 0.7 Length of cycle 0.25 0.6 Esacpe from cycles to stable states 0.5 0.00 1 2 3 4 5 6 7 8 9 101112131415 1 2 3 4 5 6 7 8 9 101112131415 Invasion times Invasion times F I G U R E 5 Explaining the predictability of community assembly using topological features. This figure illustrates the out-of-sample explanation power of the four dynamical sources (topological features) in classifying the predictability on the final composition for three species. We used 80% of 41,979 assembly dynamics as the training set, while the remaining 20% was used as the test set (the classification). We used a neural network to achieve the maximum classification power (see Appendix C for details). Panel (A) shows the explanation power for the pool of assembly histories where each species invades only once. The x-a xis represents the classified predictability (from the test set) using the four topological features, while the y-a xis represents the observed predictability. The colour of the point indicates how many assembly graphs are represented by the point. The grey line is the 45-degree line, representing that the observed predictability equals the classified predictability. Panel (B) shows the explanation power for the pool of assembly histories where each species invades 15 times. Panel (C) shows how the explanatory power increases with invasion times and reaches a plateau fast. We indicate the explanatory power using the correlation between the observed and classified predictability (using different pools of potential assembly histories). Panel (D) shows the relative importance of each topological feature (dynamical source) to the explanation power S + 1 experiments based on measures of species’ abso- combinations (Figure S7), which would make the infer- lute abundances (1 experiment is to grow the full com- ence problem more feasible. munity of S species and the other S experiments are to Focusing now on how species composition changes grow leave-one-out communities comprising S ‒ 1 spe- after the introduction of a new species (i.e. knowing the cies each). The goal of the method proposed by Michel- edges in assembly graphs), we suggest that the experi- Mata et al. (2021) is similar to Maynard et al. (2020), mental procedure can follow the standard procedures in but a major difference is that Michel-Mata et al. (2021) assessing the effects of species introductions (Friedman only requires experimental measures of species’ relative et al., 2017; Grainger et al., 2019a; Meroz et al., 2021; abundances. Moreover, if we are only interested in bot- Spaak & De Laender, 2020). In brief, one would introduce tom-up assembly (i.e. starting with no species, and then an invader species to the resident community at low abun- adding species one by one), then persistent combinations dance (relative to the abundance of the resident species) that are unreachable via introduction of single species and assess whether species composition changes. Recently do not need to be mapped to construct the correspond- developed computational tools can also facilitate the in- ing assembly graph. In Appendix E, we analysed all as- ference (Deng et al., 2021; Pande et al., 2021). If we are sembly dynamics for three species and found that more only interested in bottom-up assembly, then many edges than 90% of assembly dynamics contained unreachable do not need to be mapped to construct the corresponding
SONG et al. | 9 (a) Drake (1991) (b) Warren, Law & Weatherby (2003) A,B,C A,B,D B,C,D A,B A,C B,C A,B A,C A,D B,D B,C C,D A B C A C B D Invasion by species A Invasion by species B Invasion by species C Invasion by species D (c) Explaining empirical predictability 0.6 2 5 Observed predictability 10 Drake (1991) 15 0.5 0.4 Warren et al. (2003) 0.3 0.3 0.4 0.5 0.6 0.7 Classified predictability F I G U R E 6 Applying the theoretical framework to empirical data. Panel (A) shows the empirical assembly graph for the three algal species studied by Drake (1991). Species A, B and C are Scenedesmus quadricauda, Selenastrum bibrium and Ankistrodesmus falcatus,respectively. This assembly graph has three stable states, one transient path, one compositional cycle with length 2, from which the community can escape to a stable state. Panel (B) shows the empirical assembly graph for the four protist species studied by Warren et al. (2003). Species A, B, C and D are Blepharisma japonicum, Colpidium striatum, Paramecium caudatum and Tetrahymena pyriformis,respectively. This empirical assembly graph has two stable states, one transient path and no compositional cycles. Some lines are missing in Panel (B) because Warren et al. (2003) did not resolve those links (see table 2 in Warren et al., 2003). Panel (C) shows the application of our framework on these two empirical examples. Similar to Figure 5A–B, the x-a xis represents the classified predictability (from the test set) using the four topological features, while the y-a xis represents the observed predictability. The colour represents the number of repeated invasions (ranging from each species invades two times to 15 times). As the number of repeated invasions increases (the direction of the arrows), the observed predictability of the two communities increases and saturates to a fixed value. The grey line is the 45-degree line, representing that the observed predictability equals the classified predictability. The classified predictability is close to the observed predictability assembly graph. In Appendix E, we analysed all assembly Xiao et al., 2017). Thus, the empirical assembly graphs dynamics for three species and found that more than 80% can be computationally mapped with empirically pa- of dynamics contained unreachable edges (Figure S8). rameterised population dynamics models. However, it While alternative stable states have been the most is worth remembering that a defining feature of prior- studied consequence of priority effects, our framework ity effects is that interaction strengths likely depend on indicates that the three other consequences of prior- assembly (Fukami, 2015; Song et al., 2018), questioning ity effects can have a stronger contribution to commu- the validity of inferences based on fixed interaction nity predictability, especially in small communities. strengths. Because we know little about how variable This possibility can be tested by exploiting the strong these interaction strengths are (Park, 1954), it would be constraints between dynamical sources and predict- best to combine experimental and computational ap- ability of priority effects. Specifically, to study these proaches. That is, the differences between the observed additional dynamical sources, we can use alternative assembly graph in the experimental approach and the computational approaches based on pairwise interac- inferred graph from the computational approach may tion strengths inferred from experiments on two-species provide clues as to how variable interaction strengths communities (Case, 2000). Such empirical data are in- are due to the assembly history. A caveat, though, is that creasingly available spanning a wide range of study sys- these assembly dynamics may take many generations to tems, such as annual plants (Godoy et al., 2014; Kraft emerge, thus computational models with fitted empiri- et al., 2015), perennial plants (Song et al., 2021; Uricchio cal interactions from only a few generations risk finding et al., 2019) and microbial systems (Kehe et al., 2020; spurious dynamics that do not exist.
10 | UNTANGLING THE COMPLEXITY OF PRIORITY EFFECTS IN MULTISPECIES COMMUNITIES DI SC US SION Additionally, we have shown that the predictability of a community generally increases with repeated in- In the story ‘The Garden of Forking Paths,’ Jorge Luis vasions (Figure 3B). On a conceptual level, with more Borges envisioned a labyrinth where divergence takes species invasions, deterministic ecological processes place in time rather than space, and where different paths are more likely to overwhelm stochastic events. In the sometimes lead to the same conclusion. Similarly, an as- context of assembly graphs, the topological features of sembly history on an assembly graph can be thought of assembly graphs become apparent only when the com- as a forking path since it also creates temporal trajecto- munity trajectories are long enough. Thus, laboratory ries of species composition that can lead to the same final experiments that only allow single invasions (e.g. Drake state or different ones. By borrowing tools from graph (1991); Lawler and Morin (1993); McGrady-Steed et al. theory, we have provided a non-parametric framework (1997)), as opposed to multiple invasions (e.g. Robinson to scan the complete terrain of the labyrinth of priority and Dickerson (1987)), may underestimate community effects. This framework has allowed us to classify pri- predictability in nature, where recurrent migrations are ority effects, enumerate all possible assembly dynamics common (Newton, 2010; Secor, 2015). The predictability operating in a community, and quantify how predictable of a community does not always increase with more inva- these dynamics would be if species arrival history was sions at the same rate, but rather saturates at a constant stochastic and unknown. value (Figure 3B). Thus, laboratory studies can approxi- We have introduced the concept of topologically mate community predictability in nature with two com- unique assembly graphs. This concept has allowed us plimentary approaches. The first approach is to perform to estimate the diversity of priority effects in multispe- a few invasions. About 10 times is more than enough cies communities. We have estimated the exact diversity for up to five species (Appendix D), provided that the for two-species and three-species sets, and, as a first- time interval between invasions is large enough to allow order approximation, for sets of more than three species the communities to approach an equilibrium. The sec- (Figure 1D). We have revealed a much richer set of assem- ond approach is to empirically map the assembly graph, bly dynamics than traditional parametric approaches which can fully determine the community predictability typically capture. The parametric approaches typically with an arbitrarily given pool of assembly history, al- make two pivotal assumptions: history-independent in- though it generally takes more experimental efforts. teraction strength and invasion analysis. In invasion Moreover, we have shown that this predictability can analysis (Grainger et al., 2019b), the invasion criteria be used to classify priority effects. We have shown that assume only two possibilities for a community with n four dynamical sources can be expressed as topological species after invasion: Either it has n + 1 species (if the features within our graph-based approach to know the invasion was successful) or remains with n species (if the predictability of assembly dynamics (Figure 4). These invasion was unsuccessful). However, evidence indicates sources are the number of alternative stable states, the that the set of potential assembly dynamics in ecological number of alternative transient paths, the length of com- communities can be larger than those considered by in- positional cycles and the presence or absence of an es- vasion analyses (Amor et al., 2020; Angulo et al., 2021; cape from compositional cycles to stable states. We have Barabás et al., 2018; Carlström et al., 2019; Deng et al., found that both the frequency and combination of the 2021; Saavedra et al., 2017; Warren et al., 2003). Thus, four topological features are good predictors of how our graph-based approach may provide a more realistic many outcomes to expect (Figure 5). We have also found analysis of ecological dynamics than those approaches that the explanatory power of these four topological fea- focusing on history- independent interaction strength tures increases when species attempt to invade multiple and invasion analysis. times (Figure 5C). While the number of alternative sta- Following previous work (Fukami, 2015), we have ble states has received most of the attention (Abreu et al., shown that priority effects can differ in terms of their 2020; Amor et al., 2020; Schröder et al., 2005; Serván & contribution to community predictability. We have fo- Allesina, 2021), we show that the other three sources, es- cused this predictability on species composition given pecially in small communities, can contribute more to the uncertainty derived from the potential assembly his- community predictability (Figure 5D). The trained neu- tories (Figure 2). We have demonstrated that two types ral network has predicted the observed predictability of of predictability can be investigated: on the final com- two empirical assembly dynamics (Drake, 1991; Warren position and on the temporary changes in composition et al., 2003) well (Figure 6). (trajectories). We found that these two types are often Our non- parametric graph- based approach is not correlated and yield similar results (Figure 3A). Thus, intended to replace the parametric model- based ap- the two types of predictability have similar information, proach. Parametric models are irreplaceable tools to a phenomena that is analogous to ergodicity in statis- understand priority effects, but non- parametric ap- tics (Strogatz, 2014). However, assembly dynamics is in proaches are more flexible for accommodating differ- general not ergodic (e.g. a stable state would render the ent theoretical tools (Arnoldi et al., 2019; Barabás et al., dynamics non-ergodic) and more research is necessary. 2018; Pande et al., 2020; Spaak & De Laender, 2020).
SONG et al. | 11 Angulo, M.T., Kelley, A., Montejano, L., Song, C. & Saavedra, S. For example, while our approach uncovers three other (2021) Coexistence holes characterize the assembly and disas- types of priority effects for two species (Figure 1) that sembly of multispecies systems. Nature Ecology & Evolution, 5, are not covered in classic Lotka–Volterra approaches 1091–1101. (Fukami et al., 2016; Ke & Letten, 2018), all priority ef- Arnoldi, J.-F., Barbier, M., Kelly, R., Barabás, G. & Jackson, A.L. (2019) Fitness and community feedbacks: the two axes that drive fects can occur in parametric models by integrating pro- long-term invasion impacts. bioRxiv, 705756. cesses where assembly history affects parameters. Thus, Barabás, G., D'Andrea, R. & Stump, S.M. (2018) Chesson’s coexis- our non-parametric graph-based approach can serve as tence theory. Ecological Monographs, 88, 277–303. a roadmap for the parametric approach by motivating Capitán, J.A., Cuesta, J.A. & Bascompte, J. (2009) Statistical me- new modelling strategies of priority effects. A major chanics of ecosystem assembly. Physical Review Letters, 103, 168101. limitation of our approach is that we have focused on Carlström, C.I., Field, C.M., Bortfeld- Miller, M., Müller, B., species richness or species composition. The general Sunagawa, S. & Vorholt, J.A. (2019) Synthetic microbiota reveal concept of priority effects also covers the functional priority effects and keystone strains in the arabidopsis phyllo- properties of ecological communities, such as energy sphere. Nature Ecology & Evolution, 3, 1445–1454. flow and productivity (Dickie et al., 2012; Fukami & Carter, S.K. & Rudolf, V.H. (2019) Shifts in phenological mean and synchrony interact to shape competitive outcomes. Ecology, 100, Morin, 2003; Tan et al., 2012). A possible solution is to e02826. establish a functional map from the nodes or the links Case, T.J. (2000) An illustrated guide to theoretical ecology. Oxford: in the assembly graph onto the functional property (e.g. Oxford University Press. productivity function in Rohr et al., 2016). The flexibil- Clay, P.A., Duffy, M.A. & Rudolf, V.H. (2020) Within-host priority ity of our framework may serve as a common currency effects and epidemic timing determine outbreak severity in co- infected populations. Proceedings of the Royal Society B, 287, to characterise priority effects across study systems and 20200046. theoretical models. Deng, J., Angulo, M.T. & Saavedra, S. (2021) Generalizing game- changing species across microbial communities. ISME AC K NOW L E D GE M E N T Communications, 1, 1–8. We thank Lucas P. Medeiros, Hengxing Zou and Dickie, I.A., Fukami, T., Wilkie, J.P., Allen, R.B. & Buchanan, P.K. (2012) Do assembly history effects attenuate from species to Pengjuan Zu for insightful discussions. 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