# Uncovering the relative movements of ecological trajectories

Nicolas Djeghri, Olivier Gauthier, Anthony Sturbois, Scott J Meiners, Robert B Allen, Miquel de Cáceres

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Nicolas Djeghri, Olivier Gauthier, Anthony Sturbois, Scott J Meiners, Robert B Allen, et al.. Uncov-
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1 **Uncovering the relative movements of ecological trajectories**

2 Nicolas Djeghri¹, Olivier Gauthier¹,², Anthony Sturbois¹, Scott J. Meiners³, Robert B.

3 Allen⁴, Miquel De Cáceres⁵

4 ¹Univ Brest, CNRS, IRD, Ifremer, UMR6539 LEMAR, F-29280 Plouzané, France.

5 ²Univ Brest, CNRS, IRD, UAR3113 OSU IUEM, F-29280 Plouzané, France.

6 ³Eastern Illinois University, Department of Biological Sciences, IL 61920 Charleston,

7 USA.

8 ⁴Independent researcher, 8 Roblyn Place, 7608 Lincoln, New Zealand.

9 ⁵CSIC-CREAF, E-08193, Bellaterra, Catalonia, Spain.

10

11 **Abstract**

12 Movement analogies are often employed by ecologists to describe how ecological
13 dynamics relate to one another. For instance, two communities whose similarity
14 increases in time may be said to converge. Here we argue that the movement analogies
15 used by ecologists to compare ecological dynamics could be enriched with other notions
16 such as “pursuit” or “parallel” movements, if accompanied by appropriate statistical
17 testing. By building on the framework of Ecological Trajectory Analysis, we present here
18 Relative Trajectory Movement Assessment (RTMA), a statistical method to detect and
19 qualify the existence of relative movements in ecological dynamics defined as
20 trajectories in multivariate space. Using synthetic trajectory data, we illustrate how RTMA
21 can reveal a diverse range of relative movements beyond the convergence and divergence
22 patterns already recognized in ecology. We exemplify the use of RTMA on real ecological
23 datasets describing 1) old field successional dynamics in eastern North America and 2)
24 temporal patterns in tree size structure of a New Zealand forest. RTMA provides ecologists
25 with a new way of describing and comparing ecological dynamics that could be widely
26 applied, from plot-scale dynamics to the effects of global change.

27 **Keywords:** beta diversity, convergence, divergence, ecological dynamics, ecological
28 trajectory analysis, multivariate statistics, time series, trend analysis

29

30 **1. Introduction**

31 To describe and compare the dynamics of the systems they study, ecologists often rely
32 on movement analogies. For example, the composition of different ecological
33 communities may change over time and, if similarity increases, the communities can be
34 said to have converged (Fukami et al. 2005, Li et al. 2016, McLean et al. 2019). The notions
35 of convergence or divergence in ecological systems can thus critically inform on the
underlying ecological mechanisms (Vellend 2016). Convergence may be interpreted as a
sign of a common selective mechanism, while divergence may be seen as indicative of
drift or priority effects (Fukami et al. 2005). Here we argue that movement analogies can
be expanded beyond convergence and divergence as a richer descriptive tool for
ecological dynamics. For instance, a widespread result of global warming is the poleward
displacement of species distributions (i.e. tropicalization and deborealization, Maire et
al. 2019, McGinty et al. 2021, Chust et al. 2024). Broadly speaking, the communities
inhabiting polar latitudes increasingly resemble communities from temperate latitudes.
However, temperate latitudes communities also change, progressively increasing their
resemblance to tropical latitudes communities. The best movement analogy for such a
pattern may not be convergence or divergence but “pursuit” with polar regions pursuing
temperate regions, which are themselves pursuing tropical regions in terms of community
composition. Convergence and divergence can therefore be seen as facets of a richer
ensemble of movement analogies that can be used to describe how two ecological
entities change with respect to one another (i.e. to describe their relative movements
(Dodge et al. 2008)) including relationships such as “pursuit” or “parallel” movements.
Once this diversity of relative movements is recognized, the question of their formal
description and detection emerges.

One promising way to formalize ecological dynamics with the aim of describing their
relative movements is the framework of Ecological Trajectory Analysis (ETA, De Cáceres
et al. 2019, Sturbois et al. 2021). In ETA, a trajectory represents the temporal dynamics of
a given ecological entity (e.g. a community) as a set of temporally linked ecological states
(i.e. points) in a multidimensional space. Importantly, trajectories are considered in their
full multidimensional space without dimensionality reduction in ETA, although
ordinations are commonly used for visualization. This has the benefit of conserving all
information when studying complex multidimensional datasets with this approach.
Moreover, trajectories in ETA are computed from distances between pairs of ecological
states, allowing the framework to flexibly accommodate many types of ecological data
using adapted distance metrics (Legendre and De Cáceres 2013, De Cáceres et al. 2019).
Applications of ETA range from community ecology to stable isotopes analyses through
paleoecology (Sturbois et al. 2022, Tye et al. 2024, de la Casa et al. 2025, Fentimen et al.
2026) and extensions or adjacent frameworks allow taking account of reference
ecological conditions (Sturbois et al. 2023), work with cyclical dynamics (Djeghri et al.
submitted), or define and characterize dynamic ecological regimes (Sánchez-Pinillos et
al. 2023). Importantly for the purposes of the present work, ETA emphasizes and allows
formal description and comparison of the direction, size, speed and shape of ecological
trajectories making it fit to study trajectory relative movements.

In the original ETA framework (De Cáceres et al. 2019), relative trajectory movements are
approached through the application of symmetric and asymmetric convergence tests
based on the Mann-Kendall trend test (Mann 1945). Initially, the two convergence tests
76 were presented as alternatives that may be used if the two studied trajectories have
77 synchronous sampling times (both symmetric and asymmetric tests possible) or not
78 (asymmetric test only). However, the authors noted that these tests do not assess the
79 same aspects of convergence. The symmetric test assesses if the two trajectories
80 become closer over time, while the asymmetric test assesses if one trajectory becomes
81 closer to any part of the other trajectory.

82 Here, we argue that combining different statistical tests allows distinguishing not only
83 convergence and divergence, but also diverse relative movements relationships such as
84 "pursuit", "catch-up" or "parallel" movements. We refer to this approach as Relative
85 Trajectory Movement Assessment (RTMA). In the sections below we present the two
86 convergence tests existing in ETA and propose a new complementary test of trajectory
87 dynamic correspondence. We then show how combining those tests allows us to build
88 RTMA. We provide an evaluation of RTMA against simulated trajectories, demonstrating
89 its ability to distinguish different patterns of relative trajectory movements. We then apply
90 RTMA to two real ecological datasets on 1) old field compositional succession and 2)
91 temporal change in forest size structure to show how it can be used to describe
92 ecologically relevant patterns and processes.

93

94 **2. Assessing the Relative Movements of Trajectories**

95 **2.1. Trajectories and convergence tests in Ecological Trajectory Analysis**

96 In Ecological Trajectory Analysis (ETA), an ecological trajectory **T** is defined as the set {$x_1, ..., x_n$} of time-ordered ecological states **x**, with **n** being the size of **T** (De Cáceres et al. 2019). The ecological states are points in a multidimensional space of interest where distances between any two points $x_1$ and $x_2$ are specified using an appropriate distance metric $d(x_1, x_2)$ (see Legendre and De Cáceres 2013 for a review of some distance metrics useful in ecology and their properties). Trajectories can also be viewed as set of directed segments {$s_1, ..., s_{n-1}$} linking consecutive ecological states. In ETA, the distance between an ecological state and a trajectory $D_{pt}(x, T)$ can be computed and is defined as the shortest distance between **x** and any points or segments belonging to **T** (ecological state-to-segment distances being computed by orthogonal projection; a more detailed description is given in De Cáceres et al. 2019).

107 In the original ETA (De Cáceres et al. 2019), the relative trajectory movements of two given trajectories **T**<sub>A</sub> = {$x_{A1}, ..., x_{An}$} and **T**<sub>B</sub> = {$x_{B1}, ..., x_{Bn}$} of the same size **n** can be approached using two different convergence tests:

110 The symmetric convergence test is a Mann-Kendall test performed on the series
111 {$d(x_{A1}, x_{B1}), ..., d(x_{An}, x_{Bn})$} of pairwise distances between the successive ecological states
112 from the two trajectories (Fig. 1a). A negative tau statistic indicates convergence, while a
113 positive tau indicates divergence.
**The asymmetric convergence test** of $T_A$ approaching $T_B$ is a Mann-Kendall test performed on the series {$D_{pt}(X_{A1}, T_B), ..., D_{pt}(X_{An}, T_B)$} of distances between the ecological states of $T_A$ and $T_B$ (Fig. 1b). A negative *tau* statistic indicates that $T_A$ approaches $T_B$, while a positive *tau* indicates that $T_A$ is moving away from $T_B$. The converse asymmetric convergence test assesses whether $T_B$ is approaching $T_A$ (Fig. 1c).

<figure><img src="image_2.png" /><figcaption>**Figure 1** The four different tests available in Ecological Trajectory Analysis (ETA) to assess the relative movement of two trajectories, $T_A$ and $T_B$, of the same size. Three convergence tests were already described in De Cáceres et al. (2019): The symmetric convergence test (a), the asymmetric convergence test of $T_A$ approaching $T_B$ (b), the asymmetric convergence test of $T_B$ approaching $T_A$ (c). ETA convergence tests are based on Mann-Kendall trend tests realized on different series of distances (grey links, only first distance formula noted). We provide a new dynamic correspondence test (d and e) whose statistic epsilon (u) relies on the comparison of two sets of distances (in grey and orange).</figcaption></figure>

## 2.2. Dynamic correspondence test

In cases where two trajectories neither converge nor diverge, they may still present ecologically relevant, non-random relative movements. This importantly occurs when the two trajectories follow a similar-shaped path travelling side-by-side in the same direction or in opposite directions. We term such relative movements “parallel” and “antiparallel” respectively.

To complement the convergence tests and allow tackling the question of trajectory parallelism we propose the “dynamic correspondence test” performed between two
trajectories **T**<sub>A</sub> and **T**<sub>B</sub> with the same number *n* of ecological states. We name its test statistic *epsilon* (u) and define it as:

$$v = \sum_{i=1}^{n} d(x_{Ai}, x_{Bn-(i-1)}) - \sum_{i=1}^{n} d(x_{Ai}, x_{Bi})$$

In this statistic, the right-hand member represents the sum of the same distances as used in the symmetric convergence test. This member will be low if the two trajectories follow a similar shaped path in the same direction and high otherwise. The left-hand member represents the sum of distances between ecological states “opposed in time” in the two trajectories (i.e. distances between first and last ecological state, between second-to-first and second-to-last ecological states etc. of **T**<sub>A</sub> and **T**<sub>B</sub>). This member will be low if the two trajectories follow a similar path in opposed directions and high otherwise. Therefore, *epsilon* will be high and positive if the two trajectories follow a similar-shaped path in the same direction (Fig. 1d) and will be low and negative if the two trajectories follow a similar-shaped path in opposite directions (Fig. 1e). *Upsilon* is tested by permutation of the ecological states order within the two trajectories.

We used simple simulated trajectory pairs to assess the behavior of the dynamic correspondence test in more detail and present the results in Appendix S1. Our simulations reveal that the dynamic correspondence test yields a correct type 1 error and that it finds significant results under a broad scope of situations (varying the two trajectories relative position, shape, main direction...). Used on its own, the dynamic correspondence test should therefore not be seen as a diagnostic tool for parallel or antiparallel movements (see section 2.3 for the combination of this test with the convergence tests). Instead, the dynamic correspondence test is a way to telling whether two trajectories have broadly the same (positive *epsilon*) or opposed (negative *epsilon*) direction of travel. For more complex shaped trajectories that lack overall directionality (i.e. trajectories that present many turns), the dynamic correspondence test will be more sensitive to trajectory shapes.

## 2.3. Relative Trajectory Movement Assessment (RTMA)

Recapitulating the previous sections, four tests can be used to evaluate the relative movement between two given trajectories of same size **T**<sub>A</sub> and **T**<sub>B**: 1. the symmetric convergence test between **T**<sub>A</sub> and **T**<sub>B**, 2. the asymmetric convergence test assessing if **T**<sub>A</sub> approaches **T**<sub>B**, 3. the asymmetric convergence test assessing if **T**<sub>B** approaches **T**<sub>A** and 4. the dynamic correspondence test between **T**<sub>A** and **T**<sub>B**. We refer to the combination of those four tests as Relative Trajectory Movement Assessment (RTMA) and show how it allows assigning a relative movement relationships to pairs of trajectories, beyond classic convergence and divergence.
RTMA uses the significance against a chosen statistical significance (i.e. alpha) level and the sign of tests' statistics to distinguish between different relative movement relationships between two trajectories. Multiple testing is accounted for by applying the Šidák correction (Šidák 1967) to the alpha levels. Because of this correction, the minimum trajectory size that might yield significant result at alpha = 0.05 is n = 6. Formally, each test performed during RTMA can yield three results (either non-significant, significant with positive statistic, or significant with negative statistic) amounting to $4^3 = 64$ possible combinations. However, not all combinations are equally probable. Many are symmetric counterparts (i.e. trajectory A approaching trajectory B is conceptually the same as trajectory B approaching trajectory A) and the dynamic correspondence test becomes irrelevant once convergence or divergence has been detected. As a result, we propose a taxonomy of 10 trajectory relationships recognized by RTMA: Neutral, Parallel, Antiparallel, Convergence, Divergence, Approaching, Departing, Pursuit, Catch-up and Escape. The details of these relationships and their formal detection criteria are given in Table 1. Some relationships exist in “weak” versions, characterized by the relaxation of one of the statistical test requirements (Table1). These weak versions of relationships denote that different combinations of significant tests can occur in qualitatively similar situations. Within RTMA, the dynamic correspondence test results are only useful if no significant convergence or divergence has been found, restricting its application to distinction between Neutral and Parallel or Antiparallel relationships (Table 1) and solving the limitation posed by convergence tests (see section 2.2). Performing RTMA can thus be understood as navigating a network of statistical tests results that allow the attribution of a relationship to a pair of trajectories (Fig. 2). Some combinations are not covered in Table 1 because they are deemed unlikely from a conceptual perspective (e.g. convergence found by the symmetric test with divergence found by both asymmetric tests). RTMA labels these as “unknown” when they occur.

The relationships recognized by RTMA can further be grouped to describe some important, high-order, properties. We recognize three higher-order groups that are not always mutually exclusive (Table 1, Fig. 2):

- the **Convergence group** (Convergence and Approaching and their weak versions, as well as Catch-up) groups relationships where the distance between the two trajectories broadly decreases over time.

- the **Divergence group** (Divergence and Departing and their weak versions, as well as Escape) groups relationships where the distance between the two trajectories broadly increases over time.
<table><caption>**Table 1** Typology of relative movement relationships recognized in Relative Trajectory Movement Assessment (RTMA) and formal detection method combining results from four statistical tests. In relationships belonging to the oriented group, the illustration depicts the “back” trajectory in black and the “front” trajectory in grey. Weak versions of some relationships exist when less restrictive test combinations can emerge from situations close to the archetype.</caption><thead><tr><th>Relative movement relationship</th><th>Symmetric convergence test</th><th>Asymmetric convergence tests (x2)</th><th>Dynamic correspondence test</th><th>Group(s)</th><th>Archetype</th></tr></thead><tbody><tr><td><strong>Neutral</strong></td><td>Non-significant</td><td>Both non-significant</td><td>Non-significant</td><td>No group, Null Hypothesis for RTMA</td><td></td></tr><tr><td><strong>Parallel</strong></td><td>Non-significant</td><td>Both non-significant</td><td>Significant, positive <em>upsilon</em></td><td>No group</td><td></td></tr><tr><td><strong>Antiparallel</strong></td><td>Non-significant</td><td>Both non-significant</td><td>Significant, negative <em>upsilon</em></td><td>No group</td><td></td></tr><tr><td><strong>Convergence</strong><br>(and Weak Convergence 1 & 2)</td><td>Significant, negative tau (non-significant in Weak Convergence 2)</td><td>Both significant with negative tau (both non-significant in Weak Convergence 1)</td><td>Not applicable</td><td>Convergence group</td><td></td></tr><tr><td><strong>Divergence</strong><br>(and Weak Divergence 1 & 2)</td><td>Significant, positive tau (non-significant in Weak Divergence 2)</td><td>Both significant with positive tau (both non-significant in Weak Divergence 1)</td><td>Not applicable</td><td>Divergence group</td><td></td></tr></tbody></table>
<table><tr><td><strong>Approaching</strong><br>(and Weak Approaching)</td><td>Significant, negative tau (non-significant in Weak Approaching)</td><td>One significant with negative tau</td><td>Not applicable</td><td>Convergence group &amp; Oriented group</td><td></td></tr><tr><td><strong>Departing</strong><br>(and Weak Departing)</td><td>Significant, positive tau (non-significant in Weak Departing)</td><td>One significant with positive tau</td><td>Not applicable</td><td>Divergence group &amp; Oriented group</td><td></td></tr><tr><td><strong>Pursuit</strong></td><td>Non-significant</td><td>Both significant with opposed tau</td><td>Not applicable</td><td>Oriented group</td><td></td></tr><tr><td><strong>Catch-up</strong></td><td>Significant, negative tau</td><td>Both significant with opposed tau</td><td>Not applicable</td><td>Convergence group &amp; Oriented group</td><td></td></tr><tr><td><strong>Escape</strong></td><td>Significant, positive tau</td><td>Both significant with opposed tau</td><td>Not applicable</td><td>Divergence group &amp; Oriented group</td><td></td></tr></table>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>
-the **Oriented group** (Approaching and Departing and their weak versions, as well as Catch-up, Pursuit, and Escape) groups relationships where there is a qualitative asymmetry in the role of the two trajectories. Formally, oriented group relationships have either, only one significant asymmetric convergence test, or two significant asymmetric convergence tests with opposed tau. Many pairs of words may be adapted to describe the asymmetric roles of the two trajectories depending on the exact relationship (i.e. leader/follower, target/approacher...). For simplicity, we choose to use the terms of "back" and "front" trajectories to distinguish the two roles in the general case (see visualization in Table 1).

<figure><img src="image_8.png" /><figcaption>**Figure 2** Directed network showing how the different relative trajectory movement relationships (nodes) recognized by Relative Trajectory Movement Assessment (RTMA) are related to one another through statistical tests (arrows, pointing toward the relationship where the test is significant). Relationships "Convergence", "Divergence", "Approaching", and "Departing" exist in weak versions (lighter-colored, dotted nodes). Three groups of relationships recognized by RTMA are highlighted as large colored boxes.</figcaption></figure>

## 2.4. Simulation study

To evaluate the performance of RTMA under known relative movement relationships, we simulated pairs of trajectories for each of the ten archetypes of relative movement illustrated in Table 1. Each prototype was tested under five levels of trajectory sizes (i.e. the number of ecological states constituting the trajectory), and five levels of sampling noise (independent Gaussian error term added). For simplicity, trajectories were defined
239 in two dimensions. In each combination of simulated archetype, level of trajectory size,
240 and noise, we ran 1000 replicates and assessed whether RTMA was able to recover the
241 correct relationship given the archetype tested (Appendix S2).

242 In general, the simulation study demonstrated satisfactory behavior of RTMA for the ten
243 archetypes (Fig. 3). First, the test displays correct type 1 error with the Neutral scenario
244 approaching 95 % correct detection with alpha = 0.05 (Fig. 3). As expected, detection of
245 non-neutral relationships was in all cases impaired by small trajectory size and high levels
246 of noise (Fig. 3). Detection sensitivity of RTMA may however vary among trajectory
247 relationships with Parallel and Antiparallel relationships more consistently detected than
248 others (detection over wider noise and trajectory size conditions, Fig. 3). Our simulations
249 further showed that, when RTMA fails to detect the correct relationships at a high level of
250 noise and/or low sampling effort, it attributes them a Neutral relationship in most cases
251 (Appendix S2), a desirable property as the Neutral relationship is effectively the null
252 hypothesis of RTMA. Moreover, when an incorrect relationship attribution occurs, it
253 generally involves a similar relationship (e.g. Divergence may be wrongly detected as
254 Departing but not as Approaching or Pursuit; Catch-up may be detected as Approaching
255 or Pursuit but not as Departing: details in Appendix S2). The most commonly
256 misattributed relationships were weak versions of the Approaching and Departing
257 relationships (Appendix S2). Because these relationships are very central in the network
258 of tests that RTMA represents (they have the most connections with other relationships,
259 Fig. 2), it is likely that an error in one statistical test drives RTMA to characterize a
260 relationship as either weak approaching or weak departing. Finally, combinations that
261 were deemed unlikely and hard to interpret when designing RTMA (i.e. RTMA assigning
262 “unknown”) were, as expected, very rare in our simulations.

263

264 **2.5. Software**

265 Functions performing and providing graphical representation for RTMA (see the ecological
266 examples) have been developed in the R language (R Core Team 2025) and included in the
267 ecotraj package from version 1.2.1 available on CRAN and GitHub repositories
268 (https://emf-creaf.github.io/ecotraj/).

269
<figure><img src="image_9.png" /><figcaption>**Figure 3** Proportion of correct relationship attribution (over 1000 simulations) by Relative Trajectory Movement Assessment (RTMA) at alpha = 0.05 in pairs of trajectories simulated according to the 10 archetypes (see Table 1) with different trajectory sizes (y axes) and sampling noise (x axes). The Neutral relationship (middle) corresponds to the null hypothesis for RTMA.</figcaption></figure>

3. Ecological Applications

**3.1. Field-to-forest vegetation succession**

To show how Relative Trajectory Movement Analysis (RTMA) is useful in ecological
research, we first apply it to the classic ecological question of secondary vegetation
succession (Poorter et al. 2023). Secondary succession, in its most classic form, is a
process in which a disturbed plant community changes over time from an unvegetated
state to a forested state. Secondary succession is a well-known and fairly predictable
ecological process with species marked by different traits successively dominating a
plant community (from herbs to shrubs to trees, Meiners et al. 2015). The well-known and
predictable nature of secondary succession (Myster and Pickett 1990) makes it a good
process for illustrating RTMA.
We use data from the Buell-Small Succession Study (BSS), a monitoring program that follows 10 fields (termed C3, C4, C5, C6, C7, D1, D2, D3, E1 and E2) in New Jersey, USA, 40°30' N, 74°34' W, from their agricultural abandonment between 1958 and 1966 to a late successional stage in 2025 (Meiners et al. 2015). Each BSS field is sampled at 48 permanent locations (1 m² plots) where plant community composition is measured as species percent coverage. The sampling frequency was annual at the start of the series (1958-1979) and switched to alternate years from 1980 onwards, with half of the field sampled on odd years, the other half on even years. Here, we average plot data at the field level and express dissimilarity between plant communities as percentage difference (*alias* Bray-Curtis dissimilarity, Odum 1950, Bray and Curtis 1957), resulting in one trajectory per BSS field.

To more precisely assess how RTMA documents secondary succession, we divided each trajectory into four sub-trajectories, representing different successional stages defined by field age (Myster and Pickett 1990, Yurkonis and Meiners 2006, Meiners 2007, Fig. 4a). “Early succession” (2-14 years after abandonment) captured the transition from short-lived herbaceous plants like *Ambrosia artemisiifolia* and *Daucus carota* to perennial herbs such as *Solidago* sp. and *Fragaria virginiana*. “Middle succession 1” (years 15-28) was when opportunistic woody species such as *Rubus allegheniensis*, *Rosa multiflora* and *Lonicera japonica* became abundant displacing perennial forbs. “Middle succession 2” (years 29-42) represented canopy closure by early successional trees like *Juniperus virginiana* and *Cornus florida*. Finally, “Late succession” (years 43-56) was dominated by a canopy of *Acer rubrum*, *Carya* spp. and *Quercus* spp. With an understory dominated by *Ageratina altissima*. The sub-trajectories are built so that they have the same number of ecological states (n = 7), always separated by two years, making them all comparable with RTMA. A missing data point for field C4 at the late succession stage led us to exclude that sub-trajectory.
<figure><img src="image_10.png" /><figcaption>**Figure 4** Application of Relative Trajectory Movement Assessment (RTMA) to field-to-forest secondary succession in plant communities sampled at the Buell-Small succession study (New Jersey, USA) between 1958 and 2025. a) main trajectories and sub-trajectories corresponding to different successional stages represented using Principal Coordinate Analysis (PCoA). b) RTMA results establishing the different relative movement relationships (see legend) between the sub-trajectories. Barred cells indicate that the corresponding row sub-trajectory is the “front” trajectory in an oriented relationship.</figcaption></figure>

314

315

316

317

318

319

320

321

322
Visualization of the BSS trajectories and sub-trajectories through Principal Coordinate Analysis (PCoA) revealed a strong arch effect (Fig. 4a), as expected in long ecological gradients with important species replacement (Legendre and Gallagher 2001). The relative trajectory movement relationships detected by RTMA between the different sub-trajectories are presented in Figure 4b in a heat-map-like plot. The features revealed by RTMA are consistent with expectations of secondary succession. First, when comparing sub-trajectories representing the same successional stages (groups highlighted by black boxes along the diagonal of Fig. 4b), we see a dominance of Parallel relationships (early and mid-succession) that is replaced by Neutral relationships in later successional stages. This is consistent with communities from different fields following similar successional paths until reaching late successional stages where trajectories lose directionality as communities stabilize (see also Li et al. 2016). When comparing sub-trajectories across different successional stages (away from the diagonal in Fig. 4b) we find a large dominance of the Approaching relationship and its weak version, both belonging to the oriented and convergence groups. Importantly, in the oriented group relationships, the trajectory in front was always a later successional stage, reflecting the strong directionality of succession (Fig. 4b). Taken together, the high directionality of succession and the strong dominance of relationships from the convergence group suggest that the process slows down over time, again consistent with expectations on secondary succession and previous studies on this system (Myster and Pickett 1994, Anderson 2007, Li et al. 2016).

344

## 3.2. Forest size structure temporal dynamics

At its core, RTMA labels the relative movement of pairs of trajectories with a verbal descriptor. One limitation of such labelling may occur when studying large datasets with many trajectories where it is not obvious which trajectory comparisons are of interest. In this second ecological example, we show how this difficulty can be alleviated by combining RTMA with trajectory clustering. We revisit a dataset used in the original ETA paper (De Cáceres et al. 2019) describing temporal dynamics in the size-structure of monodominant mountain beech (Fucospora cliffortioides) forest plots in the Craigieburn Forest Park (Southern Alps, South Island, New Zealand, 43°10′ S, 171°35′ E, (Harcombe et al. 1998)). Trajectories in this dataset are primarily driven by tree growth and arrival of new recruits, particularly in response to disturbances (a snowstorm and associated pathogens in 1973 (Harcombe et al. 1998) as well as a magnitude 6.7 earthquake in 1994 (Allen et al. 1999)) that left a legacy in the size structure of those forests (Coomes and Allen 2007). Following (De Cáceres et al. 2019), we use data from nine surveys (1970-1972, 1974, 1978, 1983, 1987, 1993, 1999, 2004 and 2009), from 250 plots representatively sampling the forest. The trunk diameter of each tree was measured and binned into 13 size classes (from 2.25 to 64 cm) to represent forest size structure.
362 Abundances were log-transformed and dissimilarities between communities were computed as Manhattan distances according to their cumulative abundance profile, a method allowing to represent differences in size structure (De Cáceres et al. 2013). The dissimilarity between the trajectories of the different plots was then computed as symmetrized directed segment path dissimilarity D<sub>SDSP</sub> (De Cáceres et al. 2019), a dissimilarity measure that takes into account the shape, direction and position of ecological trajectories. We then used these dissimilarities to reproduce the 5 trajectory clusters obtained in (De Cáceres et al. 2019) by *k*-means clustering based on the SSI criterion. Instead of presenting all the RTMA relationships between individual trajectories, we focus on relationships between trajectories belonging to different clusters.

372 In Fig. 5a, trajectory relationships obtained by RTMA are presented as treemaps showing the proportions of different relationships found between trajectories belonging to distinct clusters. The clusters themselves are presented in Fig. 5b and the corresponding median tree community size structures in Figs. 5c-g. Different trajectory clusters occupy different positions in the PCoA space (Fig. 5b). The treemaps show that oriented relationships are predominant (Fig. 5a) and suggest that trajectory clusters are generally ordered A -> B -> C/D -> E from the most "in the back" to the most "in the front". This progression corresponds to an increase in tree sizes and decrease in abundance (Figs. 5c-g) consistent with forest growth and thinning. It is moreover interesting to note that trajectories from cluster E are often involved in Antiparallel relationships with trajectories from other clusters (in particular cluster A, Fig. 5a). Cluster E is characterized by many small trees and a few particularly large ones (Fig. 5g), a size structure that may be explained by high mortality rates in large trees (Hurst et al. 2011) resulting in free space and light for new recruits. In this context, antiparallel relationships can be interpreted as a "return", opposed to the main growth dynamic, and linked primarily to recruitment. RTMA therefore suggests that, in addition to the main dynamic of forest growth and thinning, an important proportion of trajectories from cluster E represent recruitment events allowing stand re-initiation.
a) Cluster A

Cluster B

<figure><img src="image_11.png" /></figure>

Cluster B

Cluster C

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

Cluster C

Cluster D

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

Cluster D

Cluster E

<figure><img src="image_16.png" /></figure>

Cluster A

<figure><img src="image_17.png" /></figure>

Cluster B

<figure><img src="image_18.png" /></figure>

Cluster C

<figure><img src="image_19.png" /></figure>

PCoA 1 (5%)

**Treemaps legend**
*   Oriented group: Row cluster in front
*   Oriented group: Column in front
*   Divergence
*   Convergence
*   Parallel
*   Antiparallel
*   Neutral
*   Unknown

<figure><img src="image_20.png" /></figure>

Cluster D

Number of stems

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

Tree diameter classes (cm)
**Figure 5** Application of Relative Trajectory Movement Assessment (RTMA) to the size structure development of monodominant mountain beech forests in the Craigieburn Range (Southern Alps, South Island, New Zealand, 1970-2009). a) Treemaps representing the proportion of different RTMA relationships found between trajectories from five different clusters. The unlabeled rectangles correspond to individual relationships composing the groups highlighted by the treemaps. b) Principal Coordinate Analysis (PCoA) displaying the trajectories and the associated clusters. c-g) median size-structure in each of the cluster over time (lighter curves: beginning of the series, darker curves: end of the series).

401

## 4. Discussion

Using simulations and ecological applications, we have shown that the movement analogies used by ecologists to compare ecological dynamics can be enriched and diversified and that Relative Trajectory Movement Assessment (RTMA) is a valuable framework for this purpose. We believe RTMA creates a unique place within the statistical and conceptual toolkit of ecologists, complementing the description of single trajectories (Bagchi et al. 2017, De Cáceres et al. 2019, Lamothe et al. 2019) and allowing further characterization of how two dynamics relate to each other. Nonetheless, we believe RTMA is not without limitations, of which the users should be well aware.

411

### 4.1. Limitations and recommendations

A first important limitation of RTMA is its categorical nature. Relationships between trajectories are expressed as verbal, categorical descriptors lacking a quantification of effect size (although the statistics of the different tests combined in RTMA are still available). Such quantification is complex as the relationships between two trajectories is multifaceted (e.g. difference in position, difference in main direction of travel (if any), difference in speed...) and how these different facets should be reflected or weighted is unclear. Although we believe the categorical labels provided by RTMA are already useful, further developments could therefore seek to make RTMA, or alternative frameworks investigating relative trajectory movements, more quantitative and continuous.

The second limitation of RTMA is that it fundamentally relies on Mann-Kendall trend tests and as such, it is not well suited to describe changing (i.e. non-monotonic) relative movements of trajectories. This is generally the case when the trajectories present complex movements with numerous turns. Even with simpler and/or straighter trajectories, non-monotonicity in relative movements will occur when two trajectories first converge and then diverge (or the opposite), for example: if they cross each other's paths, or if one trajectory undergoes a sharp directional change (e.g. a 180° turn). Non-monotonicity can cause convergence tests to become non-significant and make RTMA
overly likely to find Weak Approaching and Weak Departing relationships due to their high connectivity to other relationships (Fig. 2).

Regarding those limitations, we make three recommendations: First, when making ecological interpretations, the relationships recognized by RTMA should not be regarded as strictly different but rather as part of a continuum going from Divergence to Convergence in a U shape in Figure 2 (i.e. Divergence is alike Departing, Departing is alike Escape, and so on). In this context, and as shown in our ecological examples, working at the level of relationships groups (i.e. oriented group, convergence group and divergence group) may be valuable. Parallel and Antiparallel relationships do not fall within this continuum nor in any group and, in our opinion, are better left as qualitatively different. We recommend that detailed interpretation of RTMA relationships should be undertaken carefully and accompanied as much as feasible by visualization of the target pair of trajectories in reduced dimensions to prevent misinterpretation. Third, if the trajectories are long enough (based on long and/or high resolution time series), it is possible to apply RTMA on trajectories defined according to moving windows (de la Casa et al. 2025). By breaking down complex dynamics into shorter and simpler parts, a moving window-based RTMA has the potential to greatly reduce issues associated with non-monotonic relative trajectory movements.

Finally, the null model assumed by RTMA is a very simple model of no trajectory movement (i.e. ecological stability). However, other null models, such as random walks, could be deemed suitable to represent “neutral” trajectory movement. With the present formulation of RTMA, application to random walks may yield significant results because of autocorrelation in ecological states. Developing RTMA with more complex null-models able to consider random walks will require more research.

454

## 4.2. Potential applications and conclusions

Because it belongs to the flexible, distance-based, Ecological Trajectory Analysis framework (De Cáceres et al. 2019), RTMA can be applied on a variety of data types from community compositions and size structures (as shown here), trait-based community representations (Palacio et al. 2025) to stable isotope data (Sturbois et al. 2022). This flexibility in the data used will allow a wide range of applications for RTMA from applied ecology and restoration to more descriptive analyses of changes in the biosphere. Our ecological applications showcase potential use in succession studies (Meiners et al. 2015). In ecological restoration, RTMA may allow assessment of whether an ecosystem receiving a restoration action follows another system known to have completed satisfactory restoration. At larger scales, and for more descriptive studies, RTMA may be used in the context of global change studies. For instance, there is concern that the biosphere is undergoing homogenization (loss of beta diversity, Magurran et al. 2015, Olden et al. 2018, Li et al. 2020). Homogenization would be expected to generate
relationships from the convergence group in RTMA. Global change also leads to
tropicalization and deborealization patterns with species generally migrating polewards
to accommodate change (Maire et al. 2019, McGinty et al. 2021, Chust et al. 2024). In this
context, it could be expected that trajectories representing the taxonomic or functional
composition of communities in different latitudes would follow each other (Pursuit
relationships, or more generally, relationships belonging to the oriented group in RTMA).
Our second ecological example further suggests that RTMA may be used to piece together
different time series, obtained from different places, to reconstruct longer dynamics that
were not observed in their entirety. In that sense, RTMA may help to perform hybrid
analyses that combine time series and space-for-time substitutions (Pickett 1989).

At a more fundamental level, we believe the relationships identified by RTMA can be
interpreted as ecological patterns that were previously difficult to detect and qualify.
Having the ability to detect and describe those patterns may foster links with theory and
help uncover the underlying ecological processes, a vexingly complicated task in ecology
(Ovaskainen et al. 2019). A particularly promising way to achieve this might be to apply
RTMA simultaneously to different descriptions of communities (e.g. taxonomic,
phylogenetic, functional, Leibold and Chase 2017), as was already fruitfully done using
convergence and divergence (Fukami et al. 2005, Li et al. 2015, McLean et al. 2019).

To conclude, by providing a methodology to assess the relative movements of ecological
trajectories, RTMA opens a new way to compare and describe ecological dynamics,
enriching the notions of convergence and divergence familiar to many ecologists. RTMA
does so using simple tests that are able to consider the full complexity of ecological data
(i.e. dimensionality reduction is not needed when working on multidimensional data) and
can accommodate many different data types.

493

494 Acknowledgements

We want to acknowledge the many people and institutions that have initiated and
maintain the long-term ecological datasets used in this article (Buell-Small Succession
Study, USA; Craigieburn Forest Park mountain beech dynamics held in New Zealand's
National Vegetation Survey Databank). Our knowledge of our changing world would be
much less detailed without them. This work was supported by ISblue project,
Interdisciplinary graduate school for the blue planet (ANR-17-EURE-0015) and co-funded
by a grant from the French government under the program "Investissements d'Avenir"
embedded in France 2030.

503

504

505
506 **Code and data availability statement**

507 In addition to the new functions implemented in the ecotraj R package (https://emf-
508 creaf.github.io/ecotraj/) available on CRAN, all code and data needed to reproduce the
509 simulations and ecological examples presented in this paper are freely available at
510 Zenodo: https://doi.org/10.5281/zenodo.18381756.

511

512 **Conflict of interest statement**

513 The authors declare no conflict of interest.

514

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