## Abstract

Ecology is a science of scale, which guides our description of both ecological processes and patterns, but we lack a systematic understanding of how process scale and pattern scale are connected. Recent calls for a synthesis between population ecology, community ecology, and ecosystem ecology motivate the integration of phenomena at multiple levels of organization. Furthermore, many studies leave out the scaling of a critical process: species interactions, which may be non-local through mobility or vectors (resources or species) and must be distinguished from dispersal scales. Here, we use simulations to explore the consequences of different process scales (i.e. species interactions, dispersal, and the environment) on emergent patterns of biodiversity, ecosystem functioning, and their relationship, in a spatially-explicit landscape. A major result of our study is that the spatial scales of dispersal and species interactions have opposite effects: a larger dispersal scale homogenizes spatial biomass patterns, while a larger interaction scale amplifies their heterogeneity. We find that an interesting interplay between process scales occurs when the spatial distribution of species is heterogeneous at large scales, i.e., when the environment is not too uniform and dispersal not very strong. Interestingly, the specific scale at which scales of dispersal and interactions begin to influence landscape patterns depends on the environmental heterogeneity of the landscape – in other words, the scale of one process allows important scales to emerge in other processes. Finally, contrary to our expectations, we observe that the spatial scale of ecological processes is more clearly reflected in landscape patterns (i.e. distribution of local outcomes) than in global patterns such as Species-Area Relationships or large-scale biodiversity-functioning relationships.

## Introduction

Scale is fundamental to ecology, from the spatial and temporal scales at which we observe and manage ecosystems [1, 2, 3] to the intrinsic scales at which processes occur within and across ecosystems [4]. Much of current research efforts describe ecological patterns across scales, such as species-area or biodiversity-ecosystem functioning relationships [5, 3]. However, the scaling of ecological patterns is largely phenomenological – we can describe how patterns scale but not why [6, 5]. Although links between scales of patterns and processes have been explored in recent years [7, 8, 9], as we will discuss, a systematic and unified treatment of scale in ecology is incomplete. Thus, a critical question remains: how is the scaling of ecological patterns, such as patterns of biodiversity and ecosystem functioning, related to scales of specific processes, and why?

In answering this question, a crucial process is often overlooked: the spatial scale of species interactions. While dispersal and environmental variation are often understood to operate at various spatial scales, existing research generally assumes that species only interact locally [10, 11, 12]. Yet many species move, forage, or otherwise interact with each other at a range of spatial scales [13], even in the absence of population fluxes (dispersal). Many move daily across multiple habitat types (e.g., seabirds connecting marine and terrestrial ecosystems), for some species even at scales which exceed dispersal (e.g., salmon returning to their natal streams). Non-local competition arises from foraging across multiple localities [13]. As a result, scales of species interactions, such as competition, likely have consequences for population persistence, affecting the spatial distribution of biodiversity and ecosystem functioning in ways that are distinct from other process scales [14, 15].

How do the spatial scales of dispersal, environmental heterogeneity, and species interactions interactively influence ecological patterns? Answering this question is unlikely to be achieved via observational studies, as different combinations of ecological processes may generate identical patterns, but computational models can explore patterns that emerge as processes interact across scales. Indeed, the scale of dispersal relative to the environment has been studied most extensively, in particular within a metacommunity context [16, 7, 17]. These studies generally find that high rates of dispersal blur differences between local communities, leading to losses of biodiversity and ecosystem functioning. Although there are reasons to expect increased scales of dispersal and species interactions to have similar consequences, as both processes are influenced by some of the same variables (e.g., animal mobility) and serve to spread out the effects of species interactions, there are also reasons to expect the opposite [18]. A key difference is that large dispersal scales can allow populations to permeate through whole landscapes over a few generations, whereas individuals with large interaction scales are still bound to specific localities. As a result, increasing scales of interactions may amplify spatial heterogeneity in an ecological system [19], counter to the blurring effect of larger dispersal scales.

In addition to scales of species interactions, we will address an additional major gap which is preventing a complete knowledge of scaling in ecology: consideration of a wider range of ecological patterns within a single study than has been examined previously. Two well-recognized ecological patterns are species-area (SAR) and biodiversity-ecosystem functioning (BEF) relationships. The species-area relationship is the earliest and most widely-examined ecological pattern to explicitly consider scale [5, 20]. Although SARs have been described as one of “ ecology’s few universal regularities” [21], accumulating evidence reveals considerable variation within and among biological systems [22, 5, 23]. Likewise, BEF theory has revealed consistent patterns, typically a saturating relationship between community diversity and biomass production [24], but most work has focused on BEFs at local scales, with only recent work highlighting the importance of scale [3]. Previous studies have examined how one pattern or the other are affected by process scales [25, 23, 26], but no study has examined how SAR and BEF relationships change in tandem and if effects that are masked through one pattern are apparent through the other. As a consequence, it is unclear how both SAR and BEF relationships are affected by the interplay of processes acting at different scales, making it difficult to assess how process scales affect the overall behavior of ecosystems as different measures highlight different aspects of ecosystems. These issues will be useful to resolve, as they link basic and applied biodiversity problems, for the preservation of productive, biodiverse landscapes [27].

Here, we use a modified Lotka-Volterra metacommunity model to explore the consequences of the scaling of ecological processes for biodiversity, ecosystem functioning, and their relationship across spatial scales. Our simulations consist of species interacting in a spatially-explicit landscape, with “ patches” emerging from the environmental structure of the landscape. We first study the heterogeneity of local outcomes across the landscape: patterns of patch biodiversity, patch functioning, and relationships between them (local BEF). We can then scale up to the whole landscape scale and every scale in between. By varying the spatial scales over which metacommunity processes (abiotic environment, competitive interactions, and dispersal) play out, we test the hypothesis that ecological patterns depend on how processes interact across scales, including scales of species interactions, and lead to different patterns from those generated by commonly-assumed hierarchical process scales (i.e., scales of interactions < environment < dispersal; Fig. 1).

Species-Area relationships depend on spatial turnover in species composition, and compositional turnover is driven by ecological processes [28]. Thus, we would expect that ecological processes should strengthen SARs in scenarios where they increase compositional turnover. We predict that the strongest slopes of the SAR will occur when scales of dispersal < environment < species interactions, because (i) interactions are not constrained to abiotically suitable patches, and (ii) weaker dispersal prevents the homogenization of species composition across the landscape. Additionally, we predict that the consequences for BEF relationships will differ between local and regional scales. On local scales, we expect BEFs to weaken as interaction scales increase, given that species that are locally absent but present in nearby areas can affect local function. On regional scales, we expect BEFs to strengthen as interaction scales increase, since regional competition would only keep the most suitable species at a given location, and hence, more species would mean that multiple species are productive within a given region.

## Methods

### Model

We use a modified Lotka-Volterra metacommunity model to explore the consequences of the spatial scaling of three ecological processes – abiotic environment, species interactions, and dispersal – for biodiversity and ecosystem functioning. Our specific assumptions and parameters are motivated by two important choices. First, we focus on a classic setting of ecological assembly, i.e. the patterns that arise when many species, originating from a regional pool, come together and reach an equilibrium state, with some species going locally or regionally extinct. Furthermore, we take species interactions in the pool to be disordered, that is, heterogeneous but without a particular functional group or trophic level structure [29]. We do not exclude that different patterns could emerge for more ordered interactions (e.g. a realistic food web), or for parameter values that lead to a more complex dynamical regime (e.g. population cycles or chaos, driven by stronger species interactions or environmental perturbations).

Second, we consider the possibility of species interacting over large spatial scales. Conventional metacommunity models describe discrete local communities of habitat patches connected by dispersal, within which species interact [30]. In doing so, they implicitly assume that the spatial range of species interaction is smaller than the scale of dispersal and contained within a patch, for all species and types of interactions [14]. To relax these assumptions, we construct a metacommunity model where populations of species can disperse and interact at different spatial scales, without specifying the mechanism underlying these ecological processes. Species interactions that manifest beyond local scales are abstracted from mechanisms such as individual foraging, vector species (e.g. pathogens) [31], and spatial resource fluxes [32, 14].

The model details the dynamics of *S* different species in a community. The dynamical equation for the biomass *N*_{i} of species *i* at position in the landscape at time *t* is given by a generalized Lotka-Volterra equation of the form
where and represent vectors of spatial coordinates in the landscape. Equation (1) models the effects of three ecological processes on the abundance of species *i*: its intrinsic growth rate , which is influenced by abiotic environmental conditions at location , dispersal to and from location , which is controlled by the diffusion coefficient *δ*_{i}, and interactions with all other species *j*, including when they are located elsewhere in the landscape, .

### Environment

Abiotic conditions in each location are encoded by an environmental variable . This variable is continuous and varies smoothly over space, with parameters allowing one to tune the typical spatial scale of this variation [33]. For more details on the construction of the environment, see the Appendix.

Each species has a Gaussian fundamental niche that determines its abiotic fitness in each location, with an optimal environmental value *H*_{i} and abiotic niche width *ω*_{i}
Each fitness value is bound between 0 and 1 and reaches its maximum at an optimal environmental condition (i.e., when . We take the growth rate as . In other words, sets the actual structure of environmental conditions across the landscape, whereas is how species experience the environment and its structure.

### Interactions

We choose to limit ourselves to competitive interactions, defined by the matrix *C*_{ij}, which represents the per-capita competitive effect of species *j* on species *i*. The diagonal of the matrix (the impact of a species on itself) is set to 1, whereas all other interactions are taken independently from a random uniform distribution between 0 and . We choose to allow for moderate interactions between different species (inter-specific competition is always weaker than intra-specific), suggesting that pairwise coexistence is often possible for species with different growth rates *r*_{i}, but the total impact of many competitors is still strong enough to allow for extinctions. Previous work has shown that, in disordered communities, the outcomes of ecological assembly are robust to many details such as the nature of interactions (e.g. mutualism, predation), and depend only on a few statistical properties such as the mean and variance of interaction effects [29].

Furthermore, interactions are assumed to occur over a characteristic spatial scale encoded by a spatial kernel *K*. We use a Gaussian kernel whose standard deviation defines the interaction range such that
where indicates the norm of (distance between) the vectors and , and *γ* is the spatial range (scale) of the interactions.

We normalize the interactions by *k*_{0} such that the overall effect of the kernel is always the same (i.e. the integral over K always equals 1). This normalization means that for large-scale interactions, local competition becomes weaker. However, some amount of (especially intra-specific) competition must remain locally strong to prevent species densities from growing exponentially and exploding. Therefore, we define interactions as partially local and partially regional, with *β* governing the fraction of interactions that are regional:
We choose *β* to ensure that the effect of interactions changes with *I*, but local competition is never negligible.

### Dispersal

Finally, dispersal is modeled by the diffusion (Laplace) operator,
where *δ*_{i} is the diffusion or dispersal coefficient of the species. For simplicity, we set the dispersal coefficient to be the same for all species.

Contrary to interactions, we do not use an explicit spatial kernel here, because intensity and spatial scale are unavoidably entangled in the case of dispersal (see Discussion). Thus, as will be seen, the coefficient *δ*_{i} sets the spatial scale over which dispersal impacts ecological dynamics. We note that two aspects of our modeling choices mean that our choice of dispersal by diffusion will not be qualitatively different from applying a large dispersal kernel: our focus on the equilibrium state, and having initial conditions where all species are introduced to every point in the landscape. The former aspect of equilibrium means that any potential non-equilibrium dynamics driven by species moving quickly across space due to a large dispersal kernels do not apply. The latter aspect means that there is no dispersal limitation, i.e. a short or long-ranged dispersal kernel does not affect which parts of the landscape can be reached by a species.

### Scales

In this study we are concerned with spatial scales of three ecological processes:

*E*: environmental heterogeneity*D*: dispersal*I*: species interactions

To properly compare the interplay of different process scales, we must first compute their values for a given set of model parameters (Table 1). The scale of the environment (spatial autocorrelation) is controlled by two parameters: spectral color *ρ* and spectral cutoff *k*_{c}. In the main text, we focus on a single value for the environment scale *E* = 32, and vary the other two scales on a logarithmic scale, with values of 1, 3.2, 10, 32 and 100, where the system itself has the scale (length) of 320. See the Appendix for calculations and discussion of other values of *E*. The scale of interactions is set by, and coincides with, the width of the Gaussian kernel *γ*, such that *I* = *γ*. The scale of dispersal is mainly determined by the diffusion coefficient *δ*_{i}, and it is expected to scale as (see, e.g., [34]). The normalization constant is, however, not trivial, and as we show in the Appendix, it is approximately 10. We therefore use: .

### Parameterization and simulations

To initialize our simulations, we first add environmental structure to a two-dimensional landscape of size 320×320 (see the Appendix for details). We do not define patches explicitly, but rather allow them to emerge from the spatial structure of the environment. *S* = 20 species are initially seeded onto the landscape, with initial abundances at each location drawn from a uniform distribution between 0 and 1, resulting in roughly equal abundances at the landscape scale. For simplicity, we use periodic boundary conditions for the two-dimensional system (i.e. a torus topology), for both dispersal and interactions. We do not expect this choice to impact the results, due to the large size of the system considered.

We use 20 replicate landscapes, allowing environmental structure to vary among replicates while keeping the environmental scale constant (*E* = 32, see Appendix for other values). Each replicate landscape was used to systematically vary the spatial scale of interactions *I* and dispersal coefficient *D*. Doing so allows us to isolate the effects of interaction and dispersal scales within any one replicate without the confounding effects of different landscape structures. The generality of our findings are ascertained by comparing across replicates.

We run each simulation, where a simulation is defined as a model run with a unique combination of process scales and replicate landscape, to a maximum time of *T* = 1000, or until equilibrium is reached. For practical purposes, we define an equilibrium as when the maximal change in biomass of any species in any location over a time-span of *T* = 1 is less than 10^{−5}. A full list of parameter values can be found in Table 1. The value of *δ*_{i} was changed to control the dispersal scale *D*, the values of *ρ* and *k*_{c} were changed to control the environment scale *E*, and the value of *γ* was changed to control the interaction scale *I*. Values of *I* and *D* were chosen along a logarithmic scale. All simulations were performed using MatLab 2019a.

### Measurements

For each simulation we measure individual and total community biomass, species richness, and sample the landscape to calculate species-area relationships (SAR curves) as well as biodiversity-ecosystem functioning relationships (BEF curves). For species richness, SARs, and BEFs, we define a species to be extinct at a given location if its biomass is below than a threshold of 10^{−3}.

To calculate SAR curves, we use 40 different spatial scales from 1×1 (single pixels) to 320×320 (the entire landscape) on a logarithmic scale, and computed the species richness at each. For a given scale, we randomly choose 100 locations in the landscape, and sampled a region centered around the location chosen. We averaged over the 100 estimates of the species richness to obtain the mean richness value for a given scale.

We calculate both local and regional BEF curves, based on random sampling of the system, on average measuring each pixel once. We do this in a similar way to the SAR curves, but also measure total community biomass. For the local BEF, we use a 1×1 pixel area with 102,400 random locations chosen, while for the regional BEF we use an intermediate area of size 10×10 with 1024 locations sampled. Thus, on average, we measure every location in the system once, for both local and regional BEF.

A striking pattern we observe in our results are spatial patterns of biodiversity and functioning in landscapes that are not well captured by summary variables, such as SARs. To capture these patterns, we calculate how correlated the biomass is of a given species as distance between sampling locations increases (i.e., ‘spatial correlation’), which can be used to quantify the properties of spatial patterns we observe. To calculate species’ spatial correlations, we do the following: 1) We normalize the species’ distribution by subtracting its average biomass (taken over the whole system). 2) We obtain a correlation map by calculating the convolution of a spatial distribution with itself, using a two-dimensional Fast Fourier Transform. 3) We normalize the correlation map by dividing the resulting two-dimensional map by its maximum value (i.e., we set a correlation value of 1 at the origin).4) We define the one-dimensional correlation function as the average between a vertical and horizontal transects through the correlation map. To define the scale of correlation for a given species, we locate the distance at which the correlation function reaches half its height, i.e., the distance from the origin where its value is the average of the maximum value (which is always 1) and its minimal value (typically around 0).

## Results

### Local outcomes: functioning and diversity across localities

Our first major result is that, although they can arise from similar biological mechanisms (e.g., individual mobility), dispersal and interaction scales have opposite impacts on biodiversity and functioning patterns across the landscape (Fig. 2 and S3). We start from the case of weakly-connected communities with local interactions where all landscape patterns result from environmental variation (top-left panel, Fig. 2). Increasing the spatial scale of dispersal leads to a blurring of total community biomass over the landscape (from left to right, Fig. 2). In contrast, increasing the scale of species interactions leads to a sharpening of spatial patterns, amplifying underlying environmental heterogeneity (top to bottom, Fig. 2). The antagonism between these two effects can be seen by the fact that they counteract each other when increasing both scales at once, leading to similar-looking outcomes (along the diagonal, Fig. 2), but dispersal eventually wins out – the states along the right column are virtually identical, whereas the same is not true across the bottom row. Critically, it is not until the scales of dispersal or interactions exceed the scale of environmental heterogeneity (i.e., outside the dashed-lined boundary in Fig. 2) that the scale of either process significantly alters spatial patterns in biomass (see also Fig. S2). Larger emergent scales of total community biomass due to high *D*, and the opposite due to high *I*, can also be seen in Fig. 5, which shows how quickly patterns among locations become dissimilar as the distances between them increase.

We then focus on a subset of our scenarios above to show how process scales impact not only total biomass but also individual species distributions (Fig. 3). We observe that increasing dispersal scale predictably makes larger, more coherent domains (i.e., fairly defined areas with similar characteristics) with typically higher local diversity. Increasing interaction scale creates a more granular landscape with a broader range of diversities, including many low-diversity patches and a few high-diversity ones. Indeed, large interaction scales lead to more spotty species distributions, with rare species persisting in some locations where they would not in other scenarios (Fig. 3 bottom row). Two notable examples include species 1 (red patches) persisting only when interactions are large and dispersal is small, and species 2 (cyan patches, due to its coexistence with species 3) taking on a more clumped distribution with large interaction scales.

### Regional outcomes: functioning and diversity at the landscape scale

The outcomes described above allow us to identify spatial patterns in local outcomes in the landscape, but what are outcomes for the landscape as a whole? Given the additive nature of biomass across localities, two regions could have identical biomass at the landscape scale even if one region has high variation among localities that span extremes of high and low values, whereas another varies little with biomass values that are intermediate. Here, we see that biomass is highest when interaction scales are large (Fig. S4), an effect that is quickly eroded as dispersal scales increase. Interestingly, these high-biomass landscapes had extreme variation in biomass among localities, including areas of extremely low biomass (dark blue in Fig. 2) and extremely high biomass (red in Fig. 2). Therefore, high biomass is driven by a disproportionate subset of local communities in a landscape. Furthermore, these high biomass landscapes were unremarkable in regional species richness in the landscape, and actually had fewer species per locality on average than other scenarios (Fig. S4a).

### Cross-scale outcomes: BEF and SAR

Next, we turn to two types of cross-scale outcomes (Fig. 4). First, we consider the relationship in BEF curves (i.e. total biomass vs. species diversity), which we compute at local and regional scales. At the regional scale, we are unable to distinguish between the scenarios investigated in Fig. 3. By contrast, local BEF relationships better reflect the underlying process scales. In particular, they exhibit a hump-shaped relationship for large interaction scales, suggesting that patches with the largest total biomass are not the most diverse, but rather have a few high-performance species. This result ties into our previous observation that the interaction scale tends to amplify environmental heterogeneity, and may thus put more weight on selection effects, where abiotic conditions select the best-performing species at the exclusion of others.

We also look at a pattern aggregated over continuously increasing spatial scales – the SAR. We would expect that changes in the slope or shape of the SAR as the aggregation scale (x-axis) exceeds the spatial scales of our ecological processes, as has been demonstrated for the Stability-Area Relationships [8]. However, we do not observe a clear link between process and pattern scales, beyond the fact that the inflection point (in particular for low *D* and *I*) corresponds to the environmental scale *E* (vertical gray line in Fig. 3). The main impact of process scale is that, by amplifying landscape heterogeneity, a large interaction scale *I* leads to a stronger SAR. Specifically, as predicted, at the smallest scale the *D* < *E* < *I* scenario (magenta curve) yields the lowest species richness compared to all other scenarios, whereas at the scale of the entire landscape, its richness is very high.

However, aggregated measures of biodiversity and functioning at regional scales miss much of the information captured by local measures, such as the distribution and turnover in biomass (Fig. 2 and Fig. 3). Yet these local patterns can be quantified. Figure 5 presents the results of the spatial correlation of species biomass distributions, which measures how the biomass of a species correlates over the distance between sampling. We observe clear trends in scale, with consistent patterns of growing (shrinking) correlation with higher dispersal (interaction) scales.

## Discussion

This study focuses on a critical question: how is the scaling of ecological patterns, such as patterns of biodiversity and ecosystem functioning, related to scales of specific processes, and why? We have modelled how intrinsic scales of ecological processes align with the emergence of ecological patterns in a metacommunity, where we control the spatial scale of environmental heterogeneity, dispersal, and species interactions. We further focused on the regime where many species may coexist in a stable equilibrium. Under these assumptions, we have arrived at the following answer: characteristic scales of biodiversity and functioning, as summarized by the inflection point of species-area relationships, are set by scales of environmental heterogeneity. Scales of dispersal and species interactions do not affect those inflection points. Rather, they have opposing effects on the magnitude of differences between small-scale and large-scale patterns. Below, we expand upon and place our findings within existing knowledge, examine the mechanisms that underlie our findings, contrast results among ecological variables, and end by placing our results within a context of ecosystem preservation.

A main finding of our study is that the spatial scale of interactions amplifies environmental heterogeneity, sharpening observed spatial patterns, in contrast to dispersal scales. Importantly, observed spatial patterns did not reflect the absolute value of the spatial scale of each ecological process, but rather, their values relative to the environment; decreasing the spatial scale of the environment shifts the boundary of blurring/sharpening effects of dispersal and species interactions (Fig. S2). We find this effect because environmental conditions are quite literally the template upon which dispersal and species interactions mold species distribution. Large-scale (i.e., at scales above the template) processes are more important than small-scale ones in determining overall patterns, meaning that only when dispersal or interactions have large scales can they impact large-scale patterns.

We examined the impacts of process scales on two classes of patterns: first, on the spatial scaling of patterns (SAR and BEF), and second, on the spatial structure of species abundances in the landscape. Unexpectedly, the latter class of patterns appears to better reflect the scale of ecological processes, such as the distribution and turnover of biomass and biodiversity across the landscape. These patterns would be lost by examining mean biodiversity and function at specific aggregation scales (e.g., local vs. regional; Fig. S4), but were well captured via spatial autocorrelation (Fig. 5). From these analyses, one take-home message is that increasing the scale of species interactions actually amplifies variation on small scales. In other words, large-scale processes do not necessarily beget large-scale patterns.

The question of how process scales affect observed patterns can also be spun around: what information about process scales can be inferred from the various patterns we see? Considering the opposing effects that dispersal and interaction scales have on pattern scales (Fig.2), it is not clear that such an inference is possible. However, given that patterns scales change differently (compare Fig. 2 with Fig. S3, for instance), combining several measures together may provide an answer, for instance by finding when changes in spatial correlations of biodiversity and biomass no longer behave similarly. In this context, it is perhaps to be expected that no clear connection was found between well known patterns such as BEF and SARs, and process scales. Over the past few decades, ecologists have been cautioned from interring processes from patterns [35]. Our results demonstrate exactly why this is important: a lack of a 1:1 mapping between a pattern and any one specific process.

Indeed, our finding that the SAR curves did not exhibit transitions at particular spatial scales, that would allow us to identify the typical scales of the underlying processes (other than the environment), runs counter to other contexts, such as the invariability-area relationship [8]. In particular, we do not find a triphasic SAR curve that is often reported [36, 8], since our model does not consider individual sampling and dispersal limitation, which typically lead to stronger SAR slopes at small and large scales, respectively. We thus see the strongest slopes at intermediate spatial scales, consistent with results under similar settings [37], and hinting that we are largely seeing community dynamics typical of species-sorting [30]. Centering on the average SAR slope itself, on the one hand, we found that large interaction scales may enhance the SAR by amplifying landscape heterogeneity and creating low-diversity strips along the edges of species ranges. On the other hand, this spatial heterogeneity could also promote coexistence as a weaker competitor might thrive in the margins [38]. This suggests that edge effects may play a prevalent role in the case of long-range interactions, and deserves more extensive investigation. Overall, the scales of biotic processes (interaction and dispersal) are mainly reflected inasmuch as they change overall community properties, such as total diversity across the landscape.

In line with our expectations, dispersal tends to homogenize spatial patterns and can thus hinder our ability to infer the properties of smaller-scale processes. However, it is important to note that this generic blurring effect may disappear in some specific ecological settings, such as the well-known Turing instability arising from interactions between two reactants dispersing at different rates, an “ activator” and an “ inhibitor”. In this study we focused on many interacting species, so that finding these two very distinct behaviours of activator and inhibitor is not typical, and hence dispersal always leads to a smoothing effect. We can therefore reconcile this apparent discrepancy by noting that, within our framework, such Turing patterns [39] where regular spatial scales emerge due to dispersal and local interactions alone, appear when a fast-diffusing species induces an effective long-range self-competition for a slow-diffusing species. This has been discussed in the context of vegetation patterns, where plants compete at a range through fast-diffusing water [40, 41]. Thus, it may be that dispersal in one species effectively creates a type of ranged interaction in another species, leading to the formation of heterogeneous spatial patterns that do not reflect the underlying environmental conditions.

An interesting problem we encountered, which is worth expounding upon to aid future research in this area, is how to place dispersal on comparable scales to other processes. For both environmental factors and species interactions, we could separate the intensity of variation and the scale over which it takes place, for instance, by modelling interactions with a spatial kernel which defines the range of these interactions. For dispersal, however, this distinction does not hold in the continuum limit nor in the stable equilibrium regime that we consider in this study. This can be understood intuitively in a single dimension: organisms who disperse from site *x* to site *x* + 1 at time *t* will be counted in those that disperse from site *x* + 1 to site *x* + 2 at a later moment in time. Therefore, dispersing twice as fast between neighboring sites can be equivalent to dispersing twice as far. This equivalence breaks down when the details of individual dispersal events matter, e.g. for very rare and long-ranged dispersal events [42]. But even then, the strength of each dispersal event would still play into the spatial scale over which dispersal impacts the dynamics over longer times. As a consequence, defining dispersal scale from a spatial kernel alone might seem more intuitive, but would actually hide the importance of intensity, and we prefer to simply model nearest-neighbor dispersal and acknowledge that intensity and scale are entangled. We detail in the Appendix how we ensure that our metrics of spatial scale are dynamically meaningful and comparable for all three processes.

Knowledge of the spatial scale of ecological processes is critical to understanding the maintenance of ecosystems. To illustrate this argument, one can imagine a landscape manager interested in preserving some baseline level of functioning in a landscape at a specific spatial extent, for example, primary production. If the spatial scale of interest does not encompass the intrinsic scales of processes that govern functioning, then landscape alteration beyond that scale might impact functioning in an unanticipated and undesirable manner; these scales will differ among ecosystems based on how species’ traits and the physical landscape affect how organisms experience scales of E, D, and I. In other words, the scales important to the maintenance of ecosystem function may be mismatched from the (typically small) spatial scales at which ecosystem functioning is observed and managed, but the degree to which this is true depends on process scaling. Our results suggest that it will be difficult to manage landscapes to preserve biodiversity and ecosystem functioning simultaneously, despite their causative relationship, for two related reasons. First, the fact that increasing dispersal and interaction scales had opposing effects on either ecosystem property presents a unique management challenge, given that both scales are tied to organismal movement, albeit on distinct timescales (i.e., daily vs. once-per-generation). Second, ecosystems attained the highest biomass in scenarios which also led to the lowest levels of biodiversity, specifically, when interaction scales were large and dispersal scales were small. We note that this second issue may only be relevant when interactions are largely competitive, since our modeling, and thus results, did not consider mutualistic interactions which would likely change the observed trade-off between biodiversity and biomass. How would a manager plan a landscape to enhance interaction scales (preserving function) while simultaneously minimizing scales of dispersal (preserving biodiversity)? This might be most successful in species with body plans for long-distance movement, but that can remain philopatric for behavioural reasons (which can be environmentally determined; i.e., territorial hunters).

Our conclusions are twofold. First, we bring forward an important spatial scale – the range of species interactions – that has been largely neglected in previous analyses (e.g. metacommunity theory). This interaction range can derive from many of the same ecological mechanisms as dispersal, for instance individual mobility, yet these two scales lead to opposite ecological effects. This suggests that we must carefully distinguish whether mobility actually leads to population dispersal or to large-range interactions, and re-evaluate possible consequences of evolution or environmental change in these processes. Finally, we saw that the spatial scale of ecological processes might not appear clearly in the scale of resulting patterns such as Species-Area or Biodiversity-Ecosystem Functioning relationships, though they may sometimes be reflected in local outcomes. While we focused on a few important biodiversity and functioning patterns, our study paves the way for future work investigating systematically under which conditions various ecological pattern scales may or may not reflect the spatial scale of underlying processes.

## Conflict of interest disclosure

The authors declare they have no conflict of interest relating to the content of this article.

## Data accessibility

Script files for simulations and analysis of results shown in the manuscript are available at the open-access repository: https://doi.org/10.5281/zenodo.5543190.

## Acknowledgements

YRZ, MB, DWS and ML were supported by the TULIP Laboratory of Excellence (ANR- 10-LABX-41), and by the BIOSTASES Advanced Grant, funded by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 666971). We thank all members of the BEF Scale working group for valuable discussions and feedback.

## Appendix

### Measurement of scales

As mentioned in the main Methods section, we explicitly measure and compare three spatial scales: environmental conditions (E), dispersal (D) and species interaction (I).

In our model, the species interactions are explicitly defined with a distance over which they occur – via the Gaussian kernel function. This naturally gives us the scale of interactions *I*, as the width of the Gaussian function, such that *I* = *γ*. To estimate the dispersal scale *D*, we compare the effect of changing the dispersal coefficient *δ* with changing *γ*. In Fig. S1 we show how changing *δ* and *γ* (and thereby *D* and *I*) affects the community biomass distribution. As seen in the left panel, with low *δ* and *γ* the difference from a null scenario of no dispersal and no interaction distance is very small, but increasing either *δ* or *γ* changes the community biomass distribution considerably. In the middle and right panels we see these differences, as we change only *δ* (middle) or only *γ* (right). This clearly shows three things: 1) The effect of interaction distance scales linearly with *γ*, as expected by its definition. 2) The effect of dispersal coefficient scales with , as expected from dimensional considerations (e.g., [34]). 3) More specifically, to make these two effects comparable, the dispersal scale is missing a factor of 10, i.e. . This can be seen by the fact that for both *δ* = 1 in the middle panel and *γ* = 10 in the right panel, the y-axis values are roughly the same (10^{−1.2}).

The environment itself is generated using a combination of a spectral color and cutoff wavenumber (see subsection below), but this does not explicitly define the scale. To measure the scale of the environment, we follow the same approach as for the correlation function and measure the scale for a species biomass distribution (using a convolution based on FFT), except that we do this for the value of intrinsic growth rate , as it is directly set by the environment. For each of the 20 species, we can calculate a correlation function (in the same manner as explained in the methods), and from this we calculate the correlation scale (the point of middle height for the correlation function). We average this value over all 20 species, to calculate the environment’s scale for a given system. Since this result depend on the randomization of the environment, we repeat this for many replicates, and choose values of *ρ* and *k*_{c} that will on average give a value of *E* we want to have.

### Generating the landscape

The landscape profile is defined by a spectral color (*ρ*) and cutoff (*k*_{c}). A spectral color close to 0 corresponds to “ white” noise, i.e. noise that exhibits little or no spatial autocorrelation; a spectral color close to 1 indicates “ red” noise – noise with high spatial autocorrelation [33]. The spectral cutoff creates a point of truncation in the frequency profile that prevents high variation between adjacent pixels, in effect smoothing the noise across the landscape. Together, color and cutoff control the degree of structural fragmentation of the landscape. More weight on higher frequencies (low *ρ*, high *k*_{c}) entails smaller and less-connected fragments of similar environmental conditions. Weight on lower frequencies (high *ρ*, low *k*_{c}) creates long bands of constant environmental conditions which can act as corridors for species favoring this value.

To generate the environmental landscape , we prescribe a frequency profile for the noise:
which is a power-law with color *ρ* (*ρ* = 1 corresponds to red noise) and an exponential cutoff with wavenumber *k*_{w} = *k*_{c}*L/*2 which removes high spatial frequencies, smoothing the landscape and avoiding strong variations between adjacent pixels. Note that the cutoff wavenumber is simply the normalization of the spectral cutoff by the number of different frequencies represented by the chosen resolution of the domain, *L/*2, with *L* the number of pixels along the x and y axes, such that in the spectral domain it represents the resolution of the landscape.

Practically speaking, for a two-dimensional landscape, we generate a *L × L* matrix *R*_{ij} of uniform random numbers over [−1, 1] corresponding to amplitudes for each wave vector (*k*_{x}, *k*_{y}). We then multiply these random numbers by the profile above
with where index *i* is a natural number running over [1,L]. We set the element *M*_{L/2,L/2} corresponding to the uniform trend (*k*_{i} = *k*_{j} = 0) to 5. Finally, we apply a Fast Fourier Transform on the matrix *M*_{ij} to obtain the landscape matrix.

### Additional plots

We show below a few additional plots, to clarify issues discussed in the main text. In Fig. S2 we show the overall difference between different sets of values of *D* and *I* to the case of no dispersal and local interactions, for two values of *E*. In Fig. S3 we show the spatial distribution of species richness, for 5×5 different parameter sets with different values of *D* and *I*, corresponding to Fig. 2. In Fig. S4 and Fig. S5 we show summary statistics for each of these 5×5 parameter sets, of total community biomass, average local diversity, and total diversity. In Fig. S6 and Fig. S7 we show the spatial distributions of biomass and species richness, for a different landscape, one that has an environmental scale of *E* = 10.