# Diffusion of constructed resources and specialism
promote facilitation in spatial systems

Alice Nadia Ardichvili, Sebastien Barot, Jean-christophe Lata, Nicolas Loeuille

► To cite this version:

Alice Nadia Ardichvili, Sebastien Barot, Jean-christophe Lata, Nicolas Loeuille. Diffusion of constructed resources and specialism promote facilitation in spatial systems. Oikos, 2025, pp.e11025. 10.1002/oik.11025. hal-05019368

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OIKOS

Research article

# Diffusion of constructed resources and specialism promote facilitation in spatial systems

Alice Nadia Ardichvili ID <fnref n="1" />,<fnref n="2" />, Sebastien Barot ID <fnref n="1" />, Jean-Christophe Lata ID <fnref n="1" /> and Nicolas Loeuille ID <fnref n="1" />

Correspondence: Alice Nadia Ardichvili (alice.ardichvili@mail.mcgill.ca)

Oikos

2025: e11025

doi: 10.1002/oik.11025

Subject Editor: Pedro Peres-Neto

Editor-in-Chief: Pedro Peres-Neto

Accepted 3 March 2025

Sessile organisms, such as plants, contend with locally limiting nutrients and neighbours. By consuming nutrients or increasing local concentration by niche construction, they alter the conditions of growth in their local environment. Diffusion of resources alters the interaction sign between two neighbours usually thought to compete for resources. We investigate conditions under which the positive effect of niche construction outweighs the negative effect of competition. Following the experimental design of facilitation experiments, we model patches connected by a flow of resources and ask whether the presence of a niche constructor in one patch eases the colonisation of the other by a conspecific or heterospecific organism. The net interaction sign between a constructor and a coloniser depends on the niches of the two organisms, the diffusion of resources, and the type of niche construction. Niche constructors have a positive effect on neighbouring organisms when niches are sufficiently differentiated, and when the constructed resource diffuses more than the non-constructed resource. Our work proposes mechanisms for the emergence of facilitation, which is increasingly recognised as a key process structuring plant communities. We discuss the implications for the spatial structure of plant communities and their functioning, including in an agricultural context.

Keywords: competition, ecosystem engineering, facilitation, indirect interactions, meta-ecosystem, niche construction

## Introduction

Facilitation is widespread among plant species (Michalet and Pugnaire 2016, Yin et al. 2022) and micro-organisms (Piccardi et al. 2019), potentially occurring at a local scale. The mechanisms driving facilitation are diverse and include the creation of a favourable microclimate (Callaway et al. 2002, Michalet et al. 2014), sediment retention (Bruno 2000), the fixation of nitrogen (Koffel et al. 2018), and production of by-products used as resources in cross-feeding communities (Goldford et al. 2018). Interactions potentially switch from competition to facilitation as external stress increases (Callaway et al.).

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

<footnote n="1">¹Sorbonne Université, Université Paris Cité, Univ Paris Est Créteil, CNRS, IRD, INRAE, Institut d'Écologie et des Sciences de l'Environnement de Paris (iEES-Paris), Paris, France</footnote>

<footnote n="2">²Station d'Ecologie Theorique et Experimentale, CNRS, Moulis, France</footnote>
2002, Michalet et al. 2014, Piccardi et al. 2019), switch from facilitation to competition at different life stages (He et al. 2013, Yin et al. 2022), or at different stages of succession (Koffel et al. 2018). Recent studies show that facilitation is widespread and not always linked to environmental stress, or external conditions in general (Yin et al. 2022, Yu et al. 2024). Because the nature of interactions ultimately drives the coexistence of species and the structure of ecological communities, understanding conditions for the emergence of facilitation remains an important issue in modern ecology.

Facilitation implies that a (facilitating) organism modifies the biotic, abiotic or resource conditions of another (con-specific or heterospecific) organism (Filazzola and Lortie 2014). For example, a species with herbivory protection may indirectly protect other plant species growing in its vicinity (Danet et al. 2021). Facilitation is the result of a special case positive niche construction as defined by Odling-Smee et al. (1996), when the environment of an organism is positively affected by the presence of a niche constructor. However, niche construction by one species may not always lead to facilitation (Walker et al. 2001). Several studies show that niche construction (also called ecosystem engineering – Gurney and Lawton 1996, Cuddington et al. 2009) can have negative or positive effects on other species, including in a spatial context (Silver and Di Paolo 2006, Lehmann 2008). A mechanistic understanding of how niche construction impacts interactions between organisms is lacking.

Niche construction may influence interactions between species via indirect effects on resources. Many niche-con-structing activities modify resource flows, ultimately result-ing in indirect effects between niche-constructing organisms (Kylafis and Loreau 2011, Koffel et al. 2021). Some niche-constructing activities increase the quantity of one resource (plants performing biological nitrogen fixation (Menge et al. 2009), *Lupinus* species increasing the availability of phospho-rus (Hinsinger et al. 2003), ants raising aphids for their nec-tar). In that sense, these organisms increase the supply rate of one resource, a behaviour that we call ‘farming’ in the rest of the paper. The expectation that farming organisms may improve the conditions for others is for instance the justifica-tion for intercropping cereals with legumes (Duchene et al. 2017, Layek et al. 2018). However, when organisms also rely on other resources, a farming species may increase the quan-tity of one resource while depleting another, resulting in an unclear interaction between the farming organism and the others around it. For example, an N-fixing plant species that also consumes phosphorous may or may not facilitate the crop it is planted with (Hinsinger et al. 2011).

Niche-constructing organisms may also affect resources that are not independent (i.e. whose supply levels are not determined by different external fluxes). Organisms can take up the same nutrient from different ressources (e.g. from dif-ferent nitrogen forms coupled by degradation or chemical processes,) while others can control the rate of transforma-tion between the different ressources. Plants take up nitrogen as ammonium and nitrate (Britto and Kronzucker 2013). Due to soil nitrification (the conversion of ammonium to

nitrate by soil microorganisms), the stock of nitrate depends on the stock of ammonium. Some plants can control the rate of nitrification by blocking the activity of nitrifying commu-nities (Lata et al. 2004). In microbial communities, many species can grow from glucose and metabolic by-products of other species (crossfeeding – Goldford et al. 2018). In that case, glucose and metabolic by-products are two assimilable forms of carbon, and the quantity of by-products depends on the stock of glucose. By affecting other microbes (i.e. via the emission of specific anti-microbial substances or deplet-ing another essential nutrient), one species can influence the rate of transformation of glucose into by-products. The bal-ance between the benefits of by-products versus competition for other resources ultimately determines whether co-occur- ring microorganisms compete or cooperate (Machado et al. 2021). In the rest of the paper, we refer to niche-constructing organisms controlling transformation rates of one resource in another as ‘transformers’.

The spatial context influences how organisms interact. In a non-spatial context, where resources are shared, niche dif-ferentiation drives the strength of interaction between two organisms. Competition is minimal when niches are suffi-ciently differentiated (limiting similarity in Macarthur and Levins 1967 – stabilising mechanisms sensu Chesson 2000). Several modelling studies show that in ecosystems connected by resource flows, spatial heterogeneity reduces the compe-tition between organisms with similar niches (Huston and DeAngelis 1994), or prevents the exclusion of weaker com-petitors due to diffusion of resources (Gravel et al. 2010, Haegeman and Loreau 2015). The dispersal modality of consumers and resources (informed, symmetrical) changes the strength of competition between organisms (Gravel et al. 2010, Haegeman and Loreau 2015). We expect that in a con-text where niche construction potentially changes the type of interaction (switching it from competition to facilitation), asymmetry in resource diffusion also impacts the strength of the interaction.

While it is clear that niche construction may modify the type of interaction between species sharing resources (de Mazancourt and Schwartz 2010, Koffel et al. 2021, Machado et al. 2021), the potential increase in resource availability due to niche construction needs to be balanced by the consumption of resources by the niche constructor (Walker et al. 2001). Since the spatial context influences the strength of competition between two organisms (Huston and DeAngelis 1994, Gravel et al. 2010), we expect that the spatial context also affects the balance between resource competition and niche construction, ultimately determin-ing the type of interaction between two sessile organisms. In this study, we investigate how the interplay between resource competition, niche construction and spatial context results in facilitation or competition. We aim to provide a mechanistic understanding of the emergence of facilitation.

We proceed by analogy with experimental designs assess-ing facilitation, by testing whether the growth rate of a small population is positively or negatively impacted by the niche-constructing individual situated nearby (Callaway et al.
2002). We restrict our analysis to a small spatial scale and consider two patches, e.g. habitat space for plants, a micro-bial population, an ant colony, or any organism whose spa-tial extent is smaller than that of their resources. One of the patches is empty and about to be colonized by an organism, the other patch is occupied by a niche constructor that modi-fies resource flows, e.g. soil nutrient concentrations, presence of aphids. We test whether the presence of the niche con-structor eases the establishment of the colonizer. Although the niche constructor only influences resource dynamics in its own patch, it indirectly impacts the colonizer's pool of resources due to the movement of resources among patches. We consider two resources, which could be two different nutrients or two forms of the same nutrient. In the first sce-nario, resources are independent and the organisms simply consume the resources. In the second scenario, the niche con-structor farms one resource, increasing its rate of entry. In the third scenario, we consider two dependent resources, and the niche constructor modifies the rate of transformation of one resource to the other. With the different scenarios, we vary the type of niche construction (accumulation of one resource, control of the transformation rate), and within the scenarios, we vary the intensity of niche construction, the total flow of resources between patches (exchange rate), the propensity of one resource to diffuse more than the other (diffusion asym-metry), and the niche difference between the constructor and its neighbours.

When an organism only consumes resources, the resource levels are necessarily lower in the occupied patch than in the empty patch, and the movement of resources is directed from the empty patch to the occupied patch, resulting in competi-tion regardless of the niches of the two organisms. We expect that strong niche constructors invert this tendency and improve resource availability in the empty patch. However, the ability of a niche constructor to improve resource avail-ability in the empty patch likely depends on its preference for the constructed resource and on the asymmetry in resource flow. We expect that the effect of the niche constructor switches from competition to facilitation when niche differ-ence, niche construction, and diffusion of the constructed resource are sufficient.

## Material and methods

We explore the type of niche construction in three scenarios that share common features, which we present first before presenting each scenario in detail. We model two patches containing two types of resources. The patches are connected by the passive diffusion of resources. One of them is occu-pied by a niche constructor C, and the other is empty and is about to be colonized by a neighbour N. We assume that the two organisms are sessile and that movement or disper-sal between the two patches is not possible, so that the con-structing organism does not come into the neighbour patch during the invasion process. We assume that growth is solely limited by the modeled resources. We focus on the effect of

niche construction, therefore the two patches are otherwise identical in terms of inputs and losses of resources.

Empirically determining the sign of the interaction between two organisms requires growing the species together and separately, either by removing one of them in the field (Callaway et al. 2002) or by growing them together and sepa-rately in pot experiments (Callaway 2007). By analogy, to decide whether a niche constructor has a facilitating effect on its neighbour, we evaluate the growth rate of the neigh-bour when the patch is occupied by a niche constructor ($r_c$) and compare it to the growth rate that the neighbour would achieve without the niche constructor, i.e. if both patches were empty ($r_0$) (Fig. 1d).

The sign of $\Delta r_N = r_c - r_0$ is positive when the constructor has a facilitating effect ie. the growth rate of the neighbour increases in its presence. When $\Delta r_N$ is negative, we state that the constructor has a competitive effect on the colonizer. Graphically, facilitation occurs when resource conditions are above the $r_0$ growth isocline (Fig. 2–3). $\Delta r_N$ can be further decomposed between $\Delta r_1$ and $\Delta r_2$, the change in the growth rate due to modifications of the concentration of the first or of the second resource respectively (Supporting information). Graphically, $\Delta r_1$ and $\Delta r_2$ can be viewed as the horizontal and vertical components of $\Delta r_N$. In the following scenario, we find the expression of $\Delta r_N$ by assuming that an equilibrium is reached and time derivatives are set to zero. $\Delta r_N$ thus reads:

$$ \Delta r_N = r_c - r_0 = \frac{1}{N} \left. \frac{dN}{dt} \right|_{C=C^*} - \frac{1}{N} \left. \frac{dN}{dt} \right|_{C=0} \quad (1) $$

## Baseline scenario

In the baseline scenario, the niche constructor influences resource dynamics in its patch by simply consuming resources:

$$ \frac{dC}{dt} = C(u(\beta_{1C}R_1 + \beta_{2C}R_2) - m) \quad (2a) $$

$$ \frac{dR_1}{dt} = E_1 - u\beta_{1C}R_1C - p_1R_1 + ex(R_{1N} - R_1) \quad (2b) $$

$$ \frac{dR_2}{dt} = E_2 - u\beta_{2C}R_2C - p_2R_2 + e(1-x)(R_{2N} - R_2) \quad (2c) $$

$$ \frac{dR_{1N}}{dt} = E_1 - p_1R_{1N} + ex(R_1 - R_{1N}) \quad (2d) $$

$$ \frac{dR_{2N}}{dt} = E_2 - p_2R_{2N} + e(1-x)(R_2 - R_{2N}) \quad (2e) $$

$R_1$ and $R_2$ are the resource concentrations in the occupied patch, $R_{1N}$ and $R_{2N}$ are the resource concentrations in the empty patch (Fig. 1). The patches are connected by resources flowing at a basal rate $e$, modulated by an asymmetry
<figure><img src="image_4.png" /><figcaption>(b) Farmer: Niche construction on R₁ inputs</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) Transformer: Niche construction changing the transformation rate of R₁ into R₂</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(d) Assessing the interaction type</figcaption></figure>

<figure><img src="image_7.png" /></figure>

Figure 1. Description of the model and of the three niche construction scenarios. The niche constructor lives in the left patch (delimited by a red dashed line) and impacts, either positively or negatively, the growth of a coloniser *N* arriving in the neighbouring patch on the right (delimited by dashed blue lines). (a) Baseline scenario: the niche constructor does not actively build its niche and consumes the two resources. (b) The constructor increases the inputs of *R*₁ at a rate *γ*. The flux that differs from the baseline model is highlighted in black, other fluxes being in grey. (c) *R*₁ becomes *R*₂ at a rate *d*₀, and the constructor can modify this rate by its action (*α*). (d) The methodology used to determine whether the niche constructor facilitates or hinders the neighbour is comparable to that of experimental designs (Callaway et al. 2002); we conclude that the niche constructor has a facilitating effect when the growth rate of the coloniser is higher when the constructor is present compared to when the constructor is absent.
<figure><img src="image_8.png" /><figcaption>Figure 2. Schematic representation of the conditions that allow facilitation. The dashed grey line is the $r_0$ growth isocline, i.e. showing the resource conditions at which the neighbour sustains the growth rate that it would have if the niche constructor were absent. Diffusion of resources between the empty, to-be-colonized patch (white circle) and the patch of the niche constructor (coloured circles) takes the concentration of the empty patch closer to that of the occupied patch, on a trajectory (along the coloured arrows) that depends on diffusion asymmetry. The resulting concentrations either yield a higher growth rate for the coloniser (in the light green area of the plot) or a lower growth rate (light red area).</figcaption></figure>

coefficient $x$ between 0 and 1. When $x=1$, only $R_1$ diffuses, when $x=0$, only $R_2$ diffuses, when $x=0.5$ diffusion is purely symmetrical. In the two patches, resource dynamics are che-mostat-like (as in several plant growth models: Tilman 1982, Loreau 1998), input rates ($E_i$) are constant and losses are pro-portional to the compartment size ($p_i \times R_i$). Variables and parameters are shown in Table 1.

Both the niche constructor and the neighbour die at a rate $m$, take up resources from their patches ($R_1$ and $R_2$, $R_{1N}$ and $R_{2N}$) at a basal rate $u$ with a preference $\beta_{1C}$ and $\beta_{2C}$, $\beta_{1N}$ and $\beta_{2N}$ for the niche constructor and the neighbour respectively. We suppose that these resources are nutritionally substitutable, thus $\beta_{1C} + \beta_{2C} = 1$. We consider continuous variations of $\beta$, when $\beta_{1C} \approx 1$, the organism is specialised on $R_1$; when $\beta_{2C} \approx 1$, the organism is specialised on $R_2$; and when $\beta_{1C} = \beta_{2C} = 0.5$, the organism has no preference and is a generalist. Note that the niche constructor is established in its patch, therefore nutrients are influenced by the niche constructor ($R_1$ and $R_2$ decrease due to nutrient consumption by $C$). On the contrary, the neighbour is about to colonize the empty patch and has a low density, i.e. the neighbour does not influence the dynamics of $R_{1N}$ and $R_{2N}$. To focus on the effect of niche construction, specialism and resource diffusion, we assume that the neighbour and the constructor have the same uptake rate $u$ and mortality rate $m$.

In this system, the growth rate of a neighbour arriving in the empty patch reads:

$$r = \frac{1}{N} \frac{dN}{dt} = u(\beta_{1N} R_{1N}^* + \beta_{2N} R_{2N}^*) - m \quad (3)$$

where $R_{1N}^*$ and $R_{2N}^*$ stabilize at different concentrations in $r_C$ and $r_0$ depending on whether the niche constructor is present in the other patch or not.

We consider several possibilities regarding the preferences of the neighbours, each allowing us to reach conclusions about different organisational scales. When $\beta_{1C} = \beta_{1N}$ and $\beta_{2C} = \beta_{2N}$, the two individuals are of the same species. When $\beta_{1C} \neq \beta_{1N}$ and $\beta_{2C} \neq \beta_{2N}$, the two individuals may be from the same species but the preference of one of them may have mutated (e.g. conspecifics at evolutionary scale). Alternatively, they may be individuals from ecologically close species. To focus our study on the effect of niche construction, specialism, and resource diffusion, we assume that the neighbour and the constructor have the same uptake rate $u$ and mortality rate $m$. Note that we are focusing on preferences to define the organizational scale, which means that all other characteristics are equal, confirming that if individuals of two species are considered, they must belong to ecologically close species.

## Farmer scenario

In the ‘Farmer’ scenario (e.g. nitrogen fixation or mycorrhizal association, Fig. 1b), the niche constructor increases the input rate of $R_1$ at a per capita rate $\gamma$, such that Eq. 1b becomes:

$$\frac{dR_1}{dt} = E_1 + \gamma C - u\beta_{1C}R_1C - p_1R_1 + ex(R_{1N} - R_1) \quad (1b')$$

## Transformer scenario

In the ‘Transformer’ scenario (mineralisation of organic matter, control of nitrification rate, Fig. 1c), $R_1$ becomes $R_2$ at a basal rate $d_0$, the niche constructor influences that rate; when $\alpha < 0$, the organism inhibits the transformation (for example plants inhibiting nitrification); when $\alpha > 0$, the constructor stimulates this transformation (for example plants boosting the mineralisation rate). Eq. 1b–d become:

$$\frac{dR_1}{dt} = E_1 - u\beta_{1C}R_1C - (d_0 + \alpha C)R_1 - p_1R_1 + ex(R_{1N} - R_1) \quad (1b'')$$

$$\frac{dR_2}{dt} = E_2 + (d_0 + \alpha C)R_1 - u\beta_{2C}R_2C - p_2R_2 + e(1-x)(R_{2N} - R_2) \quad (1c'')$$

$$\frac{dR_{1N}}{dt} = E_1 - d_0 R_{1N} - p_1 R_{1N} + ex(R_1 - R_{1N}) \quad (1d'')$$
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

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

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

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

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

Figure 3. The effect of the niche constructor depends on the type of niche construction, the magnitude *e* and asymmetry *x* of diffusion. Large circles: nutrient concentration of the patch of the niche constructor (black) and the neighbouring patch (white) at equilibrium, when there is no exchange between the patches. The black line is the zero-net-growth isocline of the niche constructor. The dashed grey line is the positive growth isocline of the neighbour such that the growth rate is the same as in the empty patch without exchange. Small circles: equilibrium resource concentration within the neighbouring patch as exchange increases (from 1 to 50 along black arrows), for different diffusion asymmetries (along the different lines, *x*=0, 0.2, 0.5, 0.8 or 1 or 1). Colours indicate whether the presence of the niche constructor is
Figure 3. (Continued).

beneficial ($\Delta r_N > 0$, green points) or detrimental ($\Delta r_N < 0$, red points) to the establishment of its neighbour. (a)–(b) In the 'consumer' scenario, the consumer always has a detrimental effect on the neighbor, (c)–(d) a farmer ($\gamma = 1.8$) sometimes facilitates the establishment of the neighbour, especially when the neighbour is a $R_1$ specialist (d), (e)–(f) an inhibitor ($\alpha = -1$, $d_0 = 1$ (e)) or a stimulator ($\alpha = 5$, $d_0 = 0.5$) has a positive effect when the diffusion is biased towards the resource whose density increases due to niche construction. Parameters: $E_1 = E_2 = 5$, $m = p_1 = p_2 = 2$, $u = 5$.

$$ \frac{dR_{2N}}{dt} = E_2 - d_0 R_{1N} - p_2 R_{2N} + e(1-x)(R_2 - R_{2N}) \quad (1e'') $$

keeping $d_0 + \alpha C > 0$ positive (negative $d_0 + \alpha C$ would imply that $R_2$ becomes $R_1$).

We illustrate our findings using a graphical approach (Fig. 2) where we plot resource concentrations in the patches of the niche constructor and of the coloniser. The concentration of the empty patch without diffusion (white circle) depends on the input and loss rates of the two resources. The concentration of the occupied patch, in the absence of diffusion, settles on the zero net growth isocline (ZNGI) of the niche constructor, i.e. the curve on which resource concentrations result in a null growth rate. The type of niche construction influences resource concentration in the neighbour patch. Niche constructors either depress the concentration of both resources (pink circle) or increases the quantities of $R_1$ (yellow circle) or $R_2$ (purple circle). Diffusion homogenizes resource concentrations, bringing the resource concentration of the empty patch (white circle) closer to that of the occupied patch (coloured circles). Rectangles in Fig. 2 illustrate the possible concentrations within the empty patch given the diffusion of resources. Resulting concentrations only increase the growth rate of the coloniser when the new concentrations lie above the dashed grey line (growth isocline sustaining the growth rate $r_0$). Note that the steepness of this isocline and of the ZNGI of the niche constructor depend on their respective preferences (steeper growth isoclines for $R_1$ specialists, flatter growth isoclines for $R_2$ specialists).

# Results

## Scenario 1: no niche construction

A consumer can always establish and reach an equilibrium when its capacity to acquire the two resources is sufficient relative to its mortality (Supporting information).

The consumer always has a competitive effect on the neighbour due to resource consumption. Figure 3a–b show that the resource concentrations in the presence of the niche constructor are always the $r_0$ growth isocline (dashed grey line). It is worth noting that the magnitude of the negative effects depends on exchange rates, differences in preferences and asymmetry in resource diffusion. The effect of the consumer on its neighbour can be mathematically analyzed and captured by $\Delta r_N$ (see the Supporting information for the complete formula). It is here always negative, implying that the consumer always negatively impacts the neighbour. There are two cases where $\Delta r_N$ is null, implying a neutral interaction between the consumer and the neighbour: either when the neighbour and the consumer are specialized on different resources, or when the resource on which the neighbour is specialized does not diffuse.

Figure 3a–b illustrates these negative effects of the consumer. The axes show the environmental state of the two resources ($R_1, R_2$) in the empty neighbouring patch in the absence of exchange (large white circle with coordinates ($R_{10}^*, R_{20}^*$)) and in the patch occupied by the niche constructor (large black circle with coordinates ($R_1^*, R_2^*$)). Here, the niche constructor is a simple consumer, so that the resource concentration

<table><caption>Table 1. Variables, parameters, units and domains.</caption><thead><tr><th></th><th>Variables</th><th>Unit</th></tr></thead><tbody><tr><td>t</td><td>time</td><td>year</td></tr><tr><td>C</td><td>biomass density of the constructor organism</td><td>g m<sup>-2</sup></td></tr><tr><td>R<sub>i</sub></td><td>concentration of resource <i>i</i> in the occupied patch</td><td>g m<sup>-2</sup></td></tr><tr><td>R<sub>IN</sub></td><td>concentration of resource <i>i</i> in the neighbouring (and empty) patch</td><td>g m<sup>-2</sup></td></tr><tr><td></td><td>Parameters</td><td></td></tr><tr><td>u</td><td>uptake rate</td><td>g<sup>-1</sup> m<sup>2</sup> year<sup>-1</sup></td></tr><tr><td>E<sub>i</sub></td><td>input rate of resource <i>i</i></td><td>g year<sup>-1</sup></td></tr><tr><td>p<sub>i</sub></td><td>loss rate of resource <i>i</i></td><td>year<sup>-1</sup></td></tr><tr><td>m</td><td>loss rate of the constructor</td><td>year<sup>-1</sup></td></tr><tr><td>β<sub>i</sub></td><td>preference for resource <i>i</i></td><td>—</td></tr><tr><td>e</td><td>exchange rate between the patches (connectivity)</td><td>year<sup>-1</sup></td></tr><tr><td>x</td><td>propensity of resource 1 to diffuse more than resource 2</td><td>—</td></tr><tr><td>γ</td><td>niche construction rate by the niche constructor (scenario 2)</td><td>year<sup>-1</sup></td></tr><tr><td>d<sub>0</sub></td><td>transformation rate of resource 1 to resource 2 (scenario 3)</td><td>year<sup>-1</sup></td></tr><tr><td>α</td><td>control of the transformation rate of the constructor (scenario 3)</td><td>g<sup>-1</sup> m<sup>2</sup> year<sup>-1</sup></td></tr></tbody></table>
is lower in the occupied patch than in the empty patch (the black circle is below and left of the white circle). Consequently, diffusion lowers resources in the neighbour patch and the constructor harms the neighbour (competition, shown as red dots in Fig. 3). Each small circle, with coordinates ($R_{1N}$, $R_{2N}$), corresponds to the equilibrium state of the neighbouring patch for a given exchange rate $e$, and are shown here for different asymmetries in diffusion. Values of $e$ increase along arrows on the figure. The colour of the point depends on the value of $\Delta r_N$, which is the effect of the constructor on the neighbour. Darker red circles indicate a stronger negative effect.

A specialist neighbour is more affected by the diffusion of its preferred resource. In Fig. 3a, the consumer and the neighbour are both generalists and their growth isoclines are parallel. In Fig. 3b, the consumer is a generalist (black line) while the neighbour is specialized on $R_1$ (dashed grey line). The effect of the consumer is quantitatively different in the two cases. When the neighbour is specialised on $R_1$ and the diffusion is biased towards $R_2$ (small circles aligned vertically on panel b), the effect of the consumer is almost neutral (the circles are almost white), which is not the case when both the neighbour and the consumer are generalists (panel a).

Imagine two plant species competing for phosphorous and nitrogen. A phosphorous specialist would not be very affected by the presence of another plant in its vicinity given that phosphorous is not very mobile in the soil. In addition, in line with consumer resource theory (Tilman 1982), a starch-specialist bacteria would not be very affected by a glucose specialist, even if the two resources diffuse abundantly.

## Scenario 2: farmer

Niche constructors now increase input rates of $R_1$. In this scenario, the farmer possibly has a positive effect on its neighbour. In Fig. 3c-d, some points are above the $r_0$ isocline, indicating situations in which the presence of the niche constructor increases resource concentrations in the neighbouring patch. The niche constructor reaches an equilibrium biomass, or grows exponentially depending on the intensity of niche construction (Supporting information). The exponential growth in our model means that in an empirical setting, the niche constructor would grow (possibly become very large) until its growth is limited by another factor that is not modelled here.

Whether the farmer reaches an equilibrium or grows exponentially, facilitation can only occur due to the construction of $R_1$, i.e. $\Delta r_N$ can be positive when $\Delta r_1$ is positive. A necessary condition is therefore (see the Supporting information for derivation):

$$ \gamma > u\beta_{1C} \frac{E_1}{p_1} \qquad (4) $$

When the condition (Eq.) 4 is fulfilled, farming leads to the accumulation of $R_1$ in the local patch, fertilising the neighbour patch through diffusion. Graphically, when this condition is fulfilled, the black dot can be on the right of the white

dot, indicating that the patch of the niche constructor contains more $R_1$ than the empty patch. The niche construction rate $\gamma$ needs to be sufficient compared the absorption of the resource by the constructor ($u\beta_{1C}$, a measure of the competitive effect of the constructor) and the baseline concentration of the empty patch ($E_1/p_1$). When $R_1$ is not well supplied (low $E_1/p_1$), facilitation is likelier, which is in line with the stress gradient hypothesis (SGH, Bertness and Callaway 1994). When the farmer is not specialized on $R_1$ (e.g. a generalist as in Fig. 3c-d), facilitation is likelier. When the condition 4 is fulfilled, the large black circle in Fig. 3c-d (state of the patch occupied by the constructor) is on the right side of the large white circle (state of the empty patch). Note that a $R_1$ specialist would behave like a consumer of the previous scenario because the occupied patch would be less rich in terms of both $R_1$ and $R_2$ (Supporting information).

To sum up, condition 4 shows that a niche constructor cannot facilitate the establishment of a neighbour, including a conspecific, when the preference of the constructor for the constructed resource is too high. The rapid uptake of the farmed resource outweighs the constructing effect and the farmer only harms its neighbour. Facilitation occurs when the constructor sufficiently constructs $R_1$ and does not have a strong preference for $R_1$.

Condition 4 is necessary but not sufficient condition because the constructor also consumes $R_2$. The effect of the niche constructor on the second resource, $\Delta r_2$, is the same as in the baseline scenario. There is always less $R_2$ in the patch of the constructor due to its consumption by the constructor. The resulting competitive effect is shown in Fig. 3c-d, where the large black circle (state of the patch occupied by the constructor) is always below the large white circle (state of the empty patch). Therefore, in this scenario, the net interaction depends on the relative weight of the $R_1$ term (facilitative when condition 4 is fulfilled) and the $R_2$ term (always linked to higher competition). Facilitation outcomes are highlighted by green circles.

A constructor has a positive effect, including on a conspecific neighbour, when the resource that it farms is diffusing the most (Fig. 3). As in the consumer scenario, exchange between patches homogenises resource concentrations in a way that depends on asymmetry. Increase of the exchange rate is beneficial for the neighbour (small green circles) as long as the resulting resource concentration in the neighbour patch is above the dashed grey line. When the constructor and the neighbour have the same niche (Fig. 3c), the effect of the constructor is positive only when the exchange is strongly biased towards the constructed resource $R_1$ ($x=1$ at the horizontal green line, or $x=0.8$ at the two pale green circles). When the diffusion is not 100% asymmetric, the exchange rate has to be sufficiently weak to limit losses of $R_2$ (the two circles are the closest to the large white circle and correspond to weak exchange rate $e$). To summarize, when condition 4 is fulfilled, the exchange rate and asymmetry in diffusion determine whether facilitation occurs.

When the constructor and the neighbour have the same niche (Fig. 3c), the effect of the constructor is positive only
when the exchange is strongly biased towards the constructed resource $R_1$ ($x=1$ at the horizontal green line, or $x=0.8$ at the two pale green circles). When the diffusion rate is 100% biased towards the constructed resource ($x=1$), an increase in the diffusion rate also increases facilitation. When the diffusion is not 100% asymmetric, the exchange rate has to be sufficiently weak as to limit losses of $R_2$ (the two circles are the closest to the large white circle and correspond to weak exchange rate $e$).

When the neighbour is specialised on the constructed resource $R_1$ (Fig. 3d), more combinations of exchange and asymmetry are beneficial for the neighbour. In panel d, the farmer has a negative effect when $R_2$ only diffuses (vertical line of red circles at $x=0$). On the contrary, when the neighbour has a stronger preference for $R_2$ (as in Fig. 3c where it is a generalist), facilitation is less frequent. Facilitation by the farmer is possible for a larger range of exchange and asymmetry when species or other phenotypes are specialised on the farmed resource.

Niche differentiation also strongly promotes facilitation (Supporting information). When the niches of the neighbour and the farmer are differentiated, their competition for resources is low, so the positive effects of niche construction outweigh the effects of competition between resources.

Supporting information explores the combinations of exchange rate $e$ and asymmetry $x$ leading to facilitation. A biased diffusion towards $R_1$ leads to more beneficial effects. The effect of exchange rate is more ambiguous and interacts with asymmetry (Supporting information). When the farmed resource diffuses the most, the positive effects occur for low exchange rates. On the contrary, when the non-farmed resource diffuses most, the exchange rate needs to be sufficiently large for facilitation to happen. When the neighbour is a specialist of the farmed resource (Supporting information), increasing the exchange rate $e$ is always beneficial.

For example, a plant performing biological nitrogen fixation can increase the growth rate of its neighbours only if the plant itself does not consume all of the fixed nitrogen, if the neighbouring plants have a strong affinity for nitrogen, and if the other resources on which the two species depend are not very mobile. A plant mineralizing phosphorous would hardly have positive effects on its neighbours because of the tendency of phosphorous to diffuse very slowly.

### Scenario 3: transformer

Niche constructors now influence the transformation of $R_1$ into $R_2$ (Eq. 1b'-e'), either by inhibiting the transformation ($\alpha < 0$) or by stimulating it ($\alpha > 0$). The transformer reaches a positive equilibrium, grows exponentially or exhibits bistable dynamics (Supporting information for more details).

As in the farmer scenario, the transformer potentially facilitates its neighbour. Figure 3e–f illustrate that for example, an inhibiting $R_2$ specialist or a stimulating $R_1$ specialist both facilitate their neighbours. The expression of $\Delta r_1$ (Supporting information for the full expression) shows that a stimulator naturally negatively affects the concentration of $R_1$ in the

neighbouring patch due the combination of consumption and of transformation effects. The effect of an inhibitor on $R_1$ is more subtle because the inhibitor consumes $R_1$ (which is detrimental to the concentration of $R_1$ in the neighbouring patch) but allows for the better retention of $R_1$ due to niche construction (which is beneficial to the concentration of $R_1$ in the neighbouring patch). The expression of $\Delta r_1$ shows that $\Delta r_1$ is positive when the intensity of niche construction outweighs the consumption of $R_1$ (such that $-\alpha > u\beta_{1c}$). Graphically, this means that there are some combinations of preference, niche construction intensity and uptake rate for which the black dot is to the right of the white dot, as it is the case in Fig. 3e.

The effect of a transformer on $\Delta r_2$, ie. whether the black circle is going to be above or below the white circle in Fig. 3 – see the Supporting information for the full expression of $\Delta r_2$ – depends on several direct and indirect effects. First, the transformer has a direct negative effect due to its consumption of $R_2$. It also has a direct effect due to its niche construction activity, which either increases or decreases the concentration of $R_2$ in the neighbouring patch depending on whether the transformer stimulates or inhibits the transformation. Finally, the transformer has an indirect effect on $R_2$ due to its niche construction effect on $R_1$. Since a proportion of the first resource becomes the second resource, an inhibitor potentially increasing the concentration of $R_1$ in its patch also increases the supply of $R_2$. It is especially the case when diffusion is biased towards $\bar{R}_1$. This positive indirect effect is illustrated in Fig. 3e, where, when $x=1$, the concentration of the neighbour patch increases in $R_1$ and $R_2$ content (small green circles moving up and to the right as exchange increases). Conversely, a stimulator negatively affects $R_2$ due to its indirect effect on $R_1$.

Figure 3e shows that as in the ‘Farmer’ scenario, an inhibitor has a positive effect on the neighbour when the diffusion is biased towards $R_1$. Symmetrically, a stimulator has a positive effect when diffusion is sufficiently biased towards $R_2$ (Fig. 3f). Due to the transformation of resources, as opposed to the first two scenarios, diffusion of $R_1$ only changes the patch concentrations of both $R_1$ and $R_2$ (circles at $x=1$ are not horizontal in Fig. 3e–f).

As in the farmer scenario, for a given asymmetry in diffusion $x$, the effect of the exchange rate is subtle and may cause the type of interaction to change. For example in Fig. 3e–f, the interaction switches from competition to facilitation (ie. resource concentrations move from below to above the $r_0$ growth isocline as diffusion increases). In panel e, for $x=0.8$, the presence of the transformer is beneficial at weak diffusion and becomes detrimental when $e$ increases (the first two dots are light green, the rest of the dots are light red along the $x=0.8$ line). In Fig. 3f, for $x=0.2$, the constructor has a detrimental effect on the coloniser for a weak exchange rate and is beneficial when the exchange rate increases. The effect of the exchange rate depends on the asymmetry in resource diffusion.

As in the ‘Farmer’ scenario, whether facilitation occurs depends on the relative niches of the two species. In most
combinations explored in the Supporting information, (see the Supporting information for an inhibitor and C4 for a stimulator), the niche constructor has a positive effect if niches are sufficiently differentiated, and if the coloniser is specialised on the constructed resource ($R_1$ for an inhibitor, $R_2$ for a stimulator).

The Lamto savanna (Ivory coast) in which trees and grasses with contrasting strategies towards the control of nitrification offer examples of the effect of two transformer (Srikanthasamy et al. 2018). A grass that inhibits nitrification could increase the ammonium concentration in its patch, but due to the low diffusion of ammonium, its effect on a neighbouring grass of another species is possibly simply neutral, or even negative if the two plants compete for other nutrients. A tree that stimulates nitrification, on the other hand, potentially has a positive effect on a neighbouring tree because nitrate is quite mobile in the soil. Facilitation is also more likely if the neighbouring tree has a stronger affinity for nitrate than the stimulating tree.

## Discussion

The interplay of facilitation and competition largely influences the structure and functioning of ecological communities, as clearly stressed by documented examples in plant and microbial communities (Yin et al. 2022, D'Andrea et al. 2024). Our work aims to clarify the conditions in which the consumption of resources and niche construction result in facilitation or competition. We used a two-patch model to study whether and under which conditions a niche constructor facilitates the establishment of its neighbour via the modification of resource availability (Table 2). While we illustrate our findings using a graphical approach, the simplicity of our models allows the mathematical computation of clear criteria, detailed in the appendices. Although we did not investigate the long-term outcome of emerging interactions, according to our definition of facilitation, facilitation occurs when resource concentration in the neighbour patch is higher in the presence of a constructor. A cultivator, increasing the input rate of one resource, potentially has a facilitating effect

when niche construction is sufficient and basal resource availability is low. On top of these two necessary conditions, the exchange rate and diffusion asymmetry parameters need to be such that the quantity of constructed resources brought in the empty patch outweighs the quantity of the non-constructed resource (i.e. which is only consumed) leaving the patch due to diffusion. This happens when asymmetry is strongly biased towards the constructed resource, or when the exchange rate is weak and only slightly biased towards the constructed resource. The conditions of facilitation in a situation where one resource becomes another are qualitatively similar. When the niche constructor increases the rate of transformation of one resource into another, the constructors acts like a farmer increasing the quantity of the second resource. When the niche constructor decreases the rate of transformation, the constructor acts like a farmer increasing the concentration of the first resource.

In line with a general theory of the niche (Koffel et al. 2021), our results show that neighbour and niche constructor preferences determine whether the constructor has a facilitating effect. Neighbour specialism on the constructed resource and specialism on the non-constructed resource on behalf of the niche constructor increases the possibilities of facilitation, in line with model results on the control of nitrification (Ardichvili et al. 2024). When the organisms have the same preferences (because they belong to the same species, for example; Supporting information), facilitation is possible in rare cases: when both organisms are specialised on the non-constructed resource and when diffusion is strongly biased towards the constructed resource. For example, nitrate in the soil is much more mobile than ammonium. The stimulation of nitrification, performed for example by trees in west-African savannas (Srikanthasamy et al. 2018) could result in intra-specific facilitation. The spatial aggregation of nitrification-stimulating trees, as expected when short range facilitation is at play (Kéfi et al. 2007, Rietkerk and van de Koppel 2008), further enforces the idea that facilitation occurs between such trees.

When the two organisms have different preferences (for example when individuals are of different species), a niche constructor has a facilitating effect when the constructed

<table><caption>Table 2. Summary of results: conditions that favour competition and facilitation in the three scenarios.</caption><thead><tr><th>Constructor</th><th>Interaction</th><th>Conditions strengthening the interaction</th></tr></thead><tbody><tr><td>Consumer (scenario 1)</td><td>Competition</td><td>Similar preferences;<br>strong exchange between patches;<br>diffusion biased towards the preferred resource of the neighbour</td></tr><tr><td>Farmer (scenario 2)</td><td>Facilitation<br>Competition<br>Facilitation</td><td>Not possible<br>Rich environment<br>Poor environment</td></tr><tr><td>Farmer (scenario 2)<br>Inhibitor (scenario 3)<br>Stimulator (scenario 3)</td><td>Competition</td><td>Similar preferences;<br>neighbour specialised on the non-constructed resource;<br>diffusion biased towards the non-constructed resource.</td></tr><tr><td></td><td>Facilitation</td><td>Sufficient niche construction intensity;<br>neighbour specialised on constructed resource;<br>constructor specialised on the non-constructed resource;<br>strong diffusion biased towards the constructed resource</td></tr></tbody></table>
resource is more beneficial to the neighbour with a different preference than to the constructor. Facilitation is enhanced when niches are sufficiently differentiated. This result echoes earlier theoretical studies. A larger niche sensitivity difference sensu Koffel et al. (2021) for instance increases possibilities of mutualism (Koffel et al. 2021). Tree-grass coexistence may also increase due to tree (grass) stimulating (inhibiting) nitrification in a spatial model of West-African savannas (Konare et al. 2024). During the first steps of ecological succession, phosphorus-limited species fixing nitrogen facilitate the establishment of species that are less limited by phosphorus but require more nitrogen (Koffel et al. 2018, Bytnerowicz and Menge 2021).

Supposing that preferences are slightly different, for instance when the neighbour is a rare mutant only slightly different from the constructor population, the constructor facilitates neighbours having a stronger preference for the constructed resource. In an adaptive dynamics framework (Dieckmann and Law 1996), the successful colonisation of such mutants may result in the replacement of the resident population and lead to the directional selection of preference. While at first glance our model seems based on similar ideas, note that here, the growth rate of the ‘rare mutant’ is assessed in the neighbour patch, so that its environment is only indirectly created by the resident, as expected if the model is not well mixed. Our framework therefore allows the consideration of a partly private environment (as in Abs et al. 2020). Though our model does not allow making predictions on the long-term evolutionary dynamics of different strategies, here invaders with a stronger preference for the constructed resource have a higher growth rate. We could expect that niche constructors become specialised on the constructed resource in the long term. The joint evolution of preference and niche construction resulting in specialisation on constructed resources has also been observed in theoretical studies (Krakauer et al. 2009, Picot et al. 2021) and in the empirical system of farming ants becoming dependent on their farmed mushroom (Cafaro and Currie 2005). In a microbial system, the co-evolution of two species leads to the specialization on the resource that is constructed by the other species resulting in mutualism between species (Harcombe 2010).

We found that some conditions for facilitation are in line with the SGH (Bertness and Callaway 1994, Callaway et al. 2002). In the second scenario, in which the constructor increases the entry of one resource, facilitation is more likely when the ecosystem is poor, i.e. more stressful ($E_1/p_1$ is weak). The SGH has been empirically derived from the observation of plants that create a favourable microclimate in cold environments. Our findings provide theoretical justification for the SGH in resource gradients which has been documented in experimental systems (Rebele 2000, Maestre et al. 2009, Piccardi et al. 2019). In a microbial community experiment, Piccardi et al. (2019) link the SGH to resource conditions and show that increased resource availability results in stronger competition.

Unlike previous models of niche construction (Krakauer et al. 2009, Kylafis and Loreau 2011), we did not assume that niche construction is costly for the constructor. Such costs may however exist (Kuzyakov and Domanski 2000, Bouma et al. 2005) and could easily be implemented in the model by reducing the uptake rate *u* or by increasing mortality *m* of constructors. This would shift the ZNGI of the constructor up/right because the constructor would need more resources to compensate for the cost of niche construction. At equilibrium, the resource concentration of the constructor would increase, implying that the difference between the resource concentration of the empty patch and the occupied patch would decrease. Modelling a cost would not change the qualitative results but would mitigate the effects (positive or negative) of the constructor on the neighbour, and result in more neutral interactions.

Note that we model a particular type of niche construction that only manipulates resource availability. Niche construction includes multiple processes, including those that modify environmental conditions that are not resources (e.g. abiotic conditions such as micro-climate, pH..., Odling-Smee et al. 1996). While our work does not explicitly model such type of niche construction, the modification of environmental conditions such as temperature, pH, or humidity could potentially affect the activity of decomposers and impact resource fluxes.

Beyond immediate neighbours, conditions allowing local facilitation could influence the emergence of spatial patterns at larger scales. In spatially explicit models of niche construction (Klausmeier 1999, Lejeune et al. 2002, Rietkerk et al. 2002, Scanlon et al. 2007, Dong 2022), the level of niche construction by niche constructors determines the spatial patchiness of organisms. In another study, the level of facilitation determines the spatial aggregation of organisms (Kéfi et al. 2007). Although we did not take a spatially explicit approach, by making a direct link between niche construction and facilitation at a local scale, we suggest that the modification of resource availability could lead to the emergence of spatial patterns. Also, the emergence of spatial patterns requires the combination of short-range facilitation with longer-range competition (e.g. for space, Rietkerk et al. 2002). By highlighting the conditions under which short-range facilitation emerges, our model may help to better understand the conditions under which spatial patterns are expected.

To test the prediction of the model, we could combine data on the identity of plant species facilitating others (for example from Yin et al. (2022)) with plant traits (whether they engage in symbiotic fixation, mycorrhizal associations, their preference for different types of resources). We could therefore check that niche constructors have a facilitating effect only when their preference for the constructed resource is sufficiently weak. Experimentally, we could test the effect of diffusion using transplantation, as in plant-soil feedback experiments (van der Putten et al. 2013), and control the homogenisation between two pots by mixing different fractions of soil in each treatment.
Our results can be relevant in different fields, including in agronomy and agro-ecology. Knowing the conditions in which niche constructors have a positive effect on their neighbour could guide the choice of companion planting, cover crop plants or plants to be used in intercropping. To reduce synthetic inputs and improve nutrient cycling in agricultural landscapes, agroecologists suggest using companion plants, i.e. growing a target plant with a neighbour of a different species improving the growth conditions of the target plant (e.g. repelling predators, improving nutrient availability, Wezel et al. 2020). Examples include growing cereals with N-fixing legumes (Layek et al. 2018), legumes with vegetables (Brooker et al. 2015), maize with *Braccharia*, a genera inhibiting nitrification (Souza et al. 2024). Our results suggest that the niches of the companion plant and the target plant, as well as the mobility of the resource mobilised by the companion, need to be taken into account to evaluate whether the companion effectively improves the yield of the target plant.

*Acknowledgements* – We thank Thomas Koffel, Sonia Kefi and Vincent Calcagno for stimulating discussions on the manuscript. *Funding* – JCL and SB are supported by the GainGrass project (Global Assessment of Nitrification Inhibition by Tropical Grasses Project; https://anr.fr/Project-ANR-19-CE02-0009).

## Author contributions

**Alice Nadia Ardichvili:** Conceptualization (equal); Formal analysis (lead); Visualization (lead); Writing – original draft (equal). **Sebastien Barot:** Supervision (equal); Writing – review and editing (equal). **Jean-Christophe Lata:** Supervision (equal); Writing – review and editing (equal). **Nicolas Loeuille:** Conceptualization (equal); Supervision (equal); Writing – review and editing (equal).

## Data availability statement

There are no additional data for this paper.

## Supporting information

The Supporting information associated with this article is available with the online version.

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