Introduction

Ecosystems are interconnected through the flow of energy, materials, and organisms (Loreau et al. 2003; Polis et al. 1997). These flows, often referred to as subsidies (i.e., resources of external origin), can alter the recipient ecosystem by changing the spatial distribution of resources (Gravel et al. 2010) and cause trophic cascades (Leroux and Loreau 2008). The impact of such subsidies is influenced by the timing (Leroux and Loreau 2012) and quality (Osakpolor et al. 2023) of the subsidy, the ability of top predators to exploit the subsidy (McCary et al. 2021), the ratio of subsidies to comparable resources in the recipient system, and its spatial and temporal extent (Marczak et al. 2007).

Understanding the impact of subsidies on recipient ecosystems is hampered by the subsidy affecting multiple trophic levels. In many cases, a single subsidy can have direct effects on consumers and detritus in the recipient ecosystem through processes like direct consumption and recycling. For example, migratory salmon provide marine-derived subsidies to streams (Gende et al. 2002), which are consumed by various organisms (direct consumption) and their carcasses can end up in the stream's nutrient pool (recycling) to benefit primary producers (Wipfli et al. 1998). Terrestrial leaf subsidies are directly consumed by larval caddisflies, while also providing subsidies for algae through recycling (Holgerson et al. 2016).

The challenge of understanding subsidy impacts on recipient ecosystem is additionally complicated by the multiple indirect effects that a subsidy can exert on the recipient ecosystem. For example, emergent aquatic insects can subsidize the riparian secondary consumers (Kowarik et al. 2021), potentially resulting in increased predation pressures on the riparian herbivores, and ultimately increasing riparian plant productivity. Hence, emergent aquatic insects can indirectly affect plant productivity (Henschel et al. 2001; Holgerson et al. 2016). Simultaneously, those emergent aquatic insects can increase the input of subsidies into the riparian nutrient pool (Dreyer et al. 2015) causing an increase in herbivore biomass (Bultman et al. 2014), and potentially resulting in an increase in riparian secondary consumer biomass. Hence, measuring the degree of ecosystem coupling by a single subsidy becomes quite challenging as pathways of recycling and direct consumption involve multiple indirect and potentially contrasting effects in the recipient ecosystem. One of the challenges of understanding subsidy impacts on recipient ecosystems is therefore to determine the interaction between the direct consumption and recycling subsidy pathways (Allen and Wesner 2016). Mathematical models can be a useful tool to tackle this challenge.

Ecological understanding gained through feedbacks between theory and empirical studies can be valuable in addressing how anthropogenic stressors affect the subsidy dependent functioning of recipient ecosystems (Simberloff and Wilson 1969; Tilman 1977). The connection between empirical and theoretical research in subsidy-recipient ecosystem studies, however, has been missing due to a mismatch in what is measured. Empirical studies typically measure the interaction between subsidies and the recipient ecosystem in terms of the utilization of subsidies and their transfer in the recipient food web (e.g., using stable isotopes- Lam et al. 2013). Meanwhile, theoretical studies measure the input of subsidies into the recipient ecosystem (see review in Osakpolor et al. 2021). However, the input rate of subsidies does not necessarily reflect the actual energy consumed by the recipient ecosystem. Consumer preferences for subsidies can impact the stability (Abrams et al. 1998; Huxel and McCann 1998) and functioning (Leroux and Loreau 2008) of the recipient ecosystem, so a high subsidy input rate may be insufficient to determine subsidy impact if it is not a preferred resource. Moreover, theoretical studies often focus on endpoints such as stability (Gounand et al. 2014), which differ from the functional measures used in empirical studies like efficiency and production (Mehner et al. 2022).

Our aim is to develop theories on how subsidies affect the recipient ecosystem using metrics that are compatible with empirical research. To do this, we will derive and analyze a series of models to explore how subsidy-recipient ecosystem coupling pathways can impact, both independently and interactively, the equilibria stocks (i.e., biomass) and functions (i.e., recycling, efficiency and production) of the recipient ecosystem. First, we will examine the predictability of how the two different subsidy-recipient ecosystem coupling pathways (direct consumption and recycling) interact to impact the equilibria stocks and functions of the recipient ecosystem. Then we will investigate how different subsidy-recipient ecosystem coupling pathways affect the equilibria stocks and functions of the recipient ecosystem.

Methods

Model case studies

Riparian areas are a globally common and specific area of concern as they are important hotspots and hot moments of biogeochemical processes and biodiversity, supporting ecosystem functions (Costanza et al. 1997; Williamson et al. 2008). Riparian ecosystems are functionally linked to aquatic ecosystems through subsidy flows. The permeability of the boundary between aquatic and riparian ecosystems has been well documented, suggesting the examination of these ecosystems by analyzing cross-ecosystem resource exchange (Soininen et al. 2015). We will achieve our research goals using two case studies featuring a riparian ecosystem subsidized by aquatic resources. The riparian ecosystem is constituted by inorganic nutrient (e.g., nitrogen), plants (e.g., stinging nettles), herbivores (e.g., leafhoppers), and predators (e.g., spiders) subsidized by emergent aquatic insects. In case study 1, the aquatic subsidy (emergent aquatic insects) is directly consumed by the riparian predator (Fig. 1a), while in case study 2 the aquatic subsidy (flood-driven deposition of macrophytes, e.g., reeds) is directly consumed by the riparian herbivore (Fig. 1b). For both case studies, the subsidy can also be recycled to the nutrient pool.

Fig. 1
figure 1

Description of case studies. Both case studies include the same riparian ecosystem depicted in the green box. It consists of an inorganic nutrient (nitrogen), plant (e.g., stinging nettle), herbivore (e.g., leafhopper) and a predator (e.g., spider). The two case studies differ in how the subsidy contributes to the ecosystem. While in both case studies the subsidy can be recycled to the nutrient pool, in case study 1 (a) and 2 (b), the aquatic subsidy (e.g., emergent aquatic insect and macrophytes) is directly consumed by the riparian predator and the riparian herbivore, respectively

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Model

We derived models for each case study that allowed us to investigate how each subsidy pathway (i.e., direct consumption or recycling to the nutrient pool) and the combined pathway (i.e., both direct consumption and recycling to the nutrients pools) impacts the recipient ecosystem. Specifically, we had a total of seven models consisting of (1) base model: non-subsidized recipient ecosystem, (2) full model for case study 1, (3) full model for case study 2, (4) direct consumption model for case study 1, (5) direct consumption model for case study 2, (6) recycling model for case study 1, (7) recycling model for case study 2 (Figs. 2a-2g respectively for diagram depictions of each of these models).

Fig. 2
figure 2

Diagrams illustrating the seven models. Here, P, H, A, N, E, and L represent stock of predator, herbivore, plant, nutrient, predator subsidy, and herbivore subsidy, respectively. Each model has an influx of nutrients (I) to the nutrient pool, and a loss of nutrients (l) from that pool. The arrows describe the direction of nutrient flow with solid arrows indicating consumption links while dashed arrows indicate recycling links. For the descriptions of w4, w5, dL, \(\uppi\) P, \(\uppi\) H, δE, δL, δA, δH, δP see the main text

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The complete model equations consist of one abiotic compartment: inorganic nutrients (N) and five biotic compartments in grams: plant (A), herbivore (H), predator (P), H subsidy (L), and P subsidy (E). L and E are subsidies while N, A, H, P are in the recipient ecosystem (Table S1). The recipient ecosystem model is open at the basal level through constant input of inorganic nutrient, I, and constant loss of inorganic nutrient, l. Biotic modules recycle nutrients at rates di but only a fraction, 1- δi, of recycled nutrients reach the soil nutrient pool whereas the remainder (δi) is lost from the system, where i is L, E, A, H, or P. We use Type I functional responses for each consumer with an attack rate, ai, and consumption efficiency, ei because it is parsimonious (fewer parameters than more complex functional responses e.g., Type II) and enables us to derive analytical equilibria solutions. These equilibria stocks and functions can then be solved for multiple combinations of parameter values. Consequently, the model remains simple enough to be mathematical tractable, while still demonstrating a range of dynamics from stable fixed points to stable limit cycles which allowed us to gain general insights on how subsidy pathways affect the recipient ecosystem functions.

The donor ecosystem-controlled subsidy (E) (with constant input rates wE) (Eq. 5) serves as a subsidy for P (Eq. 4) and also enters the soil nutrient pool via recycling ((1 ― δE) dEE) (Eq. 1). P, therefore, has two resources, subsidy (E) and herbivore (H). Following Huxel et al. (2002) and McCann et al. (2005), we model P preference for H as πp (where 1- πp is the preference for E) (Eq. 4). Similarly, the donor ecosystem-controlled subsidy (L) (with constant input rates wL) (Eq. 6) serves as a subsidy for H (Eq. 3) and also enters the soil nutrient via recycling ((1 ― δL) dLL) (Eq. 1). As above, we also model H preference for the local plant as πh (where 1- πh is the preference for E) (Huxel et al. 2002; McCann et al. 2005) (Eq. 3). The models are described by the dynamic equations as shown in Eqs. 1 – 6. The seven models can be derived from the complete set of model equations as stated in Table 1.

Table 1 Derivation of the models from the model equations
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$$\frac{\text{dN}}{{\text{dt}}} = \text{I} + \left(\text{1} - {\updelta}_{\text{L}}\right) \, {\text{d}}_{\text{L}}\text{L }+\left(\text{1} - {\updelta}_{\text{E}}\right) \, {\text{d}}_{\text{E}}\text{E}+\left(\text{1} - {\updelta}_{\text{A}}\right) \, {\text{d}}_{\text{A}}\text{A}+\left(\text{1} - {\updelta}_{\text{H}}\right){\text{d}}_{\text{H}}H + \left(\text{1} - {\updelta}_{\text{P}}\right){\text{d}}_{\text{P}}P - {\text{a}}_{\text{A}}\text{NA} - \text{lN}$$
(1)
$$\frac{\text{dA}}{{\text{dt}}} = {\text{a}}_{\text{A}}\text{NA}-{\text{a}}_{\text{H}}{\pi }_{h}\text{AH }-{\text{d}}_{\text{A}}{\text{A}}$$
(2)
$$\frac{\text{dH}}{{\text{dt}}}= {\text{e}}_{\text{H}}{{\text{a}}}_{\text{H}}\uppi_{h}\text{AH}+{\text{e}}_{\text{H}}{{\text{a}}}_{\text{H}}\text{LH}(1-{\pi }_{h}) - {\text{a}}_{\text{P}}{\pi }_{\text{P}}\text{HP}-{\text{d}}_{\text{H}}{\text{H}}$$
(3)
$$\frac{\text{dP}}{{\text{dt}}} = {\text{e}}_{\text{P}}{{\text{a}}}_{\text{P}}{\pi }_{\text{P}}\text{HP}+{\text{e}}_{\text{P}}{{\text{a}}}_{\text{P}}\text{EP}(1-{\pi }_{\text{P}}) -{\text{d}}_{\text{P}}{\text{P}}$$
(4)
$$\frac{\text{dE}}{{\text{dt}}}= {\text{w}}_{\text{E}}-{\text{a}}_{\text{P}}\text{EP}(1-{\pi }_{\text{P}}) -{\text{d}}_{\text{E}}{\text{E}}$$
(5)
$$\frac{\text{dL}}{{\text{dt}}}= {\text{w}}_{\text{L}}-{\text{a}}_{\text{H}}\text{LH}(1-{\pi }_{h}) -{\text{d}}_{\text{L}}{\text{L}}$$
(6)

Analysis

We compared the models (Fig. 2) using the same parameter sets. First, we solved the models for equilibrium (denoted with *) using Mathematica 12.3. With the aim of generating general predictions, we randomly selected 150,000 parameter sets whereby all parameters were simultaneously varied between 0 and 10 according to a uniform random distribution (as per Leroux and Schmitz, 2015) with the exception of δi, ei, wL, and wE, which are proportions and were simultaneously varied between 0 and 1 according to a uniform random distribution (Table S2 in supporting material). Next, using the 150,000 parameter sets and analytical equilibria solutions, we calculated equilibria stocks and functions for each parameter set retaining only feasible parameter sets (i.e., where N*, A*, H*, P* > 0). We follow Loreau (2010) in the derivation and measurement of analytical expressions for ecosystem functions including production, efficiency, and recycling (Eqs. S1—S16 in supporting material). We tested the general model predictions with an empirical case for the Lake Mývatn region in Iceland or similar systems. Mývatn is naturally eutrophic and sustains large populations of aquatic emergent insects. After emergence, they serve as resource for riparian predators (Sanchez-Ruiz et al. 2018) and riparian nutrient pool (Gratton et al. 2017). In the empirical case, the model was parameterized (15 of 21 parameters) with empirical values from McCary et al. (2021) (Table S2 in supporting material).

Subsidy coupling metrics

Based on our case studies, we have three distinctive subsidy-recipient ecosystem coupling pathways at equilibrium. Specifically, coupling by the direct consumption of subsidy (E*) by predator (P*) (Eq. 7), coupling by the direct consumption of subsidy (L*) by herbivore (H*) (Eq. 8), and recycling coupling of subsidy (E* or L*) (Eq. 9 or 10) which were derived following Tunney et al. (2012). The coupling by the direct consumption of E* by P* is defined as the ratio of subsidy (E*) to an alternative local resource (H*) consumed by P*, the coupling by the direct consumption of L* by H* is defined as the ratio of subsidy (L*) to an alternative local resource (A*) consumed by H*, while the nutrient coupling is defined as the ratio of subsidy E* or L* recycled, to alternate ecosystem compartments recycled.

$${\text{Consumption coupling for case study } 1 =\frac{({\text{e}}_{\text{p}}{{\text{a}}}_{\text{p}}{\text{E}}^{*}{\text{P}}^{*}(\text{1}-{\pi }_{\text{P}}))}{({\text{e}}_{\text{P}}{{\text{a}}}_{\text{P}}{\pi }_{\text{P}}{\text{H}}^{*}{\text{P}}^{*})}}$$
(7)
$$\text{Consumption coupling for case study }2 =\frac{({\text{e}}_{\text{H}}{{\text{a}}}_{\text{H}}{\text{L}}^{*}{\text{H}}^{*}(\text{1}-{\pi }_{h}))}{({\text{e}}_{\text{H}}{{\text{a}}}_{\text{H}}{\pi }_{h}{\text{A}}^{*}{\text{H}}^{*})}$$
(8)
$$\text{Recycling coupling for case study }2 =\frac{(\left(\text{1} - {\updelta}_{\text{L}}\right) \, {\text{d}}_{\text{L}}{\text{L}}^{*})}{(\left(\text{1} - {\updelta}_{\text{A}}\right) \, {\text{d}}_{\text{A}}{\text{A}}^{*} + \left(\text{1} - {\updelta}_{\text{H}}\right){\text{d}}_{\text{H}}{\text{H}}^{*} + \left(\text{1} - {\updelta}_{\text{P}}\right){\text{d}}_{\text{P}}{\text{P}}^{*})}$$
(9)
$$\text{Recycling coupling for case study }1 =\frac{(\left(\text{1} - {\updelta}_{\text{E}}\right) \, {\text{d}}_{\text{E}}{\text{E}}^{*})}{(\left(\text{1} - {\updelta}_{\text{A}}\right) \, {\text{d}}_{\text{A}}{\text{A}}^{*} + \left(\text{1} - {\updelta}_{\text{H}}\right){\text{d}}_{\text{H}}{\text{H}}^{*} + \left(\text{1} - {\updelta}_{\text{P}}\right){\text{d}}_{\text{P}}{\text{P}}^{*})}$$
(10)

Interactions and effects of subsidy-recipient ecosystem coupling pathways on recipient ecosystem functions

We studied the interaction of the subsidy-recipient ecosystem coupling pathways for our case studies using attenuation plots. For the attenuation plots we plotted the ratio of the full model (i.e., consisting of all pathways; Figs. 2b and 2e) to the base model vs the ratio of the sum model (i.e., a sum of the direct consumption of the subsidy only model and the direct recycling of the subsidy to the nutrient pool only model) to the base model for a given metric (e.g., recycling) for each case study. In this way, we determined the interaction between the coupling pathways by examining where the points sit relative to the one-to-one line. If the points are above the line that means that recycling and direct consumption interact synergistically, i.e., are greater than the sum of their parts (the one-to-one line), while if they sit below the one-to-one line, they interact antagonistically, i.e., are less than the sum of their parts (Fig. S1 in supporting material).

Effects of subsidy-recipient ecosystem coupling pathways on stocks and function of the recipient ecosystem

To assess the independent effect of each subsidy-recipient ecosystem coupling pathway on the recipient ecosystem, we measured the natural logarithm of each equilibrium stock and function in the independent coupling models (i.e., consumption coupling by P, consumption coupling by H, and recycling coupling models) divided by the base model equilibrium stock and function. We report these coupling metrics along a coupling gradient to determine how coupling affects recipient ecosystem functions (Fig. S2 in supporting material). Specifically, the coupling gradient measures the extent to which the recipient ecosystem relies on subsidy. In this way, a value greater than 0 on the y-axis implies that subsidy coupling led to greater stocks or functions relative to a non-subsidized ecosystem for a given subsidy-recipient ecosystem coupling (x-axis).

Results

General Overview

We obtained one feasible equilibrium solution for the models. Out of the 150,000 parameter sets, a range of 3134 – 4055 parameter sets led to feasible equilibria (i.e. N*, A*, H* and P* > 0). All parameter sets were stable for the full model and the direct consumption model for case study 1, base model, and recycling model, while some parameter sets were stable for the full model and the direct consumption model for case study 2. All models had some parameter sets with stable limit cycles (Table 2).

Table 2 Overview of the models with information on parameter sets retained, their stability and stable orbits. In bracket is the percentage of total parameter sets retained that are stable or exhibit stable limit cycles
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How do subsidy-recipient ecosystem coupling pathways interact and affect recipient ecosystem functions?

The full model incorporating both direct subsidy pathways (i.e., direct consumption of the subsidy and direct recycling of the subsidy to the nutrient pool) results in lower total ecosystem stocks and lower total ecosystem functions (i.e., sum across trophic levels) than the sum model (i.e., a sum of the direct consumption of the subsidy only model and the direct recycling of the subsidy to the nutrient pool only model) (Fig. 3). Therefore, the coupling pathways interact antagonistically, i.e., feedbacks between the coupling pathways result in lower stocks and functions relative to a model which omits these feedbacks. Moreover, this difference between the full model and the sum model is dependent on the strength of the subsidy effect. As the impact of the subsidy becomes stronger (i.e., the total stocks or functions becomes much greater in the presence of the subsidy) the difference between the full and sum models decreases. The above relationship also holds at the various trophic levels of the recipient ecosystem (Figs. S3—S6 in supporting material), and for parameter sets that exhibit stable and non-stable limit cycles (Figs. S11—S14 in supporting material). The relationship was also confirmed for the empirical case (S15 in supporting material).

Fig. 3
figure 3

Interaction of subsidy-recipient ecosystem coupling pathways and effects on total stocks and total functions of the recipient ecosystem (i.e., sum across trophic levels). The red points represent case study 1, while the black points represent case study 2. Effi., Prod., Recy., Chain represents the efficiency, production, recycling and food chain efficiency functions, respectively. The red line is the one-to-one line where points below the line indicate that direct subsidy pathways (i.e., direct consumption or recycling into nutrient pool) interact antagonistically, while points above the line indicate that direct subsidy pathways interact synergistically. See Fig. S1 in supporting material on how to interpret the figure

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How do different subsidy-recipient ecosystem coupling pathways affect recipient ecosystem stocks and functions?

Increasing the subsidy couplings led to an increase in stocks and functions of the recipient ecosystem (i.e., sum across trophic levels) (Fig. 4). But the intensity of the increase differs between recipient ecosystem functions and coupling pathways. For example, the efficiency of the recipient ecosystem increased stronger in case study 1 consumption coupling than case study 2 consumption or recycling coupling (Figs. 4a-c). This difference occurs when the top predator depends more on subsidy than on local alternate resource (i.e., C1 coupling > 0) (Fig. 4b). Recycling coupling always led to equal or higher stocks and functions of the recipient ecosystem relative to a non-subsidized ecosystem (base model) (Figs. 4a, 4d, 4g, 4j), while consumption couplings may cause lower stocks and functions of the recipient ecosystem, but the stocks and functions increase with coupling (Fig. 4) (also corroborated by the empirical result Fig. S16 in supporting material). This is because recycling coupling always led to equal or higher stocks and functions across the recipient ecosystem trophic levels, while consumption couplings had cascading effects on the recipient ecosystem trophic levels (Figs. S7—S10 in supporting material). The positive cascading effects of case study 1 consumption coupling on the stock and functions of plant is more evident when the top predator depend more on subsidy than local resource (Figs. S7e, S8b, S9b, S10b in supporting material).

Fig. 4
figure 4

Subsidy coupling effects on stocks and functions of recipient ecosystem (i.e., sum across trophic levels). The y-axis shows a comparison between the coupling and the base models for a particular function/stock. Note: because of the way the parameter sets were chosen for the two case studies the recycling models for both case study 1 and case study 2 end up being identical. R, C1, and C2 represent the recycling model, and the consumption model for case study 1 and 2, respectively, while R coupling, C1 coupling, and C2 coupling are the coupling metrics for these different models (Eqs. 9 and 10 for R coupling and Eqs. 7 and 8 for C1 and C2 coupling). Base is the base model. Effi, Prod., Recy., Chain represents the efficiency, production, recycling and food chain efficiency functions respectively. See subsidy coupling metrics section for definition of the x-axis and Fig. S2 in supporting material on how to interpret the figure

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Discussion

We derived ecosystem models that enabled us to study how subsidy-recipient ecosystem coupling pathways independently and interactively affect stocks and functions of a recipient ecosystem. This is because subsidies are often coupled to recipient ecosystems via direct consumption and recycling, with both pathways causing multiple direct and indirect effects with different strengths and directions. We aimed to understand how the different couplings interact and generate feedbacks in recipient ecosystems (Allen and Wesner 2016).

The theoretical study was conducted using metrics that are compatible with empirical research. This is motivated by the mismatch in terms of what is being measured in empirical and theoretical subsidy-recipient ecosystem research studies. For example, empirical scientists often study subsidy effects in terms of its usage (e.g., Hambäck et al. 2016), while the theoreticians study subsidy effects in terms of its availability (see review in Osakpolor et al. 2021). The assumption of most subsidy models contradicts evidence that subsidy preference over local resource has an impact on the effect of subsidy (Abrams et al. 1998; Leroux and Loreau 2008).

How do subsidy-recipient ecosystem coupling pathways interact and affect recipient ecosystem functions?

The model analysis shows that the direct consumption and recycling coupling pathways of subsidy interact antagonistically, as the feedbacks between both pathways led to lower stocks and functions of recipient ecosystem (i.e., sum across trophic levels) and the various trophic levels relative to a model which omits these feedbacks. Previous studies have attempted to study how changes in multiple trophic levels interactively affect food webs. For example, a meta-analysis found that the presence of predators leads to a decline in herbivore biomass and an increase in plant biomass, regardless of plant productivity (Borer et al. 2006). On the other hand, an increase in nutrient levels increased plant biomass, but not herbivore biomass irrespective of the presence of predators. In particular, the minimal impact of increasing nutrient levels on higher trophic levels demonstrates the importance of bottom-up effects and the difficulties in teasing out the impacts of bottom-up and top-down effects in empirical systems. These results were consistent across different ecosystems, including marine, freshwater, and terrestrial, and were independent of study size and duration (Borer et al. 2006). In another meta-analysis of experimental studies, the effect of nutrient and herbivory changes on producer biomass showed limited support for statistical interactions (Gruner et al. 2008). However, a synergism of nutrient enrichment and herbivore removal on producer biomass in marine temperate rocky reef systems was found.

The above studies demonstrate some of the fundamental difficulties in attempting to tease out the interactive effects of bottom-up and top-down effects in experimental settings. By using a modeling approach, we were able to explicitly quantify these interactive effects. Our study defined interaction similar to the definition of stressors interaction, whereby combined effects are compared to the sum of individual effects (Brook et al. 2008). An interaction is antagonistic when the sum of individual effects is greater than the combined effects and synergistic when the sum of effects is less than then combined effects (Brook et al. 2008). Our model predictions provide profound insight on how subsidies can impact recipient ecosystems and the need to study the effects of the different subsidy-recipient ecosystem pathways on the recipient ecosystem. This is because the subsidy coupling pathways are also prone to anthropogenic stressors, which can change the relative importance of the recycling link and the direct consumption link. For example, a degraded riparian habitat (i.e., less vegetation structure) can reduce the accessibility of emergent aquatic insects to riparian consumers (Hunt et al. 2020; Muehlbauer et al. 2014), which could increase the input of emergent aquatic insects into the riparian nutrient pool. A degraded riparian soil (i.e., low nutrient levels) can enhance the impacts of recycling linkage of emergent aquatic insects on the riparian food web (Burpee and Saros 2020), because nutrient enrichment represents a relative relief from nutrient limitation (Hillebrand 2002).

How do subsidy-recipient ecosystem coupling pathways affect recipient ecosystem stocks and functions?

Our models also predict that consumption and recycling coupling pathways of subsidy had differential effects on the stocks and functions of the recipient ecosystem trophic levels. Recycling coupling pathway always led to equal or higher stocks and functions across the recipient ecosystem trophic levels, whereas consumption couplings had cascading effects. Specifically, the nutrient and herbivore trophic levels of the recipient ecosystem remained similar for recycling coupling, whereas the plant and predator trophic levels slightly increased. Our predictions support the trophic theory that predicts that an increase in the nutrient levels of an ecosystem consisting of plants, herbivores and predators will cause the plant biomass to increase, whereas herbivore biomass remains unchanged (Oksanen et al. 1981). The cascading (indirect) effect predicted by our model is more pronounced when the consumer of the recipient ecosystem consumes more of the subsidy than local resource. Our results have also been corroborated by previous studies focusing on different ecological systems. For example, fish predators induced a significant increase in the biomass of phytoplankton, whereas nutrient addition promoted phytoplankton growth without significantly elevating the biomass of zooplankton (Brett and Goldman 1997). Another study revealed that predators demonstrate strong top-down effects throughout entire food chains, whereas the bottom-up influence of eutrophication primarily affects plant productivity (Borer et al. 2006). Thus, the asymmetry in the effects of direct consumption and recycling coupling pathways have far-reaching implications: reduction of the consumption coupling pathway of subsidy may have more extensive multi-trophic implications for communities than will the recycling coupling pathway.

Our study acts as a bridge between empirical and theoretical fields. In particular, our predictions on ecosystem functions (production, efficiency and recycling) can be empirically tested at a field scale. Empirical methods for measuring production exist for various types of ecosystems, including marine (Nishijima et al. 2021), terrestrial (Zheng et al. 2003), and freshwater ecosystems (Puts et al. 2022). In some instances (e.g., Eggert and Wallace 2003), empirical studies of subsidies measure these ecosystem functions and in others (e.g., Barrett et al. 2005) it would require minimal additional effort. Measuring such functions is key as we know many ecosystem functions have different responses to global changes (Giling et al. 2018; Larsen et al. 2016) and forecasting the responses of these ecosystem functions is becoming more imperative.