Coupling of green and brown food webs and ecosystem stability

Abstract Ecosystems comprise living organisms and organic matter or detritus. In earlier community ecology theories, ecosystem dynamics were normally understood in terms of aboveground, green‐world trophic interaction networks, or food webs. Recently, there has been growing interest in the role played in ecosystem dynamics by detritus in underground, brown‐world interactions. However, the role of decomposers in the consumption of detritus to produce nutrients in ecosystem dynamics remains unclear. Here, an ecosystem model of trophic food chains, detritus, decomposers, and decomposer predators demonstrated that decomposers play a totally different role than that previously predicted, with regard to their relationship between nutrient cycling and ecosystem stability. The high flux of nutrients due to efficient decomposition by decomposers increases ecosystem stability. However, moderate levels of ecosystem openness (with movement of materials) can either greatly increase or decrease ecosystem stability. Furthermore, the stability of an ecosystem peaks at intermediate openness because open systems are less stable than closed systems. These findings suggest that decomposers and the food‐web dynamics of brown‐world interactions are crucial for ecosystem stability, and that the properties of decomposition rate and openness are important in predicting changes in ecosystem stability in response to changes in decomposition efficiency driven by climate change.

A key step in the integration of the two disciplines would be to link ecosystem functions and the stability of population dynamics. Some pioneering work has been conducted in this area (DeAngelis, 1980;Loreau, 1994;Odum & Pinkerton, 1955;O'Neill, 1976). A general theory incorporating the flux of energy or biomass between species predicted that ecosystems are stabilized by an increase in flux rates (DeAngelis, 1980;Loreau, 1994). Although this prediction is supported by earlier modeling studies (Odum & Pinkerton, 1955;O'Neill, 1976), these earlier theories largely focused on aboveground primary productivity as the ecosystem function or did not explicitly consider brown-world pathways. A recent study examined belowground dynamics (Miki, Ushio, Fukui, & Kondoh, 2010) but ignored aboveground food-web dynamics. Although ecosystems include both aboveground and belowground dynamics, there is currently very little understanding about the role played by decomposers in nutrient cycling and food-web dynamics and its effect on ecosystem stability.
Very few studies have developed ecosystem models to investigate the consequences of brown-world pathways for the stability of population dynamics (Gounand et al., 2014;McCann, 2011), even though it has been predicted that primary producer-based and detritus-based food webs have qualitatively different impacts on foodweb stability (Moore et al., 2004;Moore, de Ruiter, & Hunt, 1993).
McCann (2011) created an ecosystem model wherein detritus was seen to have a tendency to stabilize the entire ecosystem. More specifically, closed systems (that lack moving materials) become more stable with increasing flux rates, as predicted by earlier theories (DeAngelis, 1980;Loreau, 1994), whereas open systems (that possess moving materials) become less stable. Given that in nature a closed ecosystem is unrealistic, the presence of detritus is suggested to be a stabilizer of ecosystems, although efficient decomposition (high flux rate) can hinder this inherent stability. However, these predictions were based on a model that did not include the brown food web with decomposer dynamics, thus leaving open the question of how the presence of decomposers can affect ecosystem dynamics.
Here, an ecosystem model was developed to investigate the impact of microbial decomposers on ecosystem stability, which is defined as resilience via equilibrium recovery from a small perturbation (DeAngelis, 1980;McCann, 2011;Pimm & Lawton, 1977); however, future works will need to use multiple stability indices (Kéfi et al., 2019;Tilman, Reich, & Knops, 2006;Wang et al., 2017). The ecosystem comprised a green-world pathway with nutrients, producers, and consumers and a brown-world pathway with detritus, decomposers, and predators. Contrary to earlier theories, the results showed that ecosystem stability tends to be higher when the system has an intermediate degree of openness. However, in such stable systems with intermediate openness, high fluxes owing to efficient decomposition by decomposers can either greatly increase or decrease ecosystem stability, depending on the degree of openness.
This suggests that an increased understanding of the properties of current ecosystems, such as decomposition rates and openness, is crucial for predicting changes in ecosystem stability in response to climate change and global warming.

| MODEL
For the model, I considered an ecosystem wherein a classic food chain (green world) and a detritus-decomposer-predator chain (brown world) were coupled (Figure 1), assuming that decomposer population growth was limited either by carbon (C-limited) or a mineral nutrient (N-limited) (Daufresne, Lacroix, Benhaim, & Loreau, 2008;Daufresne & Loreau, 2001). Although food-web structures can vary between ecosystems, here I assumed the simplest system with different functional groups. The robustness of the predictions was confirmed using a more complicated model, which included second consumers in both the green and brown worlds (Results). The ecosystem model was defined by the following ordinary differential equations: where N, P, C g , D, M, and C b are the nutrient pool size and the biomass of the producers, consumers, detritus, microbial decomposers, and predators of microbes, respectively. I is the nutrient input rate; a M is the decomposition rate; e M is the conversion efficiency of detritus into microbe production; r is the nutrient uptake rate of the producers; a Cg is the uptake rate of the producers by the consumers; e Cg is the conversion efficiency of the producers into consumer production; a Cb is the rate of uptake of microbes by predators; e Cb is the efficiency of conversion of microbes into predators; l i (i = N, P, C g , D, M, or C b ) is the rate of nutrient loss from the system (a part of dead organisms can emigrate from the system, and the immigration can be negligible if the focus of the system is on a broad spatial scale because most of the area can be source and residual can be sink); and m i is the mortality rate of a species (i = P, C g , D, M, or C b ).
where δ i is a fraction of the nutrients released from all compartments added to the N pool; this represents direct nutrient cycling via ex- released from all compartments added to the organic material pool as detritus; this represents indirect nutrient cycling by microbes via the mineralization of detritus, such as feces and dead organisms, before becoming available to producers.
The carbon/nutrient limitation of decomposers depends on differences between the C:N demand of decomposers and the C:N supplied by detritus (Bosatta & Berendse, 1984;Daufresne et al., 2008;Sterner & Elser, 2002). When there is a lower abundance of detritus and a lower C:N ratio as compared with decomposers, decomposers are C-limited. In contrast, when there is an abundance of detritus and available carbon, decomposers are N-limited (Daufresne et al., 2008;Daufresne & Loreau, 2001;Zou et al., 2016). Decomposer growth φ m and nutrient uptake by decomposers φ i are then assumed as follows: where α and β represent the C:N ratio of detritus and decomposers, respectively (normally, α > β in systems wherein detritus is the substrate for the decomposer community (Ågren & Bosatta, 1996;Andersen, 1997)). Hereafter, α/β = q (>1). r M is defined as the nutrient uptake rate of the decomposers. The left and right terms in each equation represent C-and N-limitation, respectively. From Equations (2a) and (2b), switching between C-and N-limited systems is determined by the following condition: if e M a M D * (q − 1) < r M N * , C-limited, otherwise, N-limited (Zou et al., 2016).
Efficiency was not considered in terms of the nutrient uptake of the producers or decomposers because their efficiencies were likely to be similar (McCann, 2011;Zou et al., 2016). Also, for simplicity, e i = e, δ i = δ, and l N = l D = l were assumed. By setting the right-hand sides of Equations (1a)-(1f) to zero, a nontrivial equilibrium is obtained (Appendix S1). Since this paper does not focus on under the feasible condition. Note that P * and M * are always positive constant without depending on a M , and N * , C * g , and C * b can converge to positive constant (Appendix S1). Importantly, D * approaches to zero with increasing a M . These factors indicate that the property of the system does not change at a sufficient large value of a M . For that reason, the upper limit of this parameter was determined.
Using numerical analysis, local stability was determined by examining an actual portion of the dominant eigenvalue. There were no unstable regions in this system (limit cycles could not occur); therefore, the coexistence equilibrium was assumed to be globally stable at all times. Hence, in the coexistence equilibrium, resilience (an index of ecosystem stability), defined as the capacity of a system to return to a stable equilibrium after encountering an acute disturbance, was calculated as the absolute value of the highest real part of the eigenvalues of the Jacobian matrix (Pimm & Lawton, 1977).
To reveal the roles played by decomposers in ecosystem dynamics, I compared the stability of the systems with or without decomposers.
To demonstrate the effect of decomposers alone, I assumed the flux into the nutrient pool of systems with or without decomposers to be equal. This is because productivity largely influencing stability can be different if this is not assumed (McCann, 2011). This comparison was done in systems that lacked consumers of decomposers, because it is not possible if it exists. In addition, I focused particularly on the effects of decomposition rate, which is a key parameter involved in the control of nutrient recycling within a system. By varying the degree of decomposition by microbes (a M ) and the rates of nutrient and detritus loss (l), the effects of decomposition rate and the openness of the system on ecosystem stability were investigated, to reveal the roles played by decomposers in ecosystem dynamics.

| RE SULTS
The system had coexistence equilibrium in cases of both C-and N-limitation (Appendix S1). First, consider systems without consumers of decomposers to reveal the role of decomposers in system stability (C b = 0). I analyzed the effects of decomposers on stability by comparing the systems with or without decomposers. The analysis showed that in a C-limited system (Figure 2a-c), decomposers could stabilize the system over a broad range of decomposition rates (a M ) and openness (l). In an N-limited system (Figure 2d-f), however, stabilization due to decomposers was not likely to occur in nonopen systems. In both systems, decomposers can play a role in stabilizing the system, particularly if the system is open and the decomposition rate is not low (Figure 2). This tendency is almost qualitatively held even when consumption rate of producers changes ( Figures S1 and   S2). In addition, the presence of consumers of decomposers can play a stabilizing role in an ecosystem, particularly if the system is open and the consumption rate is high (Figures S3 and S4). critically depended on the degree of openness. When openness was low or the system was closed, an N-limited system exhibited higher stability than a C-limited system, particularly where there was a high exploitation rate of nutrients by decomposers (r M ) and high decomposition rates (Figure 4a). Conversely, when there was a high degree of openness, this tendency was reversed: A C-limited system exhibited higher stability than an N-limited system, particularly when there was a high exploitation rate of nutrients by decomposers (r M ) and high decomposition rates (Figure 4c-e).

| D ISCUSS I ON
The present theory predicts that (a) the presence of decomposers can play a stabilizing role in a system, particularly when the decomposition rate and degree of openness of the system are not low; (b) increased decomposition tends to stabilize the ecosystem; Regardless of the existence of microbial decomposers, detritus decreases as the decomposition rate increases, finally becoming depleted (a M ∞lim � � � � � � � � � → D * = 0 is analytically shown in the Appendix S1). Thus, without microbes, the recovery of a system following a perturbation would be delayed, particularly when decomposition occurs relatively rapidly, because detritus does not play a role in circulating nutrients within a system. When microbes are present, however, even if detritus is less abundant, there may be a high abundance of efficient microbial decomposers (a M ∞lim � � � � � � � � � → M * > 0 is analytically shown in Appendix S1), and the nutrients become tied up in biotic biomass.
Hence, even if a perturbation occurs, microbes can play a key role in maintaining the smooth circulation of nutrients within the system. Therefore, the rate of microbial decomposition is a key parameter in stabilizing a system. can stabilize an ecosystem (e.g., effect of temperature changes on decomposition efficiency (Davidson & Janssens, 2006)). Openness plays a key role in ecosystem maintenance. Earlier theories predicted that closed systems are less stable than more open systems (DeAngelis, 1980;Loreau, 1994;McCann, 2011 To the best of my knowledge, the most similar modeling study to the present one was carried out by Zou et al. (2016). Although a careful discussion is necessary because the system modeled was not the same as that in the present study, the following comparison may be possible. Zou and colleagues analyzed the links between the green and brown worlds. More specifically, they showed how some parameters in one world can influence the productivity in the other. In their model, productivity in the green world was a monotonical function of the rate of consumption of decomposers by predators. This suggested that the regulation of decomposition is closely related to plant productivity. A key point was that the productivity level is middle at an intermediate level of regulation of decomposers. In addition, high or low productivity in one world could lead to low or high productivity in the other world, respectively. This is because, for example, a high abundance of detritus implies stagnation of the brown world, resulting in a low level of nutrients, and vice versa. Hence, as well as the discussion on stability, mid-level productivity could increase stability. This link between stability and ecosystem functioning will be the subject of future studies.
The present study highlights the potential importance of the decomposition rate and the degree of openness in ecosystem maintenance. However, empirical data linking ecosystem stability with decomposition rates are currently lacking. Consequently, additional empirical studies of the dynamics of ecosystems, along with both green and brown food webs, are warranted. A possible test of the present hypothesis would be to compare the stability of ecosystems that possess different degrees of openness (e.g., stream vs. lake (Essington & Carpenter, 2000)), under varying levels of decomposition efficiency (e.g., temperature (Davidson & Janssens, 2006)). In addition, further theoretical studies are necessary to confirm the robustness of this prediction. For instance, whether this prediction is applicable to nonequilibrium systems or meta-ecosystems (Massol et al., 2011) remains an open and important question.

ACK N OWLED G M ENTS
This study was supported by a Grant-in-Aid for Scientific Research (B) (#16K18621) from the Japan Society for the Promotion of Science. A.M. thanks the faculty of Life and Environmental Science, Shimane University, for providing financial aid for publishing this article.

CO N FLI C T O F I NTE R E S T
None declared.

DATA AVA I L A B I L I T Y S TAT E M E N T
This paper includes no data because it describes a mathematical model.