Spatial compartmentation and food web stability

An important goal in ecology has been to reveal what enables diverse species to be maintained in natural ecosystems. A particular interaction network structure, compartments, divided subsystems with minimal linkage to other subsystems, has been emphasized as a key stabilizer of community dynamics. This concept inherently includes spatiality because communities are physically separated. Nevertheless, few theoretical studies have explicitly focused on such spatial compartmentation. Here using a meta-community model of a food web, I show that compartments have less effect on community stability than previously thought. Instead, less compartmentation of a food web can greatly increase stability, particularly when subsystems are moderately coupled by species migration. Furthermore, compartmentation has a strong destabilization effect in larger systems. The results of the present study suggest that spatial limitation of species interactions rather than of community interactions plays a key role in ecosystem maintenance.

Scientific REPORTS | (2018) 8:16237 | DOI: 10.1038/s41598-018-34716-w that controls spatial coupling strength M (Methods). When M = 0, sub-food webs are isolated, whereas when M is extremely large, sub-food webs are strongly coupled, behaving as a single food web. In nature, distinct sub-food webs are linked by species interactions between local populations within a local habitat. This linkage might occur in two ways, depending on the degree of penetration of migratory species into external subsystems (penetration degree). First, it may be limited to a local (boundary) habitat (boundary-separated subsystems). In this case, sub-food webs of the main habitats are locally and partially merged by interactions between species migrating between each sub-food web (Fig. 1b,c). Second, the sub-food webs may be linked across whole habitats (globally-connected subsystems). In this case, some species in a sub-food web may pass the boundary habitat and appear in another sub-food web (Fig. 1e,f).
Here, I define "compartmentation" as the degree of spatial segregation between sub-food webs. More specifically, the degree of compartmentation is controlled by three elements: the proportion of migratory species (p), spatial coupling strength (M), and penetration degree (boundary-separated or globally-connected subsystem). When p, M, and/or penetration degree are low, compartmentation is expected to be strong (Fig. 1). In the present study, p, M, and penetration degree are controlled to examine the effects of spatial compartmentation on stability. An index of stability "community stability, " which is estimated based on local stability (the tendency for community composition to return to its original equilibrium after a small perturbation), was used.
I show that, in the sense of spatial compartmentation, compartmentation has less effect on community stability. On the contrary, less compartmentation has more stabilizing effect particularly in smaller systems, although a moderate level of spatial coupling strength is required in larger systems. This supports little evidences of compartmented subcommunities within small systems 18 . Furthermore, large systems with major habitat divisions may be maintained not by limited species interactions between subcommunities, but by a moderate species migration. The present study suggests that spatial limitation of species interactions rather than of community interactions plays a key role in community maintenance.

Results
Consider a perfect compartmented food web comprised of N species (distinct sub-food webs are comprised of N/2 species), any pair of which are connected to each other with probability C (connectance), defined as the proportion of realized interaction links of the possible maximum interaction links of a given network model, in which no species move between habitats (p = 0; Fig. 1a). In this extreme case, it is trivial that the strength of spatial coupling M does not affect community stability (Fig. 2a,b). However, if sub-food webs are coupled by migratory species, the spatial coupling strength M dramatically alters the effects of p on community stability. When habitats are weakly coupled (smaller M), less connected food webs (smaller p) have higher stability. In contrast, when habitats are tightly coupled (larger M), more connected food webs have higher stability (Fig. 2a,b). Further, more spatially compartmented food webs (smaller p and M) have lower stability than less spatially compartmented food webs (larger p and M), regardless of penetration degree. These results were qualitatively unchanged by varying the stability index (Fig. S1), type of network (Fig. S2), or connectance (Fig. S3). However, system size or species richness can greatly affect the results in four ways. First, if the system becomes extremely large and M is large or small, stability can be extremely low or zero regardless of the degree of p (Fig. 2c,d). Such a system can, however, be stable within a moderate range of M. Note that destabilization is not likely to occur when M is large in smaller systems (Figs 2a,b, and S4). Second, when habitats are moderately coupled, the degree of compartmentation can dramatically affect stability. Less compartmented food webs tend to show high stability (Fig. 2c,d). Third, both more (smaller p and M) and less (larger p and M) spatially compartmented food webs have extremely low stability. Finally, the difference in stability between boundary-separated and globally-connected subsystems becomes greater the larger the system. Globally-connected subsystems tend to be more stable than boundary-separated ones (Fig. 2c,d).

Discussion
The results of the present study suggest that in smaller systems, less spatially compartmented food webs in terms of all elements of compartmentation are likely to be stable compared with more compartmented ones. By contrast, in larger systems, the degree of compartmentation becomes more central to stability. Stabilization arises only under a moderate coupling strength; more spatially compartmented food webs will not stabilize, and stability tends to be higher in less compartmented food webs. Taken together, the results indicate that spatial compartmentation has a lesser and more negative effect on community stability in larger systems than in smaller ones.
Less spatially compartmented food webs are more likely to be stable, suggesting that compartments play a less significant role in maintaining food webs than previously thought 15,17 . This is supported by three results: First, in more complex food webs with higher species richness, as in natural ecosystems, compartmentation is less stabilizing because of strong inherent instability. Second, globally-connected sub-food webs can be more stable than boundary-separated ones. Third, sub-food webs weakly coupled by migration are highly unstable. These results suggest that less stable, more compartmented food webs are maintained only by a few highly mobile species, partially supporting the prediction of a spatially implicit food web model that a mobile higher order organism can stabilize food web dynamics when embedded in a variable and expansive spatial structure 29 . An alternative hypothesis is that, in real compartmented food webs, more species may couple sub-compartmented systems through migration than previously recognized.
The role of compartmentation depends on spatial scale. More specifically, compartmentation will become less crucial for ecosystem maintenance at a larger spatial scale. In compartmented food webs on a huge spatial scale (including ocean and continent systems), such large compartments may not work as a stabilizer, whereas the spatial limitation of species interactions within each subsystem (ocean or continent system) may be a key stabilizer for such a large compartmented system. In other words, large compartmented systems are formed by mere coupling of more stable subsystems by some species. This argument may justify research that independently studies each mechanism by which large spatially distinct sub-ecosystems are maintained. The locality of species interactions and/or weak compartmentation might play a significant role in maintaining food webs. In some systems, many species can move between habitats, building species-rich communities locally (Fig. 1c,e). In contrast, if only a small proportion of species can move, some local communities are species-poor (Fig. 1b,d). May 1 speculated that inherently stable, local, species-poor systems contribute to the stability of the whole system. However, the present study reveals a completely different result: Species-rich local communities contribute to the stability of the whole community. The least stable communities with the greatest species diversity are most affected by immigration 31,34 . This suggests that the effects of stabilizing self-regulation through migration 34 outweigh the stabilization effects of species-poor systems.
In conclusion, the present results suggest that in larger systems, spatial compartmentation may not provide stabilization; instead, the spatial limitation of species interactions plays a significant role in stabilizing systems. Although the question of whether this theory is robust to various other stability indices remains open, the present study provides a new perspective on compartments in ecological networks.

Methods
Consider a meta-food web model (Fig. 1). This model assumes a random (or cascade in Fig. S2) food web in which each pair of species, i and j (i, j = 1, …, N), are connected by a trophic interaction with probability C (connectance), which is defined as the proportion of realized interaction links L of the possible maximum interaction links L max of a given network model (L = CL max ). The maximum link number, L max , is N(N − 1)/2. The spatial food web model is defined by the following ordinary differential equation 34,35 : where H N is the number of patches) is the abundance of species i in habitat l, r il is the intrinsic rate of change of species i in habitat l, s il is the density-dependent self-regulation of species i in habitat l, and a ijl is the interaction coefficient between species i and species j in habitat l. Interaction coefficients are defined as a ijl = e ijl α ijl and a jil = -α ijl , where α ijl is the consumption rate and e ijl (<1) is the conversion efficiency. In the present analysis, it was assumed that habitats are heterogeneous and there are no within-species parameter correlations among them. The migration rate is the product of a scaling parameter, M (spatial coupling strength), and the species-habitat-specific migration rate, m ilk , where k = 1…H N but k ≠ l. For simplicity, it was assumed that m ilk = m ikl . Equilibrium species abundance X il * and parameters s il , e ijl , α ijl , and m ilk were randomly chosen from a uniform distribution, U[0, 1]. r il was calculated such that dX il /dt = 0 for all i and l 34,36 .
I assumed a three-patch model (H N = 3) in which patches 1 and 3 are the original habitats in different sub-food webs and their randomly selected local populations can interact within patch 2 between them. This is the simplest model to examine the effect of spatial compartmentation on community stability. The proportion of migratory species within each community is defined as p i (i = 1 or 3). In the main text, I assumed that p 1 = p 3 = p (see Fig. S5 for an example in which this assumption is relaxed). In boundary-separated subsystems (Fig. 1b,c), migration can occur between the original habitat (patch 1 or 3) and boundary habitat (patch 2). In globally-connected subsystems (Fig. 1e,f), migration can occur across all habitats. Given total species number N, community sizes in each original habitat are controlled by the proportion of species in sub-food web 1, q 1 (i.e., q 3 = 1 -q 1 ). Then, N 1 = Nq 1 (N 3 = Nq 3 ), where N i (i = 1 or 3) is original species number within each original habitat. In the main text, I assumed that q 1 = q 3 = q = 0.5 (see Fig. S6 for an example in which this assumption is relaxed). Individual species were randomly assigned to each subsystem and migratory species were randomly chosen in each simulation.
Following earlier studies, I calculated the stability of the systems using a standard local stability analysis based on a Jacobian community matrix 34,36 . Then, I evaluated the community stability, or the probability of local equilibrium stability, which is estimated as the frequency of locally stable systems across 1000 sample communities. Local stability is calculated based on the eigenvalues of the Jacobian matrix. If the real part of the dominant eigenvalue is negative, the system is locally stable; otherwise, it is unstable. If p = 0, stability is calculated based on a system with two patches (because the middle patch is empty). I also used another stability index, resilience (engineering resilience), or the rate of recovery of the original equilibrium after a small perturbation, which is determined using the mean magnitude of the real part of the dominant eigenvalue of J across 1000 samples of locally stable communities (Fig. S1).