Relationships among generalized positive feedback loops determine possible community outcomes in plant-pollinator interaction networks

Attractors in Boolean network models representing complex systems such as ecological communities correspond to long-term outcomes (e.g., stable communities) in such systems. As a result, identifying efficient methods to find and characterize these attractors allows for a better understanding of the diversity of possible outcomes. Here we analyze networks that model mutualistic communities of plant and pollinator species governed by Boolean threshold functions. We propose a novel attractor identification method based on generalized positive feedback loops and their functional relationships in such networks. We show that these relationships determine the mechanisms by which groups of stable positive feedback loops collectively trap the system in specific regions of the state space and lead to attractors. Put into the ecological context, we show how survival units—small groups of species in which species can maintain a specific survival state—and their relationships determine the final community outcomes in plant-pollinator networks. We find a remarkable diversity of community outcomes: up to an average of 43 attractors possible for networks with 100 species. This diversity is due to the multiplicity of survival units (up to 34) and stable subcommunities (up to 14). The timing of species influx or outflux does not affect the number of attractors, but it may influence their basins of attraction.

Figure 1

Illustration of the expanded network that incorporates the update functions, and identification of stable motifs. Panel (a) shows a bipartite network, in which a regular edge represents a beneficial (positive) effect on the target node, while the an edge with a diamond arrowhead represents a harmful (negative) effect. Panel (b) shows the update function of node B in a truth table format. The update function expresses that the simultaneous activity of node C and inactivity of node D results in the activation of node B. Nodes A, C, and D have only one positive regulator and their future state is solely determined by the current state of their positive regulator. Panels (c) and (d) show the same update functions using Boolean operators and threshold functions, respectively. Panel (e) shows the expanded network of the network shown in panel (a). For each node in panel (a), there are two virtual nodes in panel (e) that represent the two possible states of that node: the active state is labeled with the node name, and the inactive state is labeled with the node name prefixed by $\sim$. The composite node that corresponds to the “AND” operator in the functions of node B is shown with a black circle. There are two stable motifs in this network, highlighted with gray (A=0 and C=0) and gray gradient (A=1 and C=1). Locking-in of the gray stable motif yields the system's convergence to a point attractor in which all nodes are inactive. Locking-in of the gray gradient stable motif yields an attractor in which A and C are active, B and D oscillate.

Figure 2

A small plant-pollinator network depicting the mechanisms of edge loss when converting the threshold functions of the Campbell et al. model to disjunctive prime form. Panel (a) shows the bipartite network; plants are denoted with “pl” and pollinators are denoted with “po.” Edges with a regular arrowhead show positive influence, while edges with diamond arrowheads show negative influence on the target species. Panel (b) shows the truth tables for pl_1 and pl_2. The “X” in table entries means that the state of that particular species can be either 0 or 1. In this model, pl_1 becomes extinct because neither po_1 nor po_2 are able to pollinate it; this leads to the extinction of po_1. The future state of pl_2 depends only on the current state of po_2, because the weight of this positive influence is larger than the cumulative weight of the negative influences from po_3, po_4, and po_5. These observations allow us to simplify the network, as shown in panel (c), reducing computational complexity. The species that were not able to persist are highlighted with gray.

Figure 3

Average percentage of the species in the original species pool that could not establish in the community for each network size. Examples of such species are depicted in Fig. 2.

Figure 4

Finding the CSMs from the consistent cycles. Panel (a) shows an interaction network that has 4 plants and 4 pollinators after simplification. Panel (b) indicates the corresponding expanded network. There are four SMs, each identifiable as a cycle between a virtual node associated to a pollinator and a virtual node associated to a plant. In addition to the SMs, there are several consistent cycles in this expanded network, three of which are highlighted with blue, pink and green lines alongside the edges. Panel (c) shows the cycle graph that corresponds to the expanded network in panel (b). The nodes of this graph (rectangular symbols) represent consistent cycles in the expanded network, and are highlighted with the same colors. The label of each node consists of two parts separated by a vertical line: the first part is the node states in that consistent cycle, and the second part is the conditions of that consistent cycle. Consistent cycles without any conditions are SMs. A directed edge in the cycle graph shows that one or more virtual node of the source cycle satisfies one or more condition of the target cycle. The hollow edge from the light green SM to the virtual node $\sim \text{po\_2}$ (which is only added for illustration purposes) represents that $\sim \text{po\_2}$ is in the LDOI of the light green SM. The blue, pink and green cycles partially satisfy each others' conditions and make up an SCC in the cycle graph. They also form a consistent SCC in the expanded network in panel (b); as a result, they can be merged to form the yellow CSM, whose condition is $\sim \text{pl\_1}$. The green cycle has the condition $\sim \text{po\_2}$, which has two ways of being satisfied: one through the yellow CSM, and one through the light green SM. This is because po_2 can become extinct due to the extinction of pl_2, or as a consequence of the inhibitory edge from pl_4. As a result, this cycle not only participates in the yellow CSM but also qualifies as a separate supported CSM.

Figure 5

Finding the supports of a CSM. Panel (a) shows an interaction network consisting of six plants and six pollinators after simplification. Panel (b) shows the expanded network and panel (c) shows the cycle graph. The orange cycle in panels (b) and (c) is a CSM with the condition $\sim \text{po\_2}$. This condition can be satisfied using two different paths: The first one is through the LDOI of the blue SM from po_6 to pl_6, po_4, pl_4, po_5, pl_3, and $\sim \text{po\_2}$. The second is through the yellow CSM, whose condition $\sim \text{po\_4}$ is satisfied by the LDOI of the pink SM that consists of $\sim \text{pl\_5}$ and $\sim \text{po\_6}$. Both the blue SM and the union of the pink SM and yellow CSM are valid supports of the orange cycle. They are mutually exclusive in the state of pl_5 and po_6. We note that the SM made up by pl_2 and po_3 is not a support of the orange cycle: It satisfies its condition but it contradicts its node states. For a similar reason the cycle in the bottom left of panel (c) has no support.

Figure 6

Scatter plot of the number of attractors against the number of (conditionally) stable motifs. The 36 batches of networks were categorized by size into three groups: small networks (10–35 nodes), medium networks (40–65 nodes), and large networks (70–100 nodes). The area of the data points represents the relative occurrence of each data point.

Figure 7

The functional relationships among SMs and motif groups and construction of the allowed successions to find the number of attractors. Panel (a) shows an example subnetwork consisting of 2 plants and 2 pollinators. Panel (b) shows the cycle graph for the interaction network in (a). Three consistent cycles M1, M2, and M3 are SMs, while the last consistent cycle M4 is a CSM with the condition of $\sim \text{pl\_1}$, which is satisfied by M3. Thus, motifs M3 and M4 are merged into the motif group M3M4. Panel (c) illustrates the network of functional relationships among the two SMs M1 and M2 and the motif group M3M4. M2 is in the LDOI of M1, and by definition M3 is in the LDOI of the motif group M3M4. These relationships are shown by hollow arrows from M1 to M2 and from M3M4 to M3. M3M4 and M1, M3 and M1, as well as M3M4 and M2 are mutually exclusive (they have shared nodes but in different states), which is represented as diamond arrowheads between these nodes. Following the construction of this network, the LDOI of each SM and maximal motif group is calculated to determine if any of them lead to a point attractor. As a result of stabilization of M1 or M3M4, all the nodes in the interaction network stabilize, and since they are mutually exclusive, each leads to a separate attractor. The first and second figures from left in panel (d) represent the attractors resulting from stabilization of M1 and M3M4 respectively. The nodes that are in gray (white) represent the extinction (persistence) of the corresponding species in that specific attractor. In one of these attractors all species persist and in the other they all become extinct. There also exists a combination of motifs, {M2, M3}, which stabilizes all nodes and excludes both previously found attractors, and hence it leads to a new attractor. The last figure in panel (d) shows this attractor in which half the species persist and the other half become extinct.

Figure 8

Box plot of the 10 based logarithm of the number of attractors (A) as a function of network size (N). The inset on the top left represents the average of the 10 based logarithm of the number of attractors as a function of network size.

Figure 9

Average percentage of species present in the attractors (community outcomes) as a function of network size. The percentages are shown relative to the size of the original species pool (cross marker) and relative to the number species in the species pool that had a chance to establish (triangle marker). The inset represents the average percentage of species that oscillate in the attractors as a function of network size. These percentages too were calculated relative to the size of the original species pool (star marker) and relative to the number of species in the species pool that had a chance to establish (circular marker). As the network size increases, on average more species manage to survive or oscillate in the attractors, but the rate of increase slows (saturates).

Figure 10

Scatter plot of the number of possible community outcomes (attractors) versus the number of independent subcommunities (active SMs that are not mutually in the LDOI of each other) in each network. Each symbol represents a distinct pair of ($x,y$) values. The solid line shows the function $y=2^x$, which corresponds to the number of community outcomes in case of subcommunities that do not affect each other's survival states in any way. The actual number of community outcomes is less than or equal to this expectation.

Figure 11

The number of negative regulators of each arbitrary species that are kept after the simplification according to Eq. (B1). The figure show examples of the number of negative regulators kept in the simplified model versus the total number of negative regulators in the threshold model, for four values of the number of positive regulators. According to the in-degree distributions in these networks, the maximum number of regulators a species can have is 25. As the number of positive regulators increases, the number of negative regulators they can overcome increases as well. Seven positive regulators can overcome 18 negative regulators, thus in this case there is no need to ensure the inactive state of any of the negative regulators.

Figure 12

An example of an interaction network without negative edges. Panel (a) shows the interaction network consisting of 4 plants and 5 pollinators. The sole case of a source SCC is the positive feedback loop made up of pl_3 and po_3. As a result, only this SCC leads to a SM representing the inactive states of species (to an inactive SM). The rest of the SCCs lead to SMs representing active states of species or CSMs representing the inactive states of species. Panel (b) shows the disjoint expanded network. The 4 inactive SMs and CSMs are highlighted with color: The SM consisting of $\sim \text{pl\_3}$ and $\sim \text{po\_3}$ is highlighted with gray, the CSM consisting of $\sim \text{pl\_2}$ and $\sim \text{po\_2}$ is highlighted with green, the CSM consisting of $\sim \text{pl\_1}$ and $\sim \text{po\_1}$ is highlighted with yellow, and the CSM consisting of $\sim \text{pl\_4}$ and $\sim \text{po\_5}$ is highlighted with pink. Panel (c) shows the cycle graph with the cycles highlighted with the same colors. The green and yellow CSMs each have a single condition. The condition of the green CSM is a virtual node within the gray SM, and the condition of the yellow CSM is a virtual node within the green CSM. As a result, the sole support of the green CSM is the gray SM, and the sole combinatorial support of the yellow CSM is the union of the gray SM and green CSM. The pink CSM has multiple conditions that are successfully satisfied by the union of the gray SM, green CSM, and yellow CSM. This union is also the only support of the pink CSM.

Figure 13

Mutual inhibition and activation between two positive feedback loops leads to unsupported CSMs in plant-pollinator networks. Panel (a) shows the subnetwork after simplification. Note that a negative edge always comes paired with a positive edge in the opposite direction, and the species that have a negative regulator must also have a positive regulator. As a consequence, the two positive feedback loops mutually activate and inhibit each other at the same time. The expanded network in (b) shows that there is an inactive SM (highlighted with gray) and two active CSMs. These two active CSMs are not supported, since the gray SM, which can satisfy their conditions, is not consistent with them.

Figure 14

The simplest subnetworks that can yield supported active or mixed-state CSMs. Panel (a) shows a subnetwork that has 9 plants and 8 pollinators. Nodes pl_2 and po_1 each have 1 positive and 7 negative incoming edges. For each of these nodes the starting point of 6 incoming negative edges can be arbitrary nodes outside of the nodes with labels. Only one of the negative incoming edges is kept during network simplification. A single case among 49 possible choices, namely, keeping the negative edges from po_2 to pl_2 and from pl_2 to po_1 results in a network consisting of pl_1, pl_2, pl_3, po_1, and po_2 that is capable of forming a supported active CSM. Panel (b) shows the corresponding expanded network. The yellow SM consisting of po_2 and pl_3 is the support of the blue active CSM. Panel (c) shows a subnetwork that has the same number of plants and pollinators. Keeping the negative edges from po_1 to pl_1 and from pl_2 to po_1 results in a simplified network that has a mixed-state CSM, highlighted with green in the expanded network of panel (d). The purple SM consisting of pl_3 and po_3 is the support of this mixed state CSM. This case has the same probability of 1 in 49 choices.