A network model for plant-pollinator community assembly

Abstract

Community assembly models, usually constructed for food webs, are an important component of our understanding of how ecological communities are formed. However, models for mutualistic community assembly are still needed, especially because these communities are experiencing significant anthropogenic disturbances that affect their biodiversity. Here, we present a unique network model that simulates the colonization and extinction process of mutualistic community assembly. We generate regional source pools of species interaction networks on the basis of statistical properties reported in the literature. We develop a dynamic synchronous Boolean framework to simulate, with few free parameters, the dynamics of new mutualistic community formation from the regional source pool. This approach allows us to deterministically map out every possible trajectory of community formation. This level of detail is rarely observed in other analytic approaches and allows for thorough analysis of the dynamical properties of community formation. As for food web assembly, we find that the number of stable communities is quite low, and the composition of the source pool influences the abundance and nature of community outcomes. However, in contrast to food web assembly, stable mutualistic communities form rapidly. Small communities with minor fluctuations in species presence/absence (self-similar limit cycles) are the most common community outcome. The unique application of this Boolean network approach to the study of mutualistic community assembly offers a great opportunity to improve our understanding of these critical communities.

Fig. 1.

A simple example plant–pollinator interaction network consisting of three plant species and two pollinator species. A pointed (flat) tip indicates a positive (negative) interaction. Pollinator 2 is a generalist in that it successfully pollinates all three plants, although it cannot feed from plant 1, as its proboscis length (λ) is significantly shorter than the nectar depth (λ) of plant 1.

Fig. 2.

The state transition network that corresponds to the interaction network shown in Fig. 1. Dynamically, the presence or absence of a species at time is determined by the species that target it and are present at time t. Here, the effect of a negative edge is equal to that of a positive edge, and a species must have more active positive incoming edges than negative at time t to be present at time t + 1 (Materials and Methods). The 32 possible community states are identified with a binary number (present = 1, absent = 0) where the values correspond to (from left to right) plant 1, 2, 3 and pollinator 1, 2. Edges indicate the succession of the system's states in the absence of outside influence. The system admits two steady states, 00000 (presence of no species) and 11101 (absence only of pollinator 1), and one cycle of length 2, 11100 ↔ 00001 (the system oscillates between all plants and no pollinators and only pollinator 2). Note that due to the artificial nature of this example interaction network, this limit cycle is highly unrealistic, and such limit cycles are extremely uncommon in larger, more realistic interaction networks.

Fig. 3.

The average number of attractors for ensembles of 1,000 randomly generated networks with equal numbers of plants and pollinators. A–D correspond to positive edge weights of 1–4, respectively. Most limit cycles consist of two, three, or four states. Their abundance increases with network size, whereas the number of steady states is comparatively constant.

Fig. 4.

A randomly chosen starting state will lead to a steady state more often than a limit cycle for small networks, but the converse is true for larger networks. This behavior corresponds to the relative abundance of steady states (limit cycles) for small (large) networks (Fig. 3).

Fig. 5.

The average number of time steps required to reach an attractor grows with network size, albeit at a much slower rate than the total size of the state space [the average path length lies in the range (2, 8), whereas the number of states ranges from 2¹⁰ to 2¹⁰⁰]. If the positive edge weight is larger, the greater abundance of attractors (Fig. 3) results in a wider but shallower distribution of transitional states, which leads to a smaller average path length.

Fig. 6.

The average normalized agreement in the present/absent state of each species for the states in a limit cycle. The agreement is determined by computing the average of all pairwise combinations of the states in a limit cycle. We report the average over all observed limit cycles. Limit cycles in small networks tend to have greater fluctuation (although the agreement is still >60%), whereas the agreement in larger networks stabilizes above 85%.

Fig. 7.

The fraction of species present in steady states (Upper) and limit cycles (Lower). Greater values for larger networks indicate that large source pools are capable of sustaining disproportionately larger communities. We attribute the increased percentage of present nodes in small network limit cycles to minimum complexity requirements for limit cycles.

Fig. 8.

The ratio of present plants to present pollinators in steady states for positive edge weight = 1. The ratio scales with the source pool composition.