Topology of plant-pollinator networks that are vulnerable to collapse from species extinction

The ability to predict the collapse of ecological communities is of significant concern in light of global patterns of rapid species extinctions. Here, we use a recently developed dynamic Boolean network-based model of mutualistic plant-pollinator community formation to investigate the stability of simulated ecological communities in the face of sequential species extinctions. We assess communities in terms of the relative change in biodiversity after species loss, and find that communities that experience a significant loss of biodiversity differ from more robust communities according to a number of topological characteristics. Notably, we show that high nestedness, a property commonly believed to promote community stability, may in extreme circumstances promote a critical over-reliance on individual species. Furthermore, the species important to the survival of the rest of the ecosystem occupy different positions in the network than less important species. Our results suggest that network measures may be applied to real ecosystems to yield insight into both their stability and the identity of potentially critical species.

Figure 1

(a) An example community where plants (pollinators) are represented with diamonds (circles). Incoming pointed (flat) tips represent beneficial (detrimental) interactions. (b) A portion of the state transition network that corresponds to the interaction network shown in (a). Every state is a unique combination of present (1) and absent (0) species, listed in the order pollinator 1,2,3; plant 1,2,3,4. The state is also represented with a diagram of the interaction network, where “present” species are shaded. The state transition network terminates at a steady state wherein every species is present in the community. (c) If plant 3 is forcibly removed from the steady state shown in (b), the community eventually progresses to the “all dead” steady state wherein no species are able to persist, i.e., the community collapses completely.

Figure 2

(a) The distribution of the relative community size after restabilization following the forced, permanent extinction of a single species. The relative community size is obtained by dividing the size of the community after restabilization by the size of the community immediately after the forced extinction. The distribution is approximately normal, with a peak corresponding to a subsequent extinction of 14$\%$ of the remaining species. In some cases the extinction is shown to increase biodiversity (relative size is in excess of 1). Circles correspond to simulation results and the line is a best-fit Gaussian. (b) The distribution of the number of extinctions required to eliminate every species in a stable community with at least ten species originally present. The distribution is skew normal, with a maximal value slightly less than half of the total number of species in the network's regional species pool (see Sec. 2). (c) The median number of extinctions required for complete collapse of a robust community, as a function of the community's nestedness. The positive correlation indicates robustness promotes stability over the range of nestedness shown. (d) The distribution of the number of extinctions required to cause complete community collapse, shown as a proportion of the maximum number of species present in the community. Circles correspond to simulation results and the line is a best-fit Gaussian.

Figure 3

The fraction of species present in a strongly connected component (subcommunity), for communities categorized by the fraction of remaining species that are lost in the equilibration process subsequent to the permanent loss of a single species (see Sec. 2). Circles indicate average values, with circle radius corresponding to the number of communities represented. Error bars correspond to the standard error of the mean. All data points are left binned. (Inset) The distribution for the number of strongly connected components follows a similar trajectory, with near a single strongly connected community observed in instances where a single extinction drives the entire community extinct.

Figure 4

Network [(a),(b)] and node-level [(c),(d)] properties of mutualistic ecological communities when subjected to a single-species elimination, as related to the fraction of species that are lost in the subsequent equilibration process. Black circles indicate average values, with circle radius corresponding to the number of species or communities represented. Error bars correspond to the standard error of the mean. All data points are left binned. (a) Communities that experience minimal loss in biodiversity have the lowest values of nestedness, with a slight increase for communities that experience an increase in biodiversity and a significant increase for communities that experience significant loss of biodiversity. Gray triangles correspond to the nestedness $Z$ scores relative to configuration null models, which preserve the density and degree distribution of the source community. (a) (Inset) A complementary view, with the axes switched and data binned according to nestedness value rather than percent extinct. The conventional view of increasing nestedness leading to more robust networks is shown to hold for low values of nestedness. (b) Higher network density is correlated with significant changes to biodiversity. (c),(d) Species whose extinction causes greater loss in biodiversity tend to have greater betweenness and closeness centralities, c.f. species in robust communities. Values are normalized to the maximum values observed in the source network.

Figure 5

The distribution of node-network influences and susceptibilities for noncritical species in robust communities (blue circles), vulnerable communities (yellow squares), and critical species (black triangles). (a) A broad view at the 0.001$\%$, 99.99$\%$ range for species in robust communities. Adjacent curves indicate that the distributions of noncritical species in robust communities are centered near the origin for both measures. (b) A zoomed view showing the 3$\%$, 97$\%$ range for species in vulnerable communities. The dashed lines indicate the median values for critical species, and the adjacent data and curves show the distributions and best-fit Gaussians to the data in both dimensions. The distribution of critical species is shifted towards positive influences and negative susceptibilities compared to the distribution of other species.