Fig 1.
The quantities used in the analysis.
Top: an illustrative network depicting interactions between plants and pollinators. A focal link is highlighted in black. The generalisation of a link is the average degree of the two interacting species. The frequency of the link is the visitation rate between the two interacting species. Frequency and generalisation are both standardised to between 0 and 1 and then combined to determine the vulnerability of a link, such that vulnerable links are low-frequency interactions between specialists [11]. The feasibility of the network was measured twice: once in its original form, with all links, and once following the removal of a focal link. The ratio of these two values (feasibility with and without the focal link) is the feasibility contribution of the link. This is represented by the graph on the right of the figure. The feasibility of a network is defined as the parameter space of intrinsic growth rates in which all species in a community can have positive abundances [4]. The feasibility domain of the network with the focal link is represented by the dotted outline, and the feasibility domain of the network without the focal link is represented by the pink area. In this case, removing the link has reduced the feasibility of the community (reduced the size of the dotted area). A reduced feasibility means that a perturbation that moves the community from the initial state (blue circle) to a final state (green circle) will result in extinctions because the community moves outside the pink feasibility domain. However, no extinctions would occur under the same perturbation in the original community without the focal link removed, because the final state of the community (green circle) is within the dotted outline. Together, vulnerability and feasibility contribution describe the risk to a community of losing a particular link.
Fig 2.
The relationship between vulnerability (the likelihood of a link being lost) and feasibility contribution (the contribution of a link to a network’s feasibility) for all species–species links across 41 mutualistic networks.
Best-fit line is from a mixed-effects model with feasibility contribution as the response variable, vulnerability as a fixed effect, and network identity as a random effect. Grey band represents the 95% confidence interval. To focus on mutualistic effects, these results are from analyses of feasibility contribution with zero interspecific competition (ρ = 0), following [17]. See S1 Fig for results using weak competition (ρ = 0.01) which were qualitatively similar. Data underlying this figure are given in S1 Data (https://doi.org/10.6084/m9.figshare.12689258.v1).
Fig 3.
The degree of taxonomic consistency for each interaction at genus (n = 469), family (n = 466), and order (n = 151) levels, for vulnerability (likelihood of a link being lost) and feasibility contribution (contribution of a link to a network’s feasibility).
Taxonomic consistency is the tendency for properties of an interaction to be more similar across occurrences than expected by chance. Points represent individual interactions. Boxplots represent 5%, 25%, 50%, 75%, and 95% quantiles of the same data, moving from the bottom whisker to the top whisker. Number in bottom left of each panel is the proportion of interactions that exhibited positive consistency (VarianceObserved < VarianceNull). For visualisation, a small number of points with low values were removed. The percentage of points with values lower than the y-axis minimum are as follows for each panel: (a) 1.5%, (b) 1.1%, (d) 3.2%, (e) 1.5%, and (f) 1.3%. To focus on mutualistic effects, these results are from analyses of feasibility contribution with zero interspecific competition (ρ = 0), following [17]. See S3 Fig for results using weak competition (ρ = 0.01) which were qualitatively identical. Data underlying this figure are given in S9 Data (https://doi.org/10.6084/m9.figshare.12689258.v1).
Fig 4.
Schematic illustrating the steps used to assess the taxonomic consistency of vulnerability and feasibility contribution.
(a) Three networks made up of links, represented by shapes. The number on each shape is the value of a link property, such as vulnerability or feasibility contribution. Different shapes (each with different colours for visualisation) represent different interactions. (b) Using the red square interaction as an example, we calculate the variance (σ2) in its link property (e.g., vulnerability). (c) 1,000 ‘null sets’ comprising the same number of links as the observed interaction were created, by sampling links randomly without replacement. Variance in each null set was calculated. (d) Paired differences in variance were calculated by subtracting the observed variance from each of the null variances. The mean of these differences was then calculated and plotted (e).