Identifying important interaction modifications in ecological systems

Trophic interaction modifications, where a consumer-resource link is affected by additional species, are widespread and significant causes of indirect effects in ecological networks. The sheer number of potential interaction modifications in ecological systems poses a considerable challenge, making prioritisation for empirical study essential. Here, we introduce measures to quantify the topological relationship of individual interaction modifications relative to the underlying network. We use these, together with measures for the strength of trophic interaction modifications to identify modifications that are most likely to exert significant effects on the dynamics of whole systems. Using a set of simulated food webs and randomly distributed interaction modifications, we test whether a subset of interaction modifications important for the local stability and direction of species responses to perturbation of complex networks can be identified. We show that trophic interaction modifications affecting interactions with a high biomass flux, those that connect species otherwise distantly linked, and those where high trophic-level species modify to interactions lower in the web have particular importance for dynamics. In contrast, the centrality of modifications in the network provided little information. This work demonstrates that analyses of interaction modifications can be tractable at the network scale and highlights the importance of understanding the relationship between the distributions of trophic and non-trophic effects.


Introduction
each consumer was assigned a term 100 times smaller. For both approaches, these 149 values were chosen to calibrate the self-regulation to the trophic interactions 150 while introducing self-regulation at a level where the local stability of populations 151 is likely but not inevitable.

152
Interaction modifications 153 We model interaction modifications by introducing a relationship between the 154 density of the modifier species, B k and the post-modification attack rate a ij : Since the population grwoth rate of both B i and B k is now dependent on 171 the density of species B k , each TIM causes two non-trophic effects, from the 172 modifier species to each of the pair of trophic interactors. These impacts have 173 also previously be described as 'trait-mediated indirect effects' (Abrams 2008, 174 Okuyama and Bolker 2012) amongst others. We incorporate these effects into 175 the interaction matrix through the addition of two additional terms, found by 176 taking the derivative of the modified functional response terms with respect to 177 the modifier species:   We assessed the ability of metrics to identify influential TIMs by testing how 250 well a system containing only a subset of interaction modifications that had 251 been selected using a particular feature was able to capture the dynamics of the 252 complete system (Figure 2). We assess dynamics in two ways discussed below: 253 1) accuracy in estimating local stability and 2) the capacity to determine the 254 direction of species responses to sustained perturbations, which we will refer to 255 as 'directional determinancy'.
256 Figure 3: Schematic showing approach to test methods for identifying interaction modifications that are influential in the dynamics of model ecological communities. The trophic interaction modifications in the full community model A are quantified and 20% retained using a particular metric to generate a subset model S. The capacity for the subset model to represent the dynamics of the full model system are then assessed by their capacity to accurately characterise stability and the direction of response to perturbation as discussed in the main text.

257
Local asymptotic stability is an extensively studied dynamic system property set each A ii to 0. In this case, R λ A 1 is always positive but can be interpreted as 267 a degree of the self-regulation that would be necessary to stabilise the community.

268
For brevity we will refer to this quantity as 'stability'. 269 We generated  2). We then calculated the magnitude of the difference in stability between the 278 full and subset model as: The statistical significance of a metric's ability to identify a subset of TIMs 280 that estimate stability differently to a random subset of TIMs was tested with a

281
Wilcoxon signed rank test, paired by the underlying community.

303
However, since for multiple error terms the expressions become very complex, 304 we directly calculate the effect numerically.

305
Because local asymptotic stability is a prerequisite for calculating directional 306 determinacy, additional communities were generated and unstable communities 307 excluded to create 2000 locally stable communities for each of the three values 308 of g and the two approaches to including self-regulation terms. We then followed 309 the same procedure as for local stability to test each metric's ability to identify a 310 subset of valuable TIMs. For each web we compared the directional determinacy 311 of a model including the selected subset of interaction modifications to a model 312 including just the trophic interactions.

314
It was possible to identify the trophic interaction modifications that had particular 315 influence on the system dynamics. In general, those TIMs affecting directional 316 determinacy also affected the local stability of the system. Results for the case 317 Figure 4: Comparison of the capacity of different metrics to identify TIMs that contribute to the dynamics of whole systems. Arrows indicate whether TIMs were selected for inclusion based on high or low values of the given property. Median error in the estimation of the logarithm of the system stability is plotted on the x-axis, median improvement in the percentage of direction of inter-specific net effects correctly estimated on the y-axes. Grey dotted lines show a baseline when TIMs are chosen at random. Those metrics in the top-right quadrant are therefore better than average at identifying TIMs valuable for both types of dynamics, while those in the lower-left quadrant are worse for both. Points marked with symbols were not significantly different to a random draw for the stability (+), the directional determinacy (x) or both (*). Wilcoxon test (n=2000, α = 0.05), paired by underlying network.
of g=0.1 and stronger self-regulation terms on producers are shown in Figure   318 3. Results for other values of body mass-biomass density scaling and different 319 self-regulation terms (SI 3) were broadly similar except in specific cases discussed 320 below. The variability in changes to directional determinacy was related to the 321 median improvement -where a metric tended to give a larger improvement,

322
there was also a greater chance that including those TIMs would result in a 323 reduced model that was considerably worse than not including any TIMs at all 324 (SI 3).

325
In general, those metrics that addressed aspects of the strength of the identifying TIMs, with many not being significantly different to a random draw.

330
Modifications caused by species with a lower biomass density were more valuable 331 than those with high density. sults suggest that there are no easy shortcuts, but that the problem is not 467 intractable. One avenue that will need particular work is the degree of accu- values for 2D and 3D foraging behaviour. We also introduce a generality penalty 678 term term ω j , which is a simple resource preference term equal to 1/n, the 679 number of resource species of consumer j. We don't include a unit correction 680 term described in the original paper since this would be a constant term in our 681 interaction matrices.    To look at the importance of accurately understanding the TIM slope parameters 714 we conducted a simple randomisation study. Using a set of 500 stable communities 715 generated as described in the main text, using g = 0.1 and the parameterisation 716 with stronger producer competition.

717
For each community we conducted two randomisation experiments using the

774
In studies of interaction matrices there is a long history (e.g. May 1973) of 775 setting these values to constant negative numbers to represent 'self-regulation'.

776
In biological terms this is often considered to represent increased mortality at 777 higher densities. It is worth noting however that the overall 'self-regulation' term 778 is a complex mixture of direct self-regulation terms and terms arising from the