Predicting The Sign of Trophic Effects: Individual-Based Simulation Versus Loop Analysis

Food web research needs to be predictive in order to support decisions system-based conservation. In order to increase predictability and applicability, complexity needs to be reduced to simple and clear results. One question emerging frequently is whether certain perturbations (environmental effects or human impact) have positive or negative effects on natural ecosystems or their particular components. Yet, most of food web studies do not consider the sign of effects. Here, we study 6 versions of the Kelian River (Borneo) food web, representing six study sites along the river. For each network, we study the sign of the effect of a perturbed trophic group i on each other j groups. We compare the outcome of the relatively complicated dynamical simulation model and the relatively simple loop analysis model. We compare these results for the 6 sites and also the 14 trophic groups. Finally, we see if sign-agreement and sign-determinacy depend on certain structural features (node centrality, interaction strength). We found major differences between different modelling scenarios, with herbivore-detritivore sh behaving in the most consistent, while algae and particulate organic matter behaving in the least consistent way.


Introduction
In complex, multi-species ecosystems, a number of interactions connect various organisms. Predicting the community-wide effects of single-species perturbations is a challenge for both ecological research and systems-based management.
In complex ecological systems, the multiplicity of direct and indirect interactions make it uneasy to provide simple and clear predictions on the effect of single-node perturbations. The effects on other organisms and, generally, community response is the outcome of a number of interconnected pathways. Predicting whether the in uence of organism i on organism j will be positive or negative is not easy, even without considering non-trophic effects and complicated functional responses. Also the experimental results on positive inter-speci c interactions are quite recent [1,2,3]. Considering effect sign is critically important, for example, if positive feedback loops [4,5] or mutualisms [6] are to be understood.
Generally speaking, topological models provide fast and easy but not very realistic results from a quite static viewpoint. Sophisticated food web simulations provide more accurate results but need a lot of data and offer results that can be harder to interpret. Qualitative models, like loop analysis, are somewhere in between, trying to combine simplicity (without explicit dynamics) and reality (interaction sign considered).
In this paper, we study food webs at the level of individual nodes (trophic groups), interspeci c interactions and whole networks. We compare (1) the results of loop analysis and dynamical simulations predicting the sign of effects following single-node perturbations.
Beyond comparing these two methods themselves, we also compare (2) the 6 food web models as well as (3) the 14 trophic groups.
Finally, (4) we ivestigate if there is an correlation between various structural properties and sign-agreement.

Data
We used 6 versions of the Kelian River (Borneo) food web for our analysis (data from [17,18]). This makes it possible to assess spatial variability within the ecosystem. The food web is described for 6 different locations along the river [13,19], based on extensive earlier eld work [17,18]. The 6 sites represent a gradient from a pristine rainforest to a human settlement. The pristine food web (site 1) contains 14 trophic groups and 2 additional groups appear only downstream, so the total number of trophic groups is 16 (see Table 1). Most of the groups represent living organisms (e.g. PRED: invertebrate predators) but there are some non-living groups as well (e.g. POM: settled and suspended coarse and ne Particulate Organic Matter). The de ntion of the groups and the description of interactions among them are based on long-term, extensive eld work [17,18]. We quanti ed the topology of the food webs based on the topological importance (TI n ) index that considers direct and indirect effects spreading along pathways of up to n step. This approach quanti es both the centrality of nodes in networks (TI i ) and the strength of effects between node i and node j (TI ij ). It is important to note that the latter can be calculated also for binary networks of un-weighted links: strength is the pure consequence of topology [20,21].
The centrality of species (trophic groups) in food webs seems to be a systemic property with important ecological correlates (e.g. body size, mobility, see: [22]). There is a number of topological indices quantifying centrality in networks and their relationship is increasingly understood [23]. Since we must deal with indirect effects in order to assess the combinations of positive and negative impacts, we use the TI index, explicitly considering the length of direct and indirect pathways in food webs [20]. We note that TI is an useful centrality measure out of the many, according to a recent study wherer machine learning identi ed the most predictive combinations of centrality indices [24]. In this paper, we used TI 3 , i.e. we considered indirect effects up to 3 steps. We performed these calculations in order to see whether there is structural basis of (constraints on) the sign-agreement (for nodes and networks) and signdeterminacy (for interactions and networks).

Loop Analysis
Loop analysis is a qualitative method, that uses signed digraphs to illustrate networks of interacting variables [7,8]. This technique gives the opportunity to represent the structure of linkages of the variables and the patterns of their variations [25,26]. Also, the qualitative models allow to examine the effect of non-biological variables on our system (such as gold mining, shing etc.). This qualitative modeling approach provides a method that is useful where species and their natural history are well-known, but not quanti ed [27].
Signed digraphs are based on the generally accepted interactions between nodes (trophic groups). These interactions come from the previous trophic models in [13]. The gures show two types of connections: arrows ( ) for positive and circle-head links (-o) for negative effects. These links are originated from the coe cients of the community matrix [7]. The diagonal terms of the community matrices are self-effects on system variables, represented in signed digraphs as links connecting variables with themselves. These links are self-dampening (circle-headed) with self-limiting growth rate. For a simple prey-predator system, see Figure 1.
Using the qualitative modelling framework of loop analysis, one can analyze pathways and feedbacks in the system, making predictions about the response of variables to perturbations. These can be the addition (increased biomass) or deletion (decreased biomass) of other nodes. Based on feedbacks and pathways, one can qualitatively specify the direction of changes. If we analyse relatively small systems (with n<5 nodes), this can be executed through the direct analysis of the signed digraph. For larger networks, the graphical feedback analysis is di cult but one can calculate response predictions from mathematical operations on the community matrix.
We followed the method described in [26] to get the predictions for our networks. The loop formula is used for calculating the equilibrium value of the variables following a perturbation, so it can be deduced how does the abundance of a certain variable change [28]: On the left side, x j is the variable with the equilibrium value being calculated and c is the changing parameter (e.g. mortality, fecundity, abundance). On the right side, f is the growth rate, ∂f i /∂c designates whether the growth rate of the i th variable is increasing or decreasing (positive or negative input, respectively), p ji (k) is the pathway connecting the variable to the changed parameter (where the perturbation enters the system), F n-k (comp) is the complementary feedback, which buffers or reverses the effects of the pathway and F n designates the overall feedback of the system, which is a measure of the inertia of the whole system to change [26,28]. See also [8] for the discussion of the correspondence between matrix algebra and loop analysis. Community matrices show the links between the trophic groups. The elements in the rows affect the elements in the columns and the values could be -1, 0 or 1. These represent prey-predator (resouce-consumer) effects. We decided to work with self-dampening variables only for living groups (see Table 3 for the community matrix of site 1).
Direct trophic interactions are represented in such a way that the elements in the rows affect the elements in the columns and the values could be -1, 0 or 1, depending on eld data. We decided to work with self-dampening variables only for living groups.
We were interested in the effect of decreasing the biomass of each trophic group (i.e. the value of each variable), one by one. For this, we needed to simply reverse our predictions [29]. By predicting these changes, qualitative models can also predict the correlation patterns between the examined groups/variables [8, 26, 28].
Levins' loop algorithm was extended in [29] to complex ecological systems. Therefore, the adjoint of the -°A is equivalent to Levins' loop analysis algorithm and its relation with the inverse matrix (A -1 ) is: where the adjoint and inverse matrices are calculated with the negative of the community matrix, thus the positive input is read down the columns and the responses along the rows [29].
An "absolute feedback" matrix was de ned in [29] to calculate the absolute number of complementary feedback cycles in a response whether positive or negative: Where ˙A denotes the adjacency matrix (absolute values of °A).
The "weighted-predictions" matrix can be calculated from the Eq. (1)-(2): where " " is a vectorised matrix operator, what denotes element-by-element division (W ij =1, when T ij =0). The elements of W show the probability of sign determinacy of response predictions in the adjoint matrix. If all of them are of the same sign in one cell, then W ij =1.
If there is an equal number of negative and positive feedback cycles, then W ij =0 [29]. The W ij =0.5 value is a threshold for sign determinacy in models of any size [27].
Our values in the "weighted predictions" matrix were mostly under W ij =0.5. According to this, most of the prediction signs were indeterminate. We used the adjoint values to investigate if sign determinacy depends on the strength of interactions (based on both structure and simulations).

Dynamical simulations
For the dynamical simulation data on the food webs, we used earlier results of an individual-based model [13,19]. This simulation model was built in a process algebra-based framework [30]: abundance was calculated for the trophic groups and interaction strength was converted to probability, according to the kinetics of the stochastic simulation framework used routinely in systems biology [31,32,33,34].
The stochastic IBM simulation model, set up with the parameters descibed in the eld, was balanced by a genetic algorithm. Following the reference runs, sensitivity analysis was performed. The abundance of each trophic group was perturbed and the response of each other trophic group was measured. Since this was a stochastic model, both the mean and the variability were evaluated.
In this paper, we used only the sign of the responses (not their strength). For comparability, minimal changes must be made in the results of the dynamical simulations: since these categories do not exist in simulations, ?+, ?-and 0* predictions were considered as +, -and 0, respectively. Further, as the effects in dynamical simulations never result exactly in 0, we needed to de ne zero effects during the simulations. We decided to have the same number of zeroes in the simulation results as we got in the predictions of loop analysis. Values in this "corridor" of the smallest positive and negative simulation outcomes were considered 0.

Statistical methods
Sign-agreement was examined on several levels. We compared the predictions of loop analysis to the calculations made by the structural importance index TI 3 and to the results of dynamical simulations for the whole networks, for individual nodes (i.e. the rows of the matrices) and for individual interactions.
As described above, we categorized the outcome of dynamical simulation results into 3 categories (+, 0, -). This way, the table of predictions from loop analysis and the matrix of simulation effects became comparable (both containing only +, -and 0). A binary sign-agreement matrix contained 1 (yes) for similar and 0 (no) for different signs in the two matrices. The percentage of 1 values served for quantifying sign-agreement (1s in the main diagonal, corresponding to self-effects, were not considered).
Chi-square tests were applied for the effect signs for each node, determining if there is signi cant difference in either loop analysis or dynamical simulations.
In order to see if there is some structural basis for sign-determinacy and sign-agreement, we tested the correlation between (1) the adjoint matrix and the dynamic simulation results, (2) the adjoint matrix and interaction strength calculated by TI ij , (3) the dynamic simulation results and interaction strength calculated by TI ij and (4) sign prediction of loop analysis and node centrality measured by TI i . Figure 2 shows the food web for each of the 6 sites. The number of trophic groups varies between 12 and 15, starting from 14 in the pristine forest (site 1).

Results
The predicted sign of effects for each trophic group is shown in Figure 3, for each of the 6 sites (Figure 3a -3f). In site 1 (Figure 3a Sign-agreement predicted by loop analysis and by the dynamic simulations are shown in Table 4, at the level of the whole networks and for individual nodes (this latter is represented by the colours in Figure 3: sign-agreement increases from light yellow to dark green).
The mean correspondence of signs at the network level is under 50 %. Maximum sign-agreement ranged between 50 and 64.29% and the minimum was between 7.14 and 27.27%. On average, site 6 in the Kelian River is the ecosystem where loop analysis best predicts the signs of dynamial simulation results. From site 2 to site 6, there is a monotonous increase in mean (and the minimal) signagreement (the maximum values show no trend and also site 1 is out of this pattern). This suggests that the two methods provide different, complementary information on the ecosystem. Some organisms generally give consistent behaviour (e.g. HEDE), while others (ALGA, POM) behave differently in different modelling environments. Whether POM will have positive or negative impact on others is differently predicted in 5 out of 6 sites. Sign-agreement was measured also at the level of individual interactions. The mean of sign-agreement is also under 50% but the maximums are between 50-72% ( Table 5). The high sign-agreement for HEDE is seen also in the interactions in this group, being involved in 4 out of 7 of the highest-agreement interactions. It is also noted that the same group (HEDE) can be involved in minimum (HEDE-POM) and maximum (HEDE-TERR) agreement interactions (see site 2). It is noted that COLG in site 5 has several minimalagreement interactions (e.g. COLG-POM), after being the most predictable group in site 4 (with the most predictable interaction, COLG-CARN).

Conclusions
Predictive food web research would be a quantitative and holistic toolkit for systems-based conservation efforts. Good predictions on strong and weak as well as positive and negative effects are important for management and policy. While dynamical simulation exercises require a large number of parameters and complicated models, the semi-quantitative methodology of loop analysis offers simpler and faster results, generally easier to understand and interpret. Structural analyses are the simplest and least realistic ones.
The major question was to what extent simpler methods can replace the more complicated ones.
Comparing the sites, we see that the relatively consistent predictability of HEDE is seen in the sites of otherwise contrasting signs (site 2, site 3). The most consistent interaction happens between TERR and CARN in the most consistent model of site 6. This is the case when loop analysis and the dynamical simulation model mostly agree on sign prediction.
Based on Table 4, the general pattern emerges that higher sign-agreement is characteristic for trophic groups at higher trophic levels in sites 1-3, while trophic groups at lower trophic levels in sites 4-6.
Supposing that both models are relevant and correct, the question emerges why and to what extent should they be similar to each other. Similarity means that one reinforces the other, while differences may suggest that they are complementary, providing different kind of information. In the future, mixed trophic impact analysis, signed topological importance and further kinds of dynamical simulations must also be compared to the models studied here.
It is a question if experimental studies (e.g. mesocosm experiments) could help to better understand the predictability of effect signs and their differences in different modelling environments. Performing experiments only for a few species (e.g. perturbing COLG) might help a lot in calibrating the amount of expected changes and scale the models.
It needs to be better understand how trophic complexity is related to predictability, in terms of either interaction strengths or interaction signs. Earlier research, based on time series data, found quite convincing results on the dimensionality of niche space [35].
Future research can be extended, for example, towards (1) considering the asymmetry of interactions (MEPS) and (2)  Availability of data and materials The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Competing interests
The authors declare that they have no competing interests.

Figure 3
Food web of the Kelian River ecosystem in sites 1-6 (a-f). Arrows show the predictions of loop analysis: the sign of the effect of decreasing the abundance of a node on another (red is positive, blue is negative). The colour of nodes shows sign-agreement (%) between the predictions of loop analysis and the results of dynamical simulations (increasing from light yellow to dark green). Whether perturbing the light yellow nodes has positive or negative effect on others largely depends on the model chosen (see Table X). On the contrary, the perturbation of dark green nodes is similarly predicted by both approaches (see Table X). Colored by ColorBrewer [38].