Identification of unique species in the marine ecosystem based on the weighted trophic field overlap

Graphical abstract


Method details
The uniqueness of a species is based on its irreplaceability in the food web and is considered when identifying important species [1][2][3][4]. Species with unique topological positions and trophic interactions [1][2][3] that overlap less with those of other species can help maintain high trophic diversity in an ecosystem [1]. Thus, uniquely important species should be protected.
Mathematical methods and the topological network perspective [5] are required to quantify the interactions among species. The topological methodology (the sum of trophic field overlap, STO) proposed by Lai et al. [6] provides an alternative means of identifying the unique species with the lowest trophic field overlap. This method elucidates the structural uniqueness of a certain species, however, it over-depends on the topological network structure and neglects the quantitative effects of predation, which affect the accurate identification of unique species in the food web. Link weight in the food-web network means the relative importance of links and the strength of trophic interaction [7][8][9]. Therefore, prey proportion as link weight in the food web should considered. Besides, the STO index does not mention cannibalism and mutual predation that can be observed commonly in the marine ecosystem. They can affect species directly and indirectly [10,11] and drive the dynamics of populations or structures [12][13][14]. Therefore, cannibalism and mutual predation also should be considered when quantifying the interaction strength among species.
In this paper, we measured the trophic field overlap of species by considering a weighted topological network both structurally and mathematically. Cannibalism and mutual predation between species were accounted for in the calculations. Regression and non-parametric test were used for comparison. This method reflects the irreplaceability of species by considering the weighted trophic field overlap (WTO) and overcomes the drawbacks of the traditional method (STO). Thus, it will contribute to the protection of unique species and maintenance of trophic diversity and ecosystem robustness. In the following sections, we will (1) state the calculation process of the STO in brief and WTO in detail; (2) state the shortcomings of the method and future research directions; (3) compare results between indicators (WTO and STO, as well as them and other topological indicators respectively), and discuss the ecological meanings of WTO and unique species based on correlation and ecological background by case study.
Uniqueness according to the STO method of Lai et al. [6] Firstly, the direct interaction strength of species i on another species j was quantified by the reciprocal of the D value of j (i.e., the number of nodes directly connected to j) if there was direct predation between them. The indirect interaction strengths were calculated based on the direct interactions as described by Lai et al. [6]. Secondly, the maximum shortest length of the food-web network plus 1 (i.e. n) dividing the sum of the direct and indirect interactions of species i on other species was defined as the final interaction strength of a species i. The threshold variable T, which increases from 0 to 1 in increments of 0.001, was used to identify strong (S) and weak (W) interactions. The value of a for species i was designated as S when a > T and W when a T. Thirdly, the STOs of all species were counted across the entire range of T values.

Uniqueness according to WTO
Each link of the nodes was weighted by the weight percentage of each prey to illustrate the relative importance of links in the food-web network. The adjacency matrix of all species was constructed based upon the trophic relationships in the food web. We replaced all of the values in the matrix with the weight percentages of prey. Then, the sum of the weight percentages in each row was defined as the weighted out-degree WD out , and the sum of those in each column was designated as the weighted in-degree. The sum of the row and column values for species i was defined as the weighted degree WD i of species i. For the weighted adjacency matrix W, the sum of each column (i.e., weighted in-degree) equals 1, while the row sums (i.e., WD out ) corresponding to the outgoing effects of species are not equal. The latter feature reflects the importance of species as prey in the whole food web and plays a significant role in calculating interaction strength.
The direct interaction strength of species i on j can be weaker if the latter has more neighbors [15]. Lai et al. [6] quantified the direct interaction strength by defining it as the reciprocal of D, which was defined as the reciprocal of the weighted degree in the improved method. Thus, the direct interaction strength of species i on j (a ij ) is defined as where W ij represents the weight proportion of i in the total intake of j or j in the total intake of i in the food web, and WD j represents the weighted degree of species j. Note that if there is no predation between i and j,a ij = 0. When all one-step effects (i.e., direct interactions) between species are calculated, a matrix WA can be constructed, where the ijth element a ij represents the direct interaction strength of species i on j. Here, i corresponds to the row number and j corresponds to the column number.
To calculate the weighted degree, the directed network including loop chains with single-node (cannibalism) and dual-direction links (mutual predation between two species) needs to be transformed into undirected network. The prey proportions attached to loop chains were retained the same as oneway links, while the pairs of those from mutual predation between two species were added up.
Indirect interaction is defined as a trophic relationship between species that interact through one or more than one intermediate species [5], and it can be quantified by using direct interactions. In a two-step food-web network, the strength of the two-step interaction is the product of the strengths of the two direct interactions via the matrix power operation WA 2 . For example, if species i has a connection with j through species k (i.e., i can interact with j in two steps and the length of the links between the two nodes is 2), then probability theory makes it possible to quantify the two-step interaction (namely, a ikj ) of species i on j as follows where a ikj represents the two-step interaction of species i on j through species k, and a ik and a kj represent the one-step interactions of species i on k and k on j, respectively. Thus, all of the possible two-step interactions between species i and j from various paths through other intermediate species can be calculated as where sa ij represents the two-step interaction of species i on j in the food network, and N represents the number of species in the food web. The operational rule according to Eq. (3) conforms to the matrix power operation algorithm; thus, all two-step interactions can be computed as WA 2 . The interaction strengths (in matrix IM) between pairwise species for the n-step food-web network are the mean values of all interaction matrices where the ijth element IM ij in matrix IM represents the interaction strength between species i and j in the food web, and n represents the maximum shortest distance in the food network plus 1 (the step length used in the analysis should be slightly greater than the maximum shortest distance to ensure that the effect of one species can spread to all other parts of the food web; see Lai et al. [6]). The threshold variable T, which increases from 0 to 1 in increments of 0.00001, was used to identify strong (S) and weak (W) interactions. All of the interactions a ij were marked S when a > T and W when a T under different T values, and a series of matrices AM T was constructed. As mentioned previously, trophic field overlap describes the similarity of the interactions among all species in the food web. Note that the W trophic interactions are responsible for food web stability [16] and that they are as important as S interactions [6]. The number of instances in which the marks in row i and the other rows are the same, column by column, is counted as the weighted trophic field overlap (WTO) of i and other species under a given T. For example, for column j and a given threshold T 1 , the WTO ijk , T1 (WTO of i and k for j under T 1 ) can be counted as 1 when AM ij in row i and AM kj in row k are both S or both W. This process can be repeated for all of the columns to obtain the WTO ik , T1 (WTO of i and k under T 1 , which equals the difference between N and the Hamming Distance of row i and k in matrix AM T1 ) and repeated for all rows with row i to obtain the WTO i , T1 (WTO of i under T 1 , which equals the sum of the differences between N and the Hamming Distances of row i and other rows in matrix AM T1 ). After counting in every matrix AM T across the entire range of T values, add up all WTOs and the result is the new uniqueness value of i. Small WTO values indicate low trophic field overlap of one species with all others, and less overlapped species are considered unique.
All of the analyses were conducted using the Ucinet 6 and Matlab r2006a software.

The shortcomings and future study
Both the network structure and weight percentage of prey could be affected by the taxonomic category. The lower the taxonomic category chosen, the more accurate the results. Species with relatively high weight percentages in diets tend to play more important roles and have stronger interactions with other species. However, the index of weight percentage could overestimate the importance of species with high body masses or sizes [4,7]. In the future, efforts should be made to examine the diet matrix with additional objective and comparable indices, such as the importance index. Meanwhile, the identification of unique species could be improved by taking into account additional factors. Moreover, the interaction strength can be further classified into different levels, such as strong, intermediate, and weak.
This index is sensitive to changes in the diets of predators since slight fluctuations may cause it to vary considerably. In a dynamic ecosystem, the diet compositions and feeding intensities of species vary considerably and may change the strengths of the trophic interactions among species and influence the uniqueness of species in the food web; thus, they should be considered for accurate identification of unique species. Moreover, the external disturbances to ecosystems affect the trophic interaction structure and strength directly or indirectly, and the net production and internal structure will change when the ecosystem gets mature. These will also change the relative uniqueness of species and the mechanisms are complex. Thus, these need to be studied further to determine how disturbances and maturity of ecosystem will affect species uniqueness in the food web.

Case study and ecological meanings
An ecological network (Fig. 1) was constructed based on data collected from five bottom trawl surveys in Haizhou Bay that were performed during March, May, July, September, and December of 2011 (Further details are available in the literature [17][18][19]). There were 93 nodes representing each species or aggregated trophic group (Table 1) and 2042 links representing the trophic flows between these trophic groups.
Unique species exist in every food web. They hold important positions in the food web and act as connections between species with their unique positions and trophic interactions, making the web even closer and reducing the loss of energy transmitted through the food chain. Moreover, unique species with the lowest WTO were found to have higher interaction strengths with other species in    matrix IM. The interaction strength distribution showed that there were many weak interactions and few strong interactions in the food web [15,21]. Thus, strong interactors tend to be unique and irreplaceable due to their distinctive trophic interactions, which contribute to the trophic diversity and ecosystem robustness. Unique species play important roles as essential prey and have uniquely strong interaction strength with other species in the food web, given the current ecosystem configuration. The roles of unique species may differ in different ecosystems since other configurations may also provide ecosystem services, but their uniquely strong trophic interaction strengths still ensure their important positions in the food web. This is determined by the algorithm of WTO and the characteristics of ecosystem with many weak interactions and few strong interactions.