\documentclass[12pt]{article} \usepackage{amsmath} \usepackage{url} \usepackage[dvips]{graphicx} \newcommand{\expect}[1]{\left\langle #1 \right\rangle} \newcommand{\etal}{\textit{et al.\ }} \title{Interaction strength combinations and the overfishing of a marine food-web} \author{Jordi Bascompte$^\dagger$\footnote{To whom correspondence should be addressed. E-mail: bascompte@ebd.csic.es, Phone: +34 954 2323 40, Fax: +34 954 6211 25}, Carlos J. Meli\'an$^\dagger$, and Enric Sala$^\ddagger$ \\ \\ $^\dagger$Integrative Ecology Group \\ Estaci\'on Biol\'ogica de Do\~nana, CSIC \\ Apdo. 1056, E-41080 Sevilla, Spain\\ \\ $^\ddagger$Center for Marine Biodiversity and Conservation \\Scripps Institution of Oceanography\\ University of California at San Diego \\ 9500 Gilman Drive, La Jolla, CA 92093, USA} \begin{document} \maketitle \baselineskip=8.5 mm \vspace{0.2 in} \centerline{\large PNAS, In Press} \centerline{\large Biological Sciences: Ecology} \centerline{\large Number of text pages: 24} \centerline{\large Words in abstract: 123} \centerline{\large Total number of characters: 32,975} \newpage {\bf The stability of communities largely depends on the strength of interactions between predators and their prey. Here we show that interaction strengths are structured non-randomly in a large Caribbean marine food-web. Specifically, the co-occurrence of strong interactions on two consecutive levels of food chains occurs less frequently than expected by chance. Even when they do occur, these strongly interacting chains are accompanied by strong omnivory more often than expected by chance. By using a food-web model, we show that these interaction strength combinations reduce the likelihood of trophic cascades after the overfishing of top predators. However, fishing selectively removes predators that are over-represented in strongly interacting chains. Hence, the potential for strong community-wide effects remains a threat.} \newpage Quantification of the strength of interactions between species is essential for understanding how ecological communities are organized and how they respond to human exploitation. Food-webs are characterized by many weak interactions and a few strong interactions (1-5), which appears to promote community persistence and stability (6-8). However, little is known about how interaction strengths are combined to form the basic construction blocks of food-webs, i.e., the simplest representations of multitrophic relationships such as tri-trophic food chains (9, 10). Here we analyze a real, large food-web to describe how interaction strengths are combined, and explore its implications for food-web dynamics. We compiled from published studies (11, 12) the largest quantitative food-web to date: 249 species/trophic groups and 3,313 interactions. It depicts the trophic interactions of a Caribbean marine ecosystem covering approximately 1,000 $km^2$, and comprises all benthic and pelagic communities from the surface to 100 $m$ depth, including detritus, 4 primary producer groups, 35 invertebrate taxa, 208 fish species, sea turtles, and sea birds (11, 12) (see Supporting Information for the food-web, list of species, and strengths and limitations of data). To investigate the structure of the food-web we calculated a per capita, standardized measure of the strength of the interaction of predators on their prey (1, 4). Using this measure, we first look at how strong interactions are combined in this food-web. Second, we explore the implications of these combinations for trophic cascades by using a food-web model. \section{Materials and Methods} \subsection{Estimation of Per Capita Interaction Strength} The strength of the interaction between predators and their prey was estimated as the proportion of prey biomass consumed per capita (per unit biomass of predator), per day, i.e., \begin{equation} \label{eq:pcis} \frac{(Q/B)_j \times DC_{ij}}{B_i}\nonumber \end{equation} where $(Q/B)_j$ is the number of times an age-structured population of predator $j$ consumes its own weight per day (13), $DC_{ij}$ is the proportion of prey $i$ in the diet of predator $j$, and $B_i$ is biomass of prey $i$ (see Supporting Information for the detailed derivation of the above expression). Parameter values were obtained from many individual studies compiled by Opitz (see ref. 12). From the above parameters, $(Q/B)_j$ is probably the most reliable. It is estimated as the metabolic efficiency of an average individual during its growth (13). Information on this metabolic efficiency comes from experimental studies for almost all species considered in this food web (12; see Supporting Information for details on strengths and limitations of data). $DC_{ij}$ values were obtained primarily from fish stomach contents in the US Virgin Islands (11). Specifically, stomach contents of a total of 5,526 specimens of 212 fish species were analyzed (11, 12). The average number of individuals per species is 27.1. The biomass ($g m^{-2}$) of the species used here are average estimates for the US Virgin Islands - Puerto Rico region (12). Body mass is estimated as the average body mass of adults (a well-known amount) multiplied by a correction factor describing the age-structure of the population (see ref. 12 for details). Density estimates were made by Opitz synthesizing several sources. She quantified previously qualitative measures of density. The only exception are fire corals, for which no information was available. We thus obtained the relative surface of fire corals from ref. 14. Multiplying this last ratio by the biomass of total corals we obtained an estimate of fire coral biomass. Per capita interaction strengths (pcis) were classified in four quartile classes: (1) pcis $<10^{-7}$; (2) $10^{-7} \leq$ pcis $<10^{-5}$; (3) $10^{-5} \leq$ pcis $<10^{-3}$, (4) pcis $\geq 10^{-3}$. We defined (4) as strong interactions. We looked at combinations of interaction strength values within class (4) in tri-trophic food chains and food chains with omnivory. Other classifications did not qualitatively change the results here presented. \subsection{Null Model and Significance} To assess the statistical significance of the co-occurrence of strong interaction strengths within tri-trophic food chains (TFCs) and chains with omnivory, we randomized the original food-web by randomly exchanging predator-prey pairs of interaction strengths. These pairs were kept as such intact to preserve the topological structure of the matrix (9). That is, if two species interact in the real food web, they also interact in each replicate, but the algorithm assigns to this link a weight randomly chosen from the pool of interaction strength values. Similarly, if two species do not interact in the real food web, they will not interact in any replicate. We generated a total of 50,000 replicate food-webs. For each replicate food-web, we classified interaction strengths in the previous four classes and measured the number of food chains with two strong interactions. We then used the distribution of the number of food chains with two strong interactions to determine the probability that a random food-web has a smaller or larger number of such food chains than that in the real food-web. Since not all interaction strength values necessarily form a TFC (e.g. a basal species A may be eaten by species B which is not eaten by any other species), we have used a second null model in which only the interaction strengths which do belong to at least a TFC are randomized. Results are qualitatively similar. \subsection{Linking Structural and Dynamical Measures of Interaction Strength} There are two main approaches to calculate interaction strength. First, Paine's seminal paper (1) was based on a dynamical assessment of the ``absolute prey response standardized by some measure of prey abundance.'' This measure is empirically calculated for a few species. On the other hand, observational, indirect (static) information has been used to estimate interaction strength for larger communities (4). Our measure of interaction strength used to describe the structure of this food-web builds from the last one. This, unfortunately, precludes any inference about dynamical implications. The reason is that it is a static measure, and there is a lack of direct estimates of other parameters which affect the population dynamical responses. For this reason, we bridge between static and dynamic measures when relating the results on structure to the dynamical model, a model built in a way that maximizes the use of observational information. This facilitates comparisons. As a first step, we have parametrized a bioenergetic model with biologically realistic values. Combinations of the structural interaction strengths are incorporated in the dynamical model by combining preference, non-linear functional response, metabolic parameters, and body mass ratios. The above static and dynamic measures of interaction strength have the following similarities and differences: Similarities: 1) They represent a property of each individual link (15). 2) They provide a top-down measure of consumption intensity (15). 3) They do not measure prey response (15). 4) Prey preference is used in both measures: $\Omega$ in the model, and $DC_{ij}$ (i.e., relative fraction of prey $i$ in the diet of predator $j$) in the static measure of interaction strength. 5) $(Q/B)$ is essentially identical to the maximum ingestion rate $Y$, although the first is per unit biomass and the second is per unit metabolic rate. Differences: 1) Our static measure ignores functional responses (it is based on fixed biomass of prey). On the other hand, our model considers functional responses which captures the fact that interaction strength varies with prey and predator density. The first measure can be calculated for lots of species, while the latter can only be calculated for a small subset of species (the real form of the functional response is unknown for the bulk of species). \subsection{The Food-Web Model} We used a bioenergetic model of a simple tri-tophic food chain and a food chain with omnivory (7, 16, 17). Although the model describes independent trophic modules, the modules studied in this paper are embedded within the entire food-web. A first step into addressing this is by adding allochtonous inputs $A$ to the model, which captures the fact that resources and consumers feed on other species. Thus, our modules are not completely isolated from the food-web (see Supporting Information for more details). The model can be written as: \begin{eqnarray} {dR \over dt}= r R (1-\frac{R}{K})-\frac{(1-\Omega_A_c)X_{RC}Y_C R^nC}{(1-\Omega_A_c)R^n+\Omega_A_cA_c^n+(1+c_CC)R_0^n} - \nonumber\\ \nonumber \\ -\frac{\Omega_R_PX_{RP}Y_PR^nP}{\Omega_R_PR^n+\Omega_A_pA_p^n+\Omega_C_PC^n+(1+c_PP)R_{02}^n} \end{eqnarray} \vspace{0.2 in} \begin{eqnarray} {dC \over dt}= -X_CC+\frac{(1-\Omega_A_c)X_{RC}Y_CR^nC}{(1-\Omega_A_c)R^n+\Omega_A_cA_c^n+(1+c_CC)R_0^n}+ \nonumber \\ \nonumber \\ +\frac{\Omega_A_cX_{AC}Y_CA_c^nC}{(1-\Omega_A_c)R^n+\Omega_A_cA_c^n}- \nonumber \\ \nonumber \\ -\frac{\Omega_C_PX_{CP}Y_PC^nP}{\Omega_R_PR^n+\Omega_A_pA_p^n+\Omega_C_PC^n+(1+c_PP)C_0^n} \end{eqnarray} \vspace{0.2 in} \begin{eqnarray} {dP \over dt}= -X_P P -FP +\frac{\Omega_R_PX_{RP}Y_PR^nP}{\Omega_R_PR^n+\Omega_A_pA_p^n+\Omega_C_PC^n+(1+c_PP)R_{02}^n}+ \nonumber \\ \nonumber \\ +\frac{\Omega_C_PX_PY_PC^nP}{\Omega_R_PR^n+\Omega_A_pA_p^n+\Omega_C_PC^n+(1+c_PP)C_0^n}+ \nonumber \\ \nonumber \\ + \frac{\Omega_A_pX_{AP}Y_PA_p^nP}{\Omega_R_PR^n+\Omega_A_pA_p^n+\Omega_C_PC^n}\\ \nonumber \end{eqnarray} where $R$ is the resource biomass, $C$ is the consumer biomass, and $P$ is the top predator biomass. $r$ is the resource intrinsic growth rate (its production-to-biomass ratio (17). $K$ is the resource carrying capacity, $R_0$, $R_{02}$, and $C_0$ are the half saturation densities of the resource when consumed by $C$, by $P$, and of the consumer itself when consumed by $P$, respectively. $X_{ij}$ is a relative specific rate of respiration, normalized to the production-biomass ratio of the consumer population (see ref. 17 and Supporting Information for a detailed derivation). $Y_j$ is the ingestion rate per unit metabolic rate (17) of species $j$. The two previous metabolic parameters are estimated using information for vertebrate ectotherms (17), and biomass estimated from this study (see Supporting Information). $F$ is the fishing rate of the top predator. $\Omega_i_j$ represents the species $j$ preference for species $i$. Thus, model (1-3) represents a simple tri-trophic food chain when $\Omega_R_P=0$, and an omnivory food-web when $\Omega_R_P>0$. $c$ is a positive constant describing the magnitude of interference among predators (see ref. 18 for details), and $n$ is the number of encounters a predator must have with its prey before the predator is maximally efficient at feeding on that prey item (19). The role of $n$ is to shift the functional response from Type II to Type III. Specifically, the Type II response is a special case of the Type III response (for $n=1$, where a predator is always maximally efficient on the prey item, 19). We have tested Holling type II (17, 19) ($n=1$, $c_i=0$), Holling type III (17, 19) ($n=2$, $c_i=0$), and predation interference (20) ($n=1$, $c_i>0$) functional responses. All three functional responses and a range of realistic parameter combinations showing stable dynamics have given similar qualitative results (except for predator interference for certain parameter combinations, see Supporting Information). The specific parameter combination used in Fig. 3 is: Functional response is Holling type II, $\Omega_A_C=0.6, \Omega_C_P=0.4, \Omega_A_P=0.6$ (Fig. 3a and b), $\Omega_A_P=0.2, \Omega_R_P=0.4$ (Fig. 3c). $X_{ij}= 0.1$ and $Y_i=3$, corresponding to weak interactions as depicted in the inset (Fig. 3a). $X_{ij}=0.2$ and $Y_i=4$ corresponding to strong interactions (Fig. 3b and c). Other parameter values are: $r=1$, $K=1$, $R_0=R_{02}=C_0=0.75$, $n=1$, $c_C=0.005$, $c_P=0.35$, and $A_C=A_P=0.01$. \section{Results and Discussion} Fig. 1{\em a} shows a random fraction of the whole food-web for representation purposes. A few strong interactions are distributed within a matrix of weak interactions, confirming previous results (1-5). The frequency distribution of per capita interaction strengths (interaction strengths hereafter) fits a lognormal distribution with marginal significance ($P=0.06$, Lilliefors' test; Fig. 1{\em b}). It spans seven orders of magnitude, highlighting the extreme variability of predator-prey interaction strengths. The frequency distribution of interaction strengths is an adequate way to explore some fundamental properties of food-webs. However, it is only a first step towards understanding the structure of complex communities (6, 9). Here, we move beyond this statistical distribution by studying how interaction strength values are combined to form the basic construction blocks of this food-web. We describe how interaction strengths are distributed in tri-trophic food chains (TFCs) in which a top predator $P$ eats a consumer $C$, which in turn eats a resource $R$ (Fig. 2{\em a}). This basic chain can be viewed as the building block of complex food-webs (10) (Fig. 2{\em c}), or the simplest representation of multitrophic relationships frequently used in theoretical studies (7, 21, 22). We were interested in determining how strong interactions are structured within TFCs, since the co-occurrence of strong interactions on two consecutive levels of a trophic chain has the potential to modify the structure and dynamics of whole food-webs through trophic cascades (23-27). Trophic cascades are predator-prey effects that alter biomass or abundance of a species across more than one trophic link (23, 27). Specifically, reductions in the abundance of a predator through fishing would propagate through the food chain resulting in increased consumer abundance and fewer resources (24). To investigate interaction strength combinations that may induce trophic cascades, we first classified interactions into four categories on the basis of the order of magnitude of interaction strength (Materials and Methods). We counted the number of TFCs with co-occurrence of two strong interaction strengths (those belonging to the upper quartile of the log per capita interaction strength distribution, $n=3,086$; see Fig. 2{\em c}). All TFCs were considered, including those starting from primary producers. The fish species involved in most strongly interacting TFCs were sharks as top predators (see Database in Supporting Information), groupers (family Serranidae) as consumers, and fishes of the Blenniidae, Clupeidae, Engraulidae, Pomacentridae, and Scaridae families as base of the TFC. To determine whether two strong interactions co-occur more often than expected by chance, we built a null model using randomized networks (see Materials and Methods). The role of omnivory (the top predator also feeds on the resource; Fig. 2{\em b}) in food web stability has been debated for decades. While previous results concluded that omnivory destabilizes food webs (28), recent papers have shown the opposite trend (16, 29). It is unclear whether omnivory can compensate trophic cascades when top predators and consumers are strong interactors (30). Consequently, we assessed the likelihood of strong omnivory accompanying strong tri-trophic interactions in the Caribbean food-web. Our analysis showed that co-occurrence of two strong interactions in TFCs is less frequent than expected by chance ($P=0.0018$). When two strong interactions co-occur, strong TFCs have a strong omnivory link more often that expected by chance ($n=585, P=0.0001$). To assess the implications of these non-random combinations of interaction strength on trophic cascades, we used a food-web model for simple tri-trophic chains and tri-trophic chains with omnivory (7, 16, 17, see Materials and Methods). Because overfishing tends to eliminate the species in the higher levels of food chains (31, 32), we simulated the fishing of top predators and explored the subsequent change in resource biomass. As in related studies, the magnitude of the trophic cascade was measured as the log ratio of resource biomass without fishing of the top predator to resource biomass with fishing of the top predator (26). The co-occurrence of two strong interactions in the basic TFC increases the magnitude of the trophic cascade (Fig. 3, compare {\em a} with {\em b}). However, the magnitude of the trophic cascade is reduced in the presence of strong omnivory (Fig. 3, compare {\em b} with {\em c}). In addition, omnivory changes qualitatively the response of the resource, which may first increase with moderate fishing of the top predator (Fig. 3, compare {\em b} with {\em c}). These results indicate that the interaction strength combinations in the web reduce the likelihood of trophic cascades, with important implications for food-web dynamics. However, our model describes isolated modules (although coupled to some extent through the allochtonous inputs, see Materials and Methods and Supporting Information). Future work is needed to explore how results are affected by the use of modules more explicitly embedded within the whole food-web. This remains a challenge since models describing an increasing number of species become increasingly unstable for biologically realistic parameter combinations (33). The reduced tendency for trophic cascades resulting from the reported interaction strength combinations, however, does not imply that this community is buffered from the effects of exploitation. Fishing selectively targets a biased sample of species belonging to upper trophic levels (31, 32). These species, which include top predators, are over-represented in the relatively rare strongly interacting TFCs. For example, ten heavily fished top predators (sharks from seven families, see Database in Supporting Information) account for 48 $\%$ of the strongly interacting TFCs in the Caribbean food-web. The likelihood of trophic cascades after the depletion of these strong interactors will thus depend on the relative fraction of strong omnivory. 31$\%$ of these strongly interacting TFCs have the buffering effect of strong omnivory, still leaving roughly two out of three strongly interacting TFCs susceptible to trophic cascades. The dynamic consequences of the structural patterns here reported provide a framework to assess the community-level impacts of overfishing. Strongly interacting TFCs include species at the base such as parrotfishes (Scaridae) and other herbivores (see Database in Supporting Information) which are important grazers of macroalgae (11). The removal of herbivores by fishing may have been partly responsible for the shift of Caribbean reefs from coral- to algal-dominated (34). Our results suggest that overfishing of sharks may have also contributed to the depletion of herbivorous fishes through trophic cascades, thus enhancing the degradation of Caribbean reefs. The community-wide impacts of fishing are stronger than expected because fishing preferentially targets species whose removal can destabilize the food-web. \vspace{0.3 in} We thank J. Alroy, F. Ballantyne, L.-F. Bersier, P. Buston, J. Jackson, P. Jordano, N. Knowlton, N. Martinez, R.M. May, R. Paine, A.G. S\'aez, S. Sandin, E. Seabloom, G. Sugihara, and D. V\'azquez for comments on a previous draft and interesting discussions. Funding was provided by the Spanish Ministry of Science and Technology (Grant to JB and Ph.D. Fellowship to CJM), and the History of Marine Animal Populations Program of the Census of Marine Life, sponsored by the Alfred P. Sloan Foundation (Grant to ES and J. Jackson). JB and CJM thank the faculty at Scripps for their hospitality during a three month visit in 2002. Part of this work was conducted at the National Center for Ecological Analysis and Synthesis while JB was a visiting scientist (summer 2003). This paper is dedicated to the memory of Ramon Margalef (1919-2004). \section{References} \vspace{0.2 in} 1. Paine, R.T. (1992) {\em Nature} {\bf 355,} 73-75. \vspace{0.1 in} 2. Fagan, W.F. \& Hurd, L.E. (1994) {\em Ecology} {\bf 75,} 2022-2032. \vspace{0.1 in} 3. Raffaelli, D. \& Hall, S.(1995) in {\em Food Webs, Integration of Patterns and Dynamics}, eds. 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(1994) {\em Science} {\bf 265,} 1547-1551. \vspace{0.1 in} \newpage \vspace{0.4 in} \section{Figure Legends} \vspace{0.3 in} {\bf Figure 1.} ({\em a}) Random sample of the Caribbean food-web containing $30 \%$ of the species and $11\%$ of the interactions. Each node represents a species or taxon. Arrows represent trophic interactions between predators and their prey. Arrow thickness is proportional to the interaction strength. Loops represent cannibalism. ({\em b}) Frequency distribution of interaction strengths (n=3,313), spanning seven orders of magnitude. The line represents the best fit to a lognormal distribution. \vspace{0.2 in} {\bf Figure 2.} ({\em a}) Tri-trophic food chain. ({\em b}) Tri-trophic food chain with omnivory. Nodes from top to bottom represent the top predator ($P$), the consumer ($C$), and the resource ($R$). Arrows represent trophic links. ({\em c}) Schematic representation of a food-web highlighting three tri-trophic food chains (one of them with omnivory). The central food chain shows co-occurrence of two strong interaction strengths, the combination explored in this paper. \vspace{0.2 in} {\bf Figure 3.} Response of the resource as a function of the fraction of predators fished in tri-trophic food chains with two weak interactions ({\em a}), two strong interactions ({\em b}), and food chains with omnivory and three strong interactions ({\em c}), based on a bioenergetic model (see Materials and Methods). The magnitude of the trophic cascade (measured as the resource log ratio) is greater for food chains with two strong interactions (compare {\em a} with {\em b}), and it is reduced when there is a similarly strong omnivory link (compare {\em b} with {\em c}). The dotted line is used as a reference. Parameter combinations are specified on Materials and Methods. \newpage \begin{center} \includegraphics[width=8.6cm]{fig1new.eps} \end{center} \caption{Bascompte Figure 1} \newpage \begin{center} \includegraphics[width=8.6cm]{fig2new2.eps} \end{center} \caption{Bascompte Figure 2} \newpage \begin{center} \includegraphics[width=8.6cm]{fig3new.eps} \end{center} \caption{Bascompte Figure 3} \end{document}