Food web assembly rules. Jan O. Haerter, Namiko Mitarai, Kim Sneppen
Center for Models of Life, Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, DK-2100, Copenhagen, Denmark
corresponding author: Jan O. Haerter, Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen, Denmark, tel: +45 353 25352, fax: +45 353 25425, email:
[email protected]. Statement of authorship: All authors developed the theory, performed the data analysis and wrote the manuscript and appendix. Word count (excluding abstract, acknowledgments, tables, figure legends, and references): approx. 4200 words. Number of figures: 5 figures. Conflict of interest: The authors have no conflicting interests. Keywords: Food web, competitive exclusion, apparent competition, biodiversity, parasite-host, concomitant link, food web stability, sustainability, trophic levels, food chain length.
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ABSTRACT In food webs, many interacting species coexist despite the restrictions imposed by the competitive exclusion principle and apparent competition. Using the generalized Lotka-Volterra equations, we show that sustainable coexistence necessitates nonzero determinant of the interaction matrix, a requirement that is equivalent to demanding that each species is part of a non-overlapping pairing. When nonzero determinant cannot be achieved, the matrix rank can be used to quantify the lack of niches, corresponding to unpaired species. For the species richness at each trophic level, these assembly rules specify sustainable combinations. In neighboring levels, the rules allow the higher level to avert competitive exclusion at the lower, thereby incorporating apparent competition. The constraints predict high species numbers at intermediate levels and thinning at the top and bottom. Using comprehensive food web data, we demonstrate how omnivores or parasites with hosts at multiple trophic levels can loosen the constraints and increase robustness in food webs.
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INTRODUCTION Fueled by ongoing rapid decline of biodiversity (Ricciardi and Rasmussen, 1999), ecology is in the midst of a lively debate on the effect of species loss or introduction on food web stability (McCann, 2000; Montoya et al., 2006). In food webs, complexity arises from combining a large number of species (the nodes) and a large number of relations between these species (the links). Addressing the latter, recent attention was devoted to the structure of links using e.g. the random, cascade and niche models (Cohen and Newman, 1985; Williams and Martinez, 2000), stirring a prolific debate on the role of the link distribution regarding food web stability (McCann, 2000; Montoya et al., 2006; Ives and Carpenter, 2007; Allesina and Tang, 2012). Yet, fundamental rules constraining node richness in food webs are still lacking. For consumer-resource relationships, the “competitive exclusion principle” states that when two consumers compete for the exact same resource within an environment, one consumer will eventually outcompete and displace the other (Gause, 1936; Hardin, 1960). More generally, the number of consumers cannot exceed the number of distinct resources (Levin, 1970). While competitive exclusion does not specify the relative abundance of consumers, experimental studies do demonstrate strong correlations between consumer and resource diversity (Sandom et al., 2013; Armbrecht et al., 2004; Andow, 1991; Murdoch et al., 1972). These observations highlight that also the consumer plays a critical role in shaping the network of species, even when direct interaction between resource species is absent, an observation captured in Holt’s paradigm of apparent competition (Holt, 1977). Despite these empirical findings, a theoretical base for the coexistence of many species in food webs is currently lacking. This lack may be partially due to the complexity of the many-species interactions, yielding an uncontrollable number of parameters and hampering direct calculations or simulations of sufficient generality. Notwithstanding these complications, progress can be made when necessary conditions are demanded. For an ecology, consisting only of a resource and a consumer level, we have recently shown that coexistence requires that the species richness of both levels is balanced (Haerter et al., 2014). Examples of such systems may be the phage-bacteria ecology in the Atlantic Ocean (Moebus and Nattkemper, 1983; Flores et al., 2012) or laboratory ecologies. 3
However, in food webs, a subset of trophic levels can generally not be considered in isolation. A species’ niche is determined by its entire set of interactions, which generally may be composed of both beneficial and harmful interactions, i.e. the species may act both as a consumer or resource. Further, many food webs contain omnivorous interactions, i.e. those where one species preys on several other species that are located at more than one trophic level. To derive necessary conditions for coexistence in food webs, a more general starting point is required. Based on the generalized Lotka-Volterra equation, we here show that in sustainable food webs each species must be part of a non-ovelapping pairing. We define a non-overlapping pairing as a topological pattern for a directed network, where each species must contribute to a closed loop and none of the loops may overlap (Details: Methods). Mathematically, this is simply the statement that the determinant of the interaction matrix must be nonzero. Our theory allows us to quantify the species richness at different trophic levels, make predictions for secondary extinctions and assess the stabilizing effect of parasitism and omnivory.
MATERIALS AND METHODS Steady state equations. For consumer-resource interactions in food webs, the generalized Lotka-Volterra equations (MacArthur, 1967) are (1) (1) S˙ i /Si
(1) = ki 1 −
n1 X
(1) (1) pji Sj − αi −
j=1
n2 X
(2,1)
ηki
(2)
Sk
(1)
k=1
for primary producers and X
nl−1 (l) (l) S˙ k /Sk
=
X
nl+1 l,l−1 l,l−1 (l−1) βkm · ηkm · Sm −
m=1
(l)
l+1,l ηpk · Sp(l+1) − αk
(2)
p=1
for species at trophic levels l > 1. We distinguish a species by the set of links that connect it to predators and prey or nutrients and the strength of these links (Details: Sec. A4). In Eqs 1 and 2, Si (l) with i = 1, . . . , nl are the (1)
densities of species residing at trophic level l, nl is the species richness at level l, ki (1)
(1)
denote the maximal growth (l)
rates of Si , pji describe differential consumption of the basic resources by the Si , αi denote the decay rate of
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(l)
(l,l−1)
species Si , ηki
(l)
(l−1)
are the interaction coefficients between a species Sk with species Si (l)
(l−1)
efficiencies of reproduction of species Sk when consuming species Sm (1)
l,l−1 , and βkm are the
.
(l)
In the steady state, the time derivatives S˙ i
and S˙ k on the LHS of Eq. 1, respectively Eq. 2, vanish and we
have the equations n1 X j=1
X
(1)
pji Sj
+
n2 (2,1) X η
ki (1) k=1 ki
(1)
(2)
Sk =
nl−1
X
ki
(1)
− αi
(1) ki
(1) ≡ k˜i , respectively
(3)
nl+1 (l,l−1) (l,l−1) (l−1) ηkm Sm
βkm
m=1
−
(l+1,l)
ηpk
Sp(l+1)
(l)
≡ αk .
(4)
p=1
Collecting all constant coefficients (RHS of Eqs 3 and 4) in the vector k and all interaction coefficients on the LHS in the interaction matrix R, we have the linear matrix equation R · S = k, where S is the vector of all species densities. For completely shared nutrients, the competition factors pji = 1. (Details: App. Secs A1 and A2). Parasite interactions. Parasites have complex life-cycles that can demand several hosts (Huxham et al., 1996; Lafferty et al., 2008). Notwithstanding these complications, we here formally treat them as consumers, respectively resources of independently acting other species. Also, we simplify concomitant links in terms of simple linear responses (details: Appendix Sec. A9). Non-overlapping pairing. det(R) 6= 0 can be fulfilled if every species is paired with another species or nutrient (this constitutes a perfect matching (Lov´asz and Plummer, 1986)). For food webs with sharp trophic levels, the network is bipartite and therefore it is required that such a perfect matching exists. When species with variation in food chain length are present, one may generally obtain nonzero det(R) by covering the entire network with closed loops of directed pairings (i.e. cycles, Sec. A6). In all these cases, a directed pairing represents one nonzero matrix element, whereas a pairing also includes the symmetric element. Empirical food web data. We use high-resolution data on seven food webs including free-living and parasite species: The North American Pacific Coast webs Carpinteria Salt Marsh (CA), Estero de Punta Banda (PB), Bahia Falsa (BF) (Hechinger et al., 2011; Lafferty et al., 2006); the coastal webs Flensburg Fjord (FF) (Zander et al., 2011), Sylt Tidal Basin (ST) (Thieltges et al., 2011), and Otago Harbor (OH), New Zealand (Mouritsen et al.,
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2011), as well as the Ythan Estuary (YT), Scotland (Huxham et al., 1996). These food webs describe consumerresource interactions between basal, predatory and parasite species. A compilation of all seven food webs has recently been provided (Dunne et al., 2013a,b). Specifically, the data distinguish three types of links: (i) links between free-living species only (“Free”), (ii) additional links between parasites and other species (“Par”) and (iii) links from free-living consumers to the parasites of their resources (“ParCon”), i.e. so-called concomitant links. Details on data analysis: Appendix Section A9. For the empirical data, the lack of niches, i.e. nullity d ≡ S − rank(R), was computed by using random values for all nonzero entries of the respective matrix R. Basal species were each given an individual nutrient source. In “ParCon sym” a subset (20 percent) of concomitant links were randomly selected to be symmetric (Details: Sec. A9.3.3). In the data analysis and simulations (Fig. 4 and 5), the trophic level of a species is defined by its prey-averaged food chain length (Sec. A9.2.1). Simulations. An idealized food web was constructed by using the average species counts at levels ni obtained from all empirical data sets and initially assuming sharp trophic levels for all species. Sharp trophic levels were obtained by rounding each species’ chain length to the nearest integer value (Fig. 5a). With the constraint of these trophic levels, a number of links was assigned to match the empirical average for free-living food webs (Fig. 5b). When adding further species, the empirical average of parasite species count was used (47 species). To obtain Fig. 5, initially, each parasite received one link. In the cases (c) and (d), this link connected the parasite to any existing free-living species. In the cases (e) and (f), this link connected the parasite to any existing species at trophic levels 3 or 4. For any subsequent link, a parasite was chosen at random. A link was then formed in three ways: Case c: randomly to connect with another existing species; Case d: randomly, but only to species at the same trophic level as for the initial link of that parasite; Case e: randomly to other existing species at levels 3, level 4; Case f: randomly to other existing species at levels 3, level 4 or another parasite (More detail: Sec. A10).
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RESULTS Theory We describe the interaction of species on L trophic levels by the generalized Lotka-Volterra equations (MacArthur, 1967). Basal species are constrained by the system carrying capacity while the consumers are assumed not selflimiting, and trophic interactions occur through the linear type-I functional response (Methods). Such equations have been widely used in community assembly models, where food web networks are assembled by numerically analyzing the equations to find parameter sets with stable and/or permanent coexistence solutions (Post and Pimm, 1983; Law and Morton, 1996; Drossel and McKane, 2003). Here we take an alternative path to find a necessary condition for coexistence, which results in the foodweb assembly rules that constrains the network topology. In the steady state, we have the matrix equation R · S = k, where S is the vector of all species densities, R is the interaction matrix between the species, and k is the vector of growth and decay coefficients. Note that R has a block structure with nonzero entries only for interactions between neighboring trophic levels but not within the same level, and that the positions of these matrix elements are symmetric due to mutual interaction between predator and prey (Fig. 1a). Stable/permanent coexistence requires that a feasible solution S∗ ≡ R−1 k > 0 exists (Allesina and Tang, 2012; Law and Morton, 1996), necessitating det(R) 6= 0 (Sec. A2). We first specialize to the case of a single shared basic nutrient, e.g. sunlight, used by all primary producers (1)
Si , with n1 the species richness on the first trophic level and i ranging from 1 to n1 . This corresponds to setting the competition terms pji = 1 for all 1 ≤ i,j ≤ n1 in Eq. 1, thereby yielding a block of n1 × n1 unit entries in the lower right block. (Example: Sec. A4). Nonzero determinant is achieved if it is possible to identify a path of matrix elements that only contains elements from the non-zero sub-matrices bordering the diagonal (Fig. 1). This is equivalent to demand that every species is part of a consumer-resource pair connecting neighboring trophic levels and none of these pairs overlap (known as perfect matching of a bipartite graph in graph theory (Lov´asz and Plummer, 1986)). The pairing guarantees that no species share exactly the same niche, i.e. a particular set of interactions with resources and
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consumers (Sec. A4), and manifests the competitive exclusion principle (Fig. 1b). For primary producers, pairings may involve the nutrient source (Fig. 1a, inset). In that case, at least n1 − 1 species at level two are required for pairing of the remaining basal species. We term this structure resource-limited. If the pairing with nutrients is not used, n2 ≥ n1 is required — a consumer-limited configuration due to the biomass restriction imposed by consumer predation (Fig. A1). In turn, those species left unpaired at level two must be paired by species at level three. Defining No and Ne as the sums of node richness at odd and even levels, respectively, it is then easy to check that the general constraints become ∆ ≡ No − Ne ∈ {0, 1} ,
(5)
which encompasses the competitive exclusion principle (Gause, 1936; Hardin, 1960; Haerter et al., 2014). Defining L as the top trophic level, for each of the two options in Eq. 5 a set of L − 1 nested inequalities arises relating the species counts ni : (i)
n1 ≥
∆
(ii)
n2 ≥
n1 − ∆
(iii)
n3 ≥
n2 − n1 + ∆
(iv)
n4 ≥ n3 − n2 + n1 − ∆
(6)
.. . This sequence continues until nL is reached. Note that feasible solutions can be constructed for any allowed choice of species counts (Sec. A2). The case ∆ = 0 is consumer-limited, whereas ∆ = 1 signals resource limitation. In practice, there may be several basic nutrients, such as different chemical compounds or spatial or temporal subdivision (Tilman, 1994). If nS > 1 separate nutrients are available, the assembly rules yield nS +1 sets of conditions analogous to those in Eq. 6 where any ∆ ∈ {0, nS } is allowed (Sec. A7). The presented theory assumes simple predator-prey couplings. Non-linear interactions (i.e. type-II response) and cannibalism can be included by adding diagonal matrix elements in Fig. 1a, corresponding to species that pair
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with themselves (Secs A7,A8).
Food web assembly We return to a single nutrient (nS = 1). In the consumer-limited case, Eq. 6(i) restricts n1 to an upper bound given by n2 . Eq. 6(ii) conversely restricts n2 to numbers equal to the sum of the total species richness in both neighboring trophic levels, hence the limitation to the abundance of n2 is much weaker compared to that of n1 . Accordingly, the basal level cannot constitute the global maximum of node richness within a level. Moving further, n3 could again exceed n2 but possible limitations stemming from the count of top predator species become noticeable. When starting at the top predator level L, by symmetry, similar constraints as in Eq. 6 hold: nL − nL−1 ≤ 0. Together with Eq. 6(i) we have for the consumer-limited state, n1 + nL ≤ n2 + nL−1 , i.e. species at intermediate trophic levels generally dominate food web biodiversity (Fig. 2). The condition is similar for the resource-limited state, where the limit is shifted by one species. For increasing L, these equations predict further increase in the contribution of intermediate species to total biodiversity. Given these general constraints, we can now discuss food web assembly. Fig. 3a shows a simple food web and its interaction matrix. Graphically, an allowed structure is again manifested by permitting non-overlapping pairing of species (Fig. 3a). Food web growth is characterized by alternating transitions between resource- and consumer-limited states. Initial community growth requires the presence of a single primary producer (Fig. 3b), hence n1 = 1. The only possible addition is then a consumer, preying on the existing producer. This entails immediate addition of a trophic level (n1 = n2 = 1) and transition to a consumer-limited state. The assembly rules subsequently permit additions at different trophic levels, but alternation between consumer and resource limitation must be preserved. Our assembly rules are easily generalized to food webs containing omnivores or parasites with hosts at several trophic levels (Sec. A6). Consider again Fig. 3a, but imagine that another species is added on trophic level 4, causing a violation of the assembly rules. This violation can be rectified by a single generalist omnivore (Fig. 3c), corresponding to an additional row and column of nonzero coefficients in the interaction matrix. The omnivore
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can be interpreted as a consumer preying on all trophic levels. Hence, one can choose a level i and let its species count ni be increased by one unit to again satisfy the assembly rules. In general omnivory or parasitism can make the graph non-bipartite. If so, the non-overlapping paring to achieve det(R) 6= 0 is extended to covering the entire network with closed loops of directed pairings (Sec. A6).
Data Analysis and simulations What do the assembly rules teach us about real food webs? For seven detailed empirical food webs (Details: Materials and Methods) containing both free-living and parasite species, we determine the difference between the respective total number of species S and the maximum number of linearly independent rows, d ≡ S − rank(R) (Details: Methods). Linear dependence can be seen as the sharing of a specific niche by several species, therefore d measures the lack of niches in the given food web. When d = 0, the assembly rules are satisfied (det(R) 6= 0). As done previously(Dunne et al., 2013a), we distinguish webs formed by: links between free-living species only; links between all free-living and parasite species; with additional concomitant links (Methods). The free-living webs show substantial structure, an example is shown in Fig. 4a and others in Sec. A9. Many species are located at sharp trophic levels (Fig. A6), a feature that manifests itself by the blocks of white spaces, i.e. absent interactions, shown in Fig. 4a. The organization into trophic levels is associated with a lack of niches in most free-living webs, i.e. d > 0. Additional analysis shows that all free-living webs are in a consumer-limited state, i.e. species richness in even trophic levels dominates (Sec. A9). We contrast these findings with the cascade (Cohen and Newman, 1985) and niche models(Williams and Martinez, 2000). Using number of species and links from the empirical datasets, we generate network samples (Fig. 4b,c). The resulting interaction matrices are characterized by very little structure in terms of trophic levels (the white blocks are all but missing). When repeating the simulations for all seven webs (Sec. A9) and obtaining the corresponding rank deficiencies, we find that the modeled networks consistently give d ≈ 0 and show less structure than empirical data (Sec. A9). We further quantify the organization of species by the chain length distributions for the empirical and modeled
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networks, where much broader distributions are found for the models. Quantifying omnivory by the standard deviation of chain lengths (Lafferty and Morris, 1996), modeled networks consistently yield substantially higher fractions of omnivory. We now consider the webs involving parasites (Fig. 4d,e). At the edge of the panels we indicate by a colorcoding, where, in terms of trophic level, parasites enter and how the free-living species are re-organized. Notably, parasites predominantly enter at high trophic levels, the lower section (approximately levels one and two, red to green colors) remains nearly unaltered by the inclusion of parasites. Specifically, interactions of a given parasite generally involve several free-living species at multiple trophic levels (Sec. A9), acting to loosen the structure at the higher levels (n3 and n4 , compare Fig. 4a,d) and reaching agreement with the assembly rules (Sec. A6). In other words, some of the parasites can be seen as effectively acting as odd-level species, thereby relaxing the initially consumer-dominated free-living webs to a more balanced state. Concomitant links (Fig. 4e) cause further entanglement of trophic levels, open additional options for possible pairings and systematically increase sustainability of food webs. Concomitant links require additional consideration, as they generally are directed links where a parasite is consumed by its host’s predator, i.e. they denote a detrimental effect on the parasite population. A positive impact on the predator’s population may however not always result. We call such links asymmetric concomitant links. If the predator’s population does benefit from the consumption, such as observed in some studies (Lafferty and Morris, 1996; Johnson et al., 2010), we use the term symmetric concomitant link. Nonetheless, such directed links can lead to additional non-overlapping pairings, when a closed loop of directed links is formed, e.g. a triangle (loop of length three). In the empirical webs it is noticeable that inclusion of asymmetric concomitant links only rarely yields rank improvement. Investigating this further, we find that many food webs with parasite interactions already contain sufficient numbers of loops to allow pairings between parasites and free-living species. The limitation arises because a surplus of parasites exists. Each loop will involve at least two free-living species but only one parasite, making it impossible to find non-overlapping pairings for all parasites. For those webs, only the inclusing of symmetric concomitant links leads to an additional improvement of rank, since then each parasite can be paired with a single free-living species. In two webs, where
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loops are rare, even asymmetric concomitant links help improve the rank. Details and simulations are available in appendix sections A9.3.3 and A10.4. For all empirical food webs, we summarize the effect of the different link additions on rank deficiency d (Fig. 4f). The results show a general decrease of d as more parasite links are added. Notably, full rank is sometimes not achieved, even when all available concomitant links are included. These findings may serve as a hint of missing data, most notably in the Ythan Estuary food web, where overall link density is low and parasite-parasite links are completely absent (further details on individual food webs: Sec. A9.3.2 and A10.3). As mentioned above, existing food web models generally produce network structures lacking rank deficiency (d ≈ 0), even for the free-living webs. When instead starting from model networks with similar link density as the empirical webs but sharp trophic levels, we obtain d substantially larger than zero (Fig. 5). Adding species that each interact with species at multiple trophic levels (Fig. 5c), i.e. mimicking the addition of parasites, our modeled webs show systematic decrease of d as links are added (black line in Fig. 5g). When adding species that each interact with a single (Fig. 5d), or exclusively higher trophic levels (Fig. 5e), saturation at d > 0 occurs (blue and red lines in Fig. 5b), When simulating also parasite-parasite interaction, d is also found to decrease (Fig. 5f, Simulation details: Methods and Sec. A10). Overall, these simulations show that addition of random species without the feature of interaction with multiple trophic levels is not sufficient to explain removal of rank deficiency. We have further explored addition of omnivorous links to the free-living web. The rank deficiency d is reduced rapidly if the addition happens randomly for all trophic levels, but the reduction is limited if omnivorous links occur mostly at the trophic level 3 as in the real data (Sec. A10.2). We have also performed extensive simulations on the effect of concomitant predation, which further emphasizes the importance of parasite-parasite interaction in achieving coexistence for some webs (Sec. A10.4).
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DISCUSSION Our results generalize the competitive exclusion principle to food webs of any number of species and trophic levels. The assembly rules imply that sustainable food web structures are far from random and far from the structure assumed in the niche and cascade models. The requiment of the full rank of the foodweb interaction matrix expresses the simple notion that each species must occupy a unique niche and leads to biologically plausible combinations of species richnesses at the different trophic levels. The requirement, i.e. non-overlapping pairing or equivalently nonzero determinant, is simple and directly allows us to evaluate the self-consistency of empirical data. Indeed, some food web datasets do fulfill the requirements, and others, known the lack interactions, e.g. the Ythan Estuary web (Huxham et al., 1996), stand out as particularly far from reaching agreement with the rules. One immediate consequence for food webs with species predominantly organized according to trophic levels, e.g. many free-living webs, is that species richness at the basal and top-predator levels should be limited by the species richness of the respective neighboring levels (compare Eq. 6). This can explain the observations, both for terrestrial (Martinez, 1991) and marine food webs (Dunne et al., 2004), which report greatest species richness at intermediate trophic levels while top predators and basal species contribute little. Another consequence for such webs is that additions of species are generally not possible at any trophic level, if sustainable ecologies are to be achieved. Even when the addition satisfy the assembly rule, it might shift the system between consumer and resource limited states. In practice, it may be precisely the dramatic transitions between the consumer and resource limit that explain the profound and abrupt impacts on species biomass and energy flow patterns which are sometimes observed in the field. E.g. the introduction of opossum shrimp into a lake caused a cascade of trophic disruptions by reduction of salmon numbers and subsequent depletion of eagle and grizzly bear (Spencer et al., 1991). Conversely, our rules describe the circumstances, under which removal of a species must trigger additional extinctions. The assembly rules also allow predictions of secondary extinction, resulting either from addition or removal of species. If the modified food web obeys the assembly rules, the food web might potentially be stable. Indeed, in some observed cases, ecological release of new species into a habitat has had relatively gentle effects (Williamson 13
and Fitter, 1996). However, a violation of the assembly rules (Eq. 6) by addition of a new species can have either of two effects: Either the new species will not be competitive and collapse, or a number of species will collapse (possibly including the species itself) to restore the food web to a permitted state. For removal of a species that leads to violation of the assembly rules, secondary extinctions (Ekl¨of and Ebenman, 2006) must be triggered to re-gain a sustainable state. We find a consistent pattern, when considering species removal in empirical food webs: E.g. in consumer-limited webs, such as the free-living empirical webs, secondary extinctions are more likely triggered by removal of resource than consumer species (Sec. A9 and Fig. A11). Community omnivory (Fagan, 1997; McCann, 2000; Bascompte et al., 2005; Thompson et al., 2007) and parasitism (Hudson et al., 2006; Lafferty et al., 2008; Dunne et al., 2013a) have been suggested as contributing to food web stability. Our approach provides theoretical support for this claim. In larticular we found that the full rank of a foodweb interaction matrix is difficult to achive for species that are organized at strict trophic positions. Species that consume resources at different trophic positions, e.g. omnivores or some parasites, are shown to loosen the constraints and make it easier to comply with the assembly rules, i.e. finding a non-overlapping pairing of species.
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ACKNOWLEDGMENTS We thank I. Dodd and S. Semsey for fruitful discussions. The authors acknowledge financial support by the Danish National Research Foundation through the Center for Models of Life.
Competing Interest Statement The authors have no competing interests to declare.
References Stefano Allesina and Si Tang. Stability criteria for complex ecosystems. Nature, 483(7388):205–208, 2012. David A Andow. Vegetational diversity and arthropod population response. Annual review of entomology, 36(1): 561–586, 1991. Inge Armbrecht, Ivette Perfecto, and John Vandermeer. Enigmatic biodiversity correlations: ant diversity responds to diverse resources. Science, 304(5668):284–286, 2004. Jordi Bascompte, Carlos J Meli´an, and Enric Sala. Interaction strength combinations and the overfishing of a marine food web. Proceedings of the National Academy of Sciences of the United States of America, 102(15): 5443–5447, 2005. JE Cohen and CM Newman. A stochastic theory of community food webs: I. models and aggregated data. Proceedings of the Royal society of London. Series B. Biological sciences, 224(1237):421–448, 1985. Barbara Drossel and Alan J McKane. Modelling food webs. In S Bornholdt and Hans G Schuster, editors, Handbook of Graphs and Networks,, pages 218–247. Wiley-VCH GmbH & Co., 2003. Jennifer A Dunne, Richard J Williams, and Neo D Martinez. Network structure and robustness of marine food webs. Marine Ecology Progress Series, 273:291–302, 2004.
15
Jennifer A Dunne, Kevin D Lafferty, Andrew P Dobson, Ryan F Hechinger, Armand M Kuris, Neo D Martinez, John P McLaughlin, Kim N Mouritsen, Robert Poulin, Karsten Reise, et al. Parasites affect food web structure primarily through increased diversity and complexity. PLoS biology, 11(6):e1001579, 2013a. Jennifer A. Dunne et al. Data from: Parasites affect food web structure primarily through increased diversity and complexity. dryad digital repository. http://dx.doi.org/10.5061/dryad.b8r5c, 2013b. Anna Ekl¨of and BO Ebenman. Species loss and secondary extinctions in simple and complex model communities. Journal of animal ecology, 75(1):239–246, 2006. William F Fagan. Omnivory as a stabilizing feature of natural communities. The American Naturalist, 150(5): 554–567, 1997. C. O. Flores, S. Valverde, and J. S. Weitz. Multi scale structure and geographic drivers of cross-infection within marine bacteria and phages. The ISME journal, pages 1–13, 2012. GF Gause. The struggle for existence. Soil Science, 41(2):159, 1936. Jan O Haerter, Namiko Mitarai, and Kim Sneppen. Phage and bacteria support mutual diversity in a narrowing staircase of coexistence. The ISME journal, 2014. Garrett Hardin. The competitive exclusion principle. Science, 131(3409):1292–1297, 1960. Ryan F Hechinger, Kevin D Lafferty, John P McLaughlin, Brian L Fredensborg, Todd C Huspeni, Julio Lorda, Parwant K Sandhu, Jenny C Shaw, Mark E Torchin, Kathleen L Whitney, et al. Food webs including parasites, biomass, body sizes, and life stages for three california/baja california estuaries: Ecological archives e092-066. Ecology, 92(3):791–791, 2011. Robert D Holt. Predation, apparent competition, and the structure of prey communities. Theoretical population biology, 12(2):197–229, 1977.
16
Peter J Hudson, Andrew P Dobson, and Kevin D Lafferty. Is a healthy ecosystem one that is rich in parasites? Trends in Ecology & Evolution, 21(7):381–385, 2006. M Huxham, S Beaney, and D Raffaelli. Do parasites reduce the chances of triangulation in a real food web? Oikos, pages 284–300, 1996. Anthony R Ives and Stephen R Carpenter. Stability and diversity of ecosystems. science, 317(5834):58–62, 2007. Pieter TJ Johnson, Andrew Dobson, Kevin D Lafferty, David J Marcogliese, Jane Memmott, Sarah A Orlofske, Robert Poulin, and David W Thieltges. When parasites become prey: ecological and epidemiological significance of eating parasites. Trends in Ecology & Evolution, 25(6):362–371, 2010. Kevin D Lafferty and A Kimo Morris. Altered behavior of parasitized killifish increases susceptibility to predation by bird final hosts. Ecology, 77(5):1390–1397, 1996. Kevin D Lafferty, Andrew P Dobson, and Armand M Kuris. Parasites dominate food web links. Proceedings of the National Academy of Sciences, 103(30):11211–11216, 2006. Kevin D Lafferty, Stefano Allesina, Matias Arim, Cherie J Briggs, Giulio De Leo, Andrew P Dobson, Jennifer A Dunne, Pieter TJ Johnson, Armand M Kuris, David J Marcogliese, et al. Parasites in food webs: the ultimate missing links. Ecology letters, 11(6):533–546, 2008. Richard Law and R Daniel Morton. Permanence and the assembly of ecological communities. Ecology, pages 762–775, 1996. Simon A Levin. Community equilibria and stability, and an extension of the competitive exclusion principle. American Naturalist, pages 413–423, 1970. L´aszl´o Lov´asz and Michael D Plummer. Matching theory. New York, 1986. Robert H MacArthur. The theory of island biogeography, volume 1. Princeton University Press, 1967.
17
Neo D. Martinez. Artifacts or attributes? effects of resolution on the little rock lake food web. Ecol. Monographs, 61(4):367–392, 1991. Kevin Shear McCann. The diversity–stability debate. Nature, 405(6783):228–233, 2000. K. Moebus and H. Nattkemper. Taxonomic investigations of bacteriophage sensitive bacteria isolated from marine waters. Helgol/inder Meeresunters, 36:357–373, 1983. Jos´e M Montoya, Stuart L Pimm, and Ricard V Sol´e. Ecological networks and their fragility. Nature, 442(7100): 259–264, 2006. Kim N Mouritsen, Robert Poulin, John P McLaughlin, and David W Thieltges. Food web including metazoan parasites for an intertidal ecosystem in new zealand: Ecological archives e092-173. Ecology, 92(10):2006– 2006, 2011. William W Murdoch, Francis C Evans, and Charles H Peterson. Diversity and pattern in plants and insects. Ecology, pages 819–829, 1972. WM Post and SL Pimm. Community assembly and food web stability. Mathematical Biosciences, 64(2):169–192, 1983. Anthony Ricciardi and Joseph B Rasmussen. Extinction rates of north american freshwater fauna. Conservation Biology, 13(5):1220–1222, 1999. Christopher Sandom, Lars Dalby, Camilla Flojgaard, W Daniel Kissling, Jonathan Lenoir, Brody Sandel, et al. Mammal predator and prey species richness are strongly linked at macroscales. Ecology, 94(5):1112–1122, 2013. Craig N Spencer, B Riley McClelland, and Jack A Stanford. Shrimp stocking, salmon collapse, and eagle displacement. BioScience, pages 14–21, 1991.
18
David W Thieltges, Karsten Reise, Kim N Mouritsen, John P McLaughlin, and Robert Poulin. Food web including metazoan parasites for a tidal basin in germany and denmark: Ecological archives e092-172. Ecology, 92(10): 2005–2005, 2011. Ross M Thompson, Martin Hemberg, Brian M Starzomski, and Jonathan B Shurin. Trophic levels and trophic tangles: the prevalence of omnivory in real food webs. Ecology, 88(3):612–617, 2007. David Tilman. Competition and biodiversity in spatially structured habitats. Ecology, 75(1):2–16, 1994. Richard J Williams and Neo D Martinez. Simple rules yield complex food webs. Nature, 404(6774):180–183, 2000. Mark Williamson and Alastair Fitter. The varying success of invaders. Ecology, 77(6):1661–1666, 1996. C Dieter Zander, Neri Josten, Kim C Detloff, Robert Poulin, John P McLaughlin, and David W Thieltges. Food web including metazoan parasites for a brackish shallow water ecosystem in germany and denmark: Ecological archives e092-174. Ecology, 92(10):2007–2007, 2011.
19
a
top predators
basal species
n6
n4
n2
...
n3 ...
Trophic Levels
n5
n3 n2
n1
n1 nutrients
n6
n5
n4
n3
n2
n1
Trophic Levels
b
single niche competitive exclusion
two niches possible coexistence
Figure 1: Food web interaction matrix and application of perfect matching. a, White (gray) boxes indicate nonzero (zero) matrix elements, orange boxes are unity matrix elements for the primary producers; dark and light blue squares indicate a possible path chosen, allowing det(R) to be nonzero. Here, No = n1 + n3 + n5 = 13 and Ne = n2 + n4 + n6 = 12, and Eq. 6 is fulfilled with ∆ = 1. Inset: Schematic of a possible pairing for the chosen path. Note that the invariance property of det(R) was used, yielding only n1 non-vanishing matrix elements in the lower right block (Details: Appendix). b, Perfect matching(Lov´asz and Plummer, 1986) applied to simple food webs where competitive exclusion rules out coexistence due to lack of niches (left) and where enough niches are available for coexistence (right).
20
a
n1=n2
b
c
Numbers of species in trophic level
n1 0
n2=n1+n3 n1 0 n2+n4=n1+n3 n1+n3≥n2≥n1 n2+n4≥n3≥n4
n1 0 1
2 3 4 Trophic level
Figure 2: Species richness for different trophic levels. a, Food web with two trophic levels only; a staircase of coexistence with balanced species richness at levels 1 and 2 (Haerter et al., 2014). b, Three trophic levels. The number of intermediate species must equal the total number of basal and predator species. Intermediate species dominate ecosystem biodiversity. c, Four trophic levels. n2 (n3 ) must at least match basal (predator) species richness n1 (n4 ), indicated by thin black lines in green and orange bar. Solid green (orange) bars show the minimal upper bound to species richness in trophic level one (two). Species richness n2 and n3 can increase even further by co-evolution of intermediate species (shaded region). Note the applicable assembly rules shown for the different cases.
21
a
n4
n3
n2
x
n3
x
x x x x x
x x
n2
x x x x 1 1
n1
b
n1
x
n4
(1,1)
(2,0)
(3,1)
(4,0)
(5,1)
(6,0)
(7,1)
(8,0)
c NG
n4
n3
NG
n2
n1
x x
n4
x x
x
n3
x x
x x x
n2 n 2
x x x x x x x x 1 1
n1
Figure 3: Food web assembly. a, Consumer limited food web and its interaction matrix. The symbols “×” mark nonzero entries and circles a path through the matrix. Basic nutrient is shown as a gray hexagon, whereas species on subsequent trophic levels are shown with coloured circles. Small circles highlight limitations by consumers. Shaded ovals indicate possible pairing of species. b, Possible assembly of the food web in (a) with labels for No +Ne and No −Ne . Note the transitions between biomass limiting states. c, Food web with a single generalist omnivore where we only show its paired link.
22
a
b
c
(df=16)
e (df+p=3)
(dc,sym=0)
f
BF CA PB FF OH ST YE
20
Lack of Niches (d)
d
10
0
Free
Par ParCon ParCon asym sym
Figure 4: Comparison to existing models and data analysis. a, Bahia Falsa free-living food web, with blue and red matrix elements for predator respectively prey dependency. Trophic levels indicated by gray and black bars, whereas color coding along the left and upper edge labels chain length of free-living species (increase from red to blue shades). b, Niche model simulation of the Bahia Falsa free-living food web. c, Cascade model simulation of the Bahia Falsa free-living food web. d, As (a) but with parasites (“Par”) and with colors for free-living species carried over from (a). Remaining species are parasites. e, As (d) but additionally including symmetric concomitant links (“ParCon sym”). f, Lack of niches (d) for seven empirical food webs (Dunne et al., 2013a). Labels mark the sub-webs of free-living species (“Free“), including also parasite links (”Par“), asymmetric (”ParCon asym“) and several symmetric concomitant links (”ParCon sym“). Dashed line connects averages in these categories. Rank deficiencies for free living (df ), free-living and parasite species (df +p ) as well as additional concomitant links (dc,sym ) marked in (a), (d), and (e), respectively. (Analysis details and abbreviations: see Methods).
23
Number of Species
a
b
empirical webs
40 20 0
n4 n3
generic web
40 20 0
1
2
3
4
n2 n1
5
Approxim ate T rophic Level
d
e
f
g
n3
n4
c
n2
n1
Lack of Niches
40
e
20
d c 0 0
f 10 L p / Sp
Figure 5: Simulations of different food web matrices. a, Barplots indicate distributions of node richness for each approximate trophic level in the seven empirical foodwebs and a generic foodweb derived by averaging the empirical node richnesses in each trophic level. b, Interaction matrix corresponding to the generic food web, containing 110 free-living and 47 parasite species (Details: Methods and Sec. A10). c, Addition of parasites that form random links to any existing free-living species. d, Addition of parasites that are confined to consumer at a specific trophic level. e, Similar to (c) but with the restriction of parasites consuming only free-living species at levels 3 and 4. f, Similar to (e) but with additional parasite-parasite interactions (hyperparasitism), approximately 5 percent of parasite links are from parasite to parasite. Note the color coding along the edges of the matrices in (c)—(f), chosen as in Fig. 4a,d,e. g, The lack of niches, i.e. rank deficiency, as a function of the number of links per perasite for each of the four cases described in (c)—(f).
24
Appendix 1.
February 12, 2015
Abstract This appendix provides additional technical detail which may be interesting to specialist readers as well as introductory examples for non-specialists, but is not required for the understanding of the main text. We provide more detail on the derivation of our main equations (Sec. A1) and their steady state (Sec. A2). We give a detailed justification of the terms resource-limited and consumer-limited state as used in the main text (Sec. A3), a simple example demonstrating the coexistence between two consumers and two producers (Sec. A4), as well as a brief summary of the assembly rules (Sec. A5). We offer in-depth discussion of food webs containing omnivores (Sec. A6) and the case of food webs where basal species feed on a set of separate nutrients (Sec. A7). We also discuss a modification of the assembly rules for the case of the type-II functional response (Sec. A8). Details on the data analysis performed with empirical food webs, including a discussion on concomitant predation, are given in Sec. A9. Simulation results for a model food web are provided in Sec. A10.
1
Contents A1
Food web ecology
4
A2
Steady state
5
A2.1 Steady state equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
A2.2 Feasible solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
A3
Resource versus consumer limitation
6
A4
Species definition and example of two shared niches
9
A5
Assembly rules for a food web with strict trophic levels
11
A6
Omnivory and parasitism
12
A6.1 Example of omnivory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
A6.2 Generalist omnivory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
A6.3 Pairing requirements for food-webs with omnivores . . . . . . . . . . . . . . . . . . . . .
15
A7
Basal species with separate nutrients
16
A8
Species link requirement to include predator saturation
17
A9
A8.1 Model equation with type-II response . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
A8.2 Constraints from steady state condition and extended species pairing . . . . . . . . . . .
18
A8.2.1 Top predators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
A8.2.2 Intermediate species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
A8.2.3 Basal species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
A8.2.4 Summary of the extended species pairing requirement . . . . . . . . . . . . . . . .
20
Data Analysis
21
A9.1 Data considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
A9.1.1 Food web data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
A9.1.2 Technical considerations on parasites . . . . . . . . . . . . . . . . . . . . . . . . . .
21
A9.1.3 Matrix rank, basal species and nutrient sources . . . . . . . . . . . . . . . . . . . .
22
A9.1.4 Treatment of detritus and particulate organic matter (POM) in the data . . . . . .
22
2
A9.1.5 Approximate trophic levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
A9.2 Analysis of food web matrices of free-living species . . . . . . . . . . . . . . . . . . . . .
23
A9.2.1 Trophic levels and omnivory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
A9.2.2 Trophic structure of food web matrices . . . . . . . . . . . . . . . . . . . . . . . . .
24
A9.2.3 Discussion of individual matrices (details) . . . . . . . . . . . . . . . . . . . . . . .
25
A9.3 Analysis of food web matrices including parasite species . . . . . . . . . . . . . . . . . .
27
A9.3.1 Trophic positions with parasites . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
A9.3.2 Discussion of individual food webs (details) . . . . . . . . . . . . . . . . . . . . . .
27
A9.3.3 Concomitant predation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
A9.4 Removal of species from food web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
A10
Modeling
33
A10.1 Idealized food web structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
A10.2 Simulations of omnivore addition within the free-living web . . . . . . . . . . . . . . . .
35
A10.3 Simulations of parasite addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
A10.4 Simulations of the effect of concomitant predation . . . . . . . . . . . . . . . . . . . . . .
39
3
A1
Food web ecology
To model food web ecology, we employ the widely-used, generalized Lotka-Volterra equations (MacArthur, 1967) which assume a well-mixed system, i.e. a system where spatial fluctuations are negligible. The links in a food web are descriptions of predominantly predator-prey relationships between species on neighboring trophic levels (Montoya et al., 2006) but could also constitute mutualist interactions (Bascompte et al., 2005). The lowest trophic level of the food chain, primary producers such as plants or microbes, draw energy from the sun and elementary chemical compounds. We define a distinct species as an organism of distinguishable phenotype. We first separate species with regard to their trophic levels and direct interactions between species occur only between those on neighboring trophic levels. To model the food chain, we require two different types of equations. For n1 primary producers with (population or mass) densities Si (1) , i = 1, . . . , n1 , residing at the first trophic level, the general form of the time-evolution equations is (MacArthur, 1967; Gross et al., 2009)
1
(1) dSi
(1) Si
dt
(1) = ki 1 −
n1 X
(1) (1) pji Sj − αi −
j=1
n2 X
(2,1)
ηki
(2)
Sk .
(A1)
k=1
In Eq. A1, competition between the primary producers for basic resources is described by a normalized (1)
overall carrying capacity in the first term on the RHS (Smith and Slatkin, 1973). ki
(1)
(αi ) denote the
(1)
(1)
growth (decay) rates of Si , pji describe differential consumption of the basic resources by the Si (2,1)
ηki
(2)
are the interaction coefficients between a species Sk (2,1)
trophic level. For positive ηki (1)
(2,1)
Si . For negative ηki
(2)
the consumer Sk
(1)
at the second with species Si
and
at the first
yields a net increase of decay rate of basal species (1)
(weak mutualism), a positive effect on the growth rate of Si
results. Primary
producers compete for resources when pji > 0 for i 6= j, but the equation also allows for the description of the case of direct competition, cooperation, as well as growth on non-overlapping resources (pji = δji ). (l)
For species Sk at any trophic level l > 1 we write
k
X
=
X
nl+1
nl−1
(l)
1 dSk (l) dt S
l,l−1 βkm
·
l,l−1 ηkm
·
(l−1) Sm
−
(l)
l+1,l ηpk · Sp(l+1) − αk ,
(A2)
p=1
m=1
(l)
(l−1)
l,l−1 where βkm are the efficiencies of reproduction of species Sk when consuming species Sm
and the
other coefficients are defined analogously to those in Eq. A1. For top predators, residing on the L’th L+1,L trophic level, there are no enemies, hence the coefficients ηpk will vanish. Recycling of the top (L)
predators’ biomass will then only occur through their natural death rates αk . For a species’ phenotype to be distinguishable, it must be characterized by a unique combination of the parameters. In the above equations, all parameters and observables have been made dimensionless by normalizing population
4
densities to the primary producer carrying capacity, growth and decay rates to the maximal primary producer growth rate, and interaction rates to the ratio of maximal growth rate and carrying capacity. Note that Eqs A1 and A2 differ by the respective consumption of resources: Primary producers consume supplies of physical or chemical energy which are replenished at a constant flux. Species at any higher trophic level depend on the availability of prey at the neighboring trophic level below.
A2
Steady state
The ecological dynamics are taken to occur without rapid evolution, i.e. we are considering systems where species richness is not a consequence of transient rise and decay of species.
A2.1
Steady state equations
To describe sustainable food webs, we demand the time-derivatives on the LHS of Eqs A1 and A2 to vanish and arrive at a set of equations that are linear in the densities of any of the species:
n1 X j=1
X
(1)
pji Sj
+
n2 (2,1) X η
ki (1) k k=1 i
(1)
(2)
Sk =
nl−1
X
ki
(1)
− αi
(1) ki
(1) ≡ k˜i
(A3)
nl+1 (l,l−1) (l,l−1) (l−1) ηkm Sm
βkm
m=1
−
(l+1,l)
ηpk
Sp(l+1)
(l)
≡ αk .
(A4)
p=1
Collecting all constant coefficient (LHS of Eqs A3 and A4) in the vector k and all interaction coefficients on the RHS in the interaction matrix R, we have the linear matrix equation R · S = k, where S is the vector of all species densities. For completely shared nutrients, the competition factors pji = 1. In the opposite extreme, where each basal species draws energy from a distinct nutrient source, pji = δji with δji the Kronecker delta function. When re-organizing the terms in the sub-matrix of interactions between species on the basal trophic level, the square matrix A1 is obtained by subtracting rows 2 to n1 from row 1. After this linear transformation we yield the modified matrix R0 , shown in Fig. 1a (main text). Apart from the first row, R0 has a block structure with nonzero entries only for interactions between neighboring trophic levels (Fig. 1a, main text). Steady-state solutions are then given by S∗ ≡ R−1 k, thus a unique solution S∗ requires that R be invertible, hence det(R) 6= 0 (Otto and Day, 2007). In the following we first consider the case of shared nutrients (pji = 1) but in Sec. A7 also separate resources are discussed.
A2.2
Feasible solutions
The condition det(R) 6= 0 is a necessary condition for nonzero population densities S∗ . In food webs, all densities should however be positive, i.e. feasible. We here outline, how to obtain feasible solutions 5
for all population densities, once a set {n1 , . . . , nL } of diversities — satisfying the assembly rules — has been obtained. For simplicity, we take the biomass within a given trophic level to be distributed equally among all species in that level and a species’ connectivity to be proportional to the number of possible links to the neighboring trophic level, i.e. the number of species at trophic level 1 preyed upon by a (l,l−1)
specific species at trophic level 2 will be proportional to n1 . We take all coefficients ηkm (l,l−1) βkm
≡ η and
≡ β to be equal.
Using these coefficients, we re-write Eq. A4 and obtain X
nl−1
β
X
nl+1 (l−1) Sm
m=1
−
(l)
Sp(l+1) =
p=1
αk >0, η
(A5)
(l)
where the RHS is required to be a positive constant. We here let Sm denote the biomass of the corresponding species. The two sums on the LHS are then just the total biomass S (l−1) , respectively S (l+1) , at trophic levels l − 1, respectively l + 1, we have a condition on β:
β>
S (l+1) . S (l−1)
(A6)
Realistic food webs show trophic biomass pyramids (Hafner, 1978), with biomass decreasing by approximately a factor of 10 when moving up to a higher trophic level. We therefore assume total biomass in each trophic level l to be a factor 10 less than that at level l − 1. β is then constrained to be larger than βmin = 0.01, hence the efficiency of food uptake must be more than one percent. Eq. A5 gives one possible construction for feasible solutions. Naturally, the values of η will vary as does the probability of links forming between different species. Also, decay rates α will not be arbitrarily close to zero. All these variations will likely force β to be substantially larger than the limit of βmin . Lastly, β is generally species dependent and will have variations as well. Hence, there will be other ways, many of them more realistic then the idealized one presented here, to achieve feasible solutions for the population densities S ∗ .
A3
Resource versus consumer limitation
Trophic cascades are changes in the population densities of consumers and resources in food chains when a top predator is added or removed, but can also be triggered by changes in nutrient availability (Heath et al., 2014). Exemplified by the growth of a simple food chain — starting with a single species — we discuss the alternating concentration of biomass on odd, respectively even, trophic levels. (1)
A freely growing basal species, S1 (1)
S1
(1)
according to Eq. A3 will reach the steady state population density
(1)
= 1−α1 /k1 and can reach the system carrying capacity (here set to unity) by increasing its growth (1)
rate k1
(1)
relative to its decay rate α1 . Consider now a simple ecosystem consisting of only two trophic 6
levels with a single species on each level, i.e. a basal and a consumer species (Fig. A1). Eqs A3 and A4 then become (2,1)
(1)
1−
α1
(1)
= S1 +
(1)
k1
(2)
η11
(2)
(1)
k1
(2,1) (2,1)
α1
S1 (1)
= β11 η11 S1
(A7)
.
(A8)
Hence, the basal species population density (2)
(1)
S1
α1
=
(A9)
(2,1) (2,1)
β11 η11
(1)
is entirely determined by the trophic interaction and the decay rate of its consumer and S1
will not
primarily be limited by the carrying capacity of the system. In its simplest form this exemplifies what is meant by the transition from the resource to the consumer-limited state.
(1,1)
(2,0)
(3,1)
(4,0)
Figure A1: Simple food web. Food web containing only a single species at each trophic level. Schematic shows re-distribution of biomass upon addition of trophic levels. Large (small) circles indicate population with large (small) biomass. Labels show the total number of species and the difference No − Ne . (3)
Continuing further, we consider a single species S1 (1)
1−
α1
(1) k1 (2) α1 (3)
α1 (2)
Hence, S1 (3) α1 .
(3)
(3,2) (3,2)
= α1 /β11 η11
on the third trophic level and obtain the solution
(2,1)
(1)
η11
(2) S1 (1) k1 (2,1) (2,1) (1) β11 η11 S1 −
= S1 + =
(3,2) (3,2)
(2)
= β11 η11 S1
(A10) (3,2)
(3)
η11 S1
(A11)
.
(A12) (3)
is now entirely specified by the interaction with S1
and the decay rate
In turn, the basal species density (1)
(1) S1
=1−
α1
(1)
1+
k1
(3)
(2,1)
1
(1)
(3,2)
β11
α1 η11 α1 η11
(3,2)
! (A13)
again can approach the carrying capacity by increasing its growth rate relative the the decay rate. Note 7
that Eq. A13 describes a reduction of the producer species both due to its own death rate and the death of its indirect support on level 3. A similar analysis of the 4-level system reveals that in that case the population of the producer increases from its previous level (Eq. A9) to a larger level because its predator (the species at level 2) has to “carry“ the death of its support (level 4), i.e. (1) S1
=
1 (2,1) (2,1)
β11 η11
(3,2)
(2) α1
+
η11
!
1
(4) α (4,3) (4,3) 1 η11 β11
.
(A14)
It is easy to continue the sequence by adding trophic levels with single species and the alternation between resource and consumer limit will be obeyed. More generally, with species diversities {n1 , . . . , nL } on the L trophic levels, in the consumer-limited case, the decay rates for consumers and their respective interactions with producer species will set the population densities of the No = Ne resource species. In the resource limited state, one degree of freedom for basal species will be preserved. In principle, sustainable networks could then be formed where one basal species again approaches the system carrying capacity by optimizing its own growth and decay rates, i. e. this basal species could be freed from exploitation by consumers. Conversely, in a sustainable consumer-limited state, all species belonging to the resource support must be exploited by consumers. None of the resource species can — even in principle — be freed. This can be made more explicit: Consider a system of only two trophic levels with diversities n1 and n2 . If n2 = n1 , the n2 equations (A4) define n2 weighted sums of the n1 = n2 species at trophic level 1. Pn1 (1) Sm of total population density It is then always possible to yield an expression for the sum S (1) ≡ m=1 (l)
on the basal trophic level which depends only on the decay rates αk of and interaction coefficients with species on the second trophic level. Hence, the population density S (1) will not depend on the growth or decay rates of species on the basal level. In other words, species fitness does not improve by increasing its growth rate, but remains solely governed by the species’ predators (Thingstad, 2000). Conversely, if n2 = n1 − 1, only the sum of n1 − 1 species on level 1 can be expressed in terms of the respective decay and interaction coefficients. The remaining degree of freedom will then be fixed by Eq. A3 where the carrying capacity will be felt by the basal species. Generalizing to L trophic levels, n2 − n1 is the number of degrees of freedom on level 1 controllable by level 2. However, as n3 constraints can be imposed by level 3 onto level 2, the net limitation is then n2 − n1 − n3 . This sequence will continue analogously until nL is reached and we are left with the familiar conditions No − Ne ∈ {0, 1}, a difference that defines the number of degrees of freedom remaining for the basal level.
8
A4
Species definition and example of two shared niches
A trophic species is defined as all species sharing a set of predators and prey. An example is shown in Fig. A2a and b. The strength of the links connecting the respective consumers and resources is thereby not relevant, as long as the link exists. I.e., one consumer may predominantly feed on one prey while the other consumer predominantly feeds on the other prey (Fig. A2c).
a
b
c trophic species
differential interaction
Figure A2: Food web with two niches. a, Example of food web with two basal and two consumer species. Basal species share nutrients and consumer species and can be combined to a single trophic species. Consumer species share resources and can also be combined to a single trophic species. b, Food web with species aggregated into trophic species. c, As in (a) but showing differential predation rates.
In our approach, we also distinguish species by the interaction strength with other species. For example, when two species feed on the same two prey species, but they do so at differential rates, we will still consider them distinguishable species. The reasoning behind this is twofold: First, for the competitive exclusion principle to be relevant, we need to be able to discuss ”perfect“ competitors, i.e. those that share a resource. Aggregating the two consumers into a single trophic species would immediately remove this situation, the discussion of competitive exclusion would hence be meaningless from the outset. Second, to discuss how the competitive exclusion principle can be generalized, we do need to discuss situations where species, despite sharing prey and predator species, may still coexist. This is at the core of our discussion. In the specific case of Fig. A2a, by allowing complementary interaction rates, the two species occupy distinct niches, even though they principally share both prey species. Coexistence between two competing species can be rationalized as a form of trade-off (Tilman, 1990), where one species may be better at preying on one resource than the other but thereby inevitably sacrificing efficiency in preying on the second resource species. To make this more explicit, we now discuss in detail the example shown in (Fig. A2c). In terms of its steady state condition:
9
(2,1)
(1)
(1)
S1 + S2 +
η11
(2,1)
(2)
S1 +
(1)
k1
(2,1)
(1)
(1)
S1 + S2 +
η12
(1)
k2
η21
(1)
k1
(2)
(1) = k˜1
(A15)
(2)
(1) = k˜2
(A16)
(2)
(A17)
(2)
(A18)
S2
(2,1)
(2)
S1 +
η22
(1)
k2
S2
(2,1) (2,1)
(1)
(2,1) (2,1)
(1)
= α1
(2,1) (2,1)
(1)
(2,1) (2,1)
(1)
= α2
β11 η11 S1 + β12 η12 S2
β21 η21 S1 + β22 η22 S2 (2,1)
To simplify the discussion, we now set all βij (2,1)
where η12
(2,1)
= η21
(2,1)
≡ η× and η11
(2,1)
= η22
(l)
≡ β, αi ≡ α and ask only for symmetric solutions
≡ η|| . Note that we have also dropped the superscripts in
these coefficients. Using these simplifications, we arrive at solutions (1)
S1
(1)
=
(2)
=
(2)
=
= S2 S1
S2
1 α (A19) β η|| + η× η|| 1 α η× η|| (k1 − α) − η× (k2 − α) + 2 k2 − k1 2 2 η|| − η× β η|| + η× η|| + η× η|| α η× 1 η (k − α) − η (k − α) + 2 k − k , × 1 1 2 || 2 2 η||2 − η× β η|| + η× η|| + η×
where the first equation should be compared to Eq. A9. From these solutions we see that the basal species (1)
are now completely dependent on the interaction rates with the consumer species. S1
(1)
and S2
become
inversely proportional to the efficiency of predation, β, and inversely proportional to the sum of contact (2)
rates with the predatory species. From the densities of the consumer species, S1
(2)
and S2
we see that
they become singular as the difference between (squared) link strengths vanishes, i.e. η|| − η× → 0. This corresponds to the case of the competitive exclusion principle, when generalizing to a niche consisting of two basal species. Equal link strength is equivalent to perfect competition between the two consumer species. The singularity could only be lifted by choosing k1 = k2 , a combined situation where we would consider the basal species to be identical. Conversely, solutions do exist when link strengths are not equal, e.g. η|| η×. In that case, feasible solutions imply inequalities between the parameters, e.g. α−1 > k1−1 + (η|| β/2)−1 , respectively α−1 > k2−1 + (η|| β/2)−1 .
(A20)
This is to say that the ”free survival time“, α−1 must be sufficiently long to outweigh the production time −1 k1/2 and effective ”processing time“ (η|| β/2)−1 . Solutions with a component resulting from η× would
yield contributions to the consumers densities from both basal species (compare Eq. A20). The above example, with each consumer species developing a weak and a strong link to a resource species, could in practice be generated by functional relations between interaction parameters, known as
10
trade-offs Tilman (1990). Infinite solutions: We return to Eqs A17 and A18. Consider the case where (2,1) (2,1)
(2,1) (2,1)
β11 η11
(2,1) (2,1)
=
β21 η21
β12 η12
(2,1) (2,1)
(2)
=
β22 η22
α1
(2)
.
(A21)
α2
This choice of parameters would make Eqs A17 and A18 equivalent and thus allow infinite solutions (1)
for S1
(1)
and S2 . While mathematically possible, complete equivalence of parameters is biologically
exceedingly unlikely. In natural systems, even a small imbalance in the magnitude of parameters would allow one of the species to become dominant. Further, it has previously been shown that the case of very similar parameter values leads to very small probabilities of coexistence (Haerter and Sneppen, 2012), simply because the population redistribution between the two species would perform a random walk, until one is exposed to the absorbing state of zero population.
A5
Assembly rules for a food web with strict trophic levels
The assembly rules state the minimal conditions for a sustainable steady state of a food web, which in turn requires that the determinant of its interaction matrix R is nonzero. Computationally, this can be examined by assigning random values for all nonzero entries, and subsequently calculating the rank of R. If the rank r is equal to the total number of species S, then the known interactions are sufficient to explain the sustainability of the food web. If the rank is less than S, the rank deficiency d ≡ S − r measures a minimal bound for the number of additional links needed to sustain the food web. The determinant-equal-zero rule can be reformulated in terms of the network structure. We here consider the case of fully bi-directional networks without species with varying food chain lengths, such as omnivores (i.e. all species must have strict trophic levels). The assembly rules can then be summarized as:
1) Food webs without omnivores require perfect matching to be sustainable. Perfect matching is the ability to cover a non-directed network with non-overlapping pairings (Lov´asz and Plummer, 1986). The determinant for symmetric networks can only be nonzero if there is perfect matching. In a food web, perfect matching is said to be obtained when all species participate in pairings, and each nutrient source can, but need not, participate in a pairing. The perfect matching rule will later be generalized when we consider omnivores or parasites (Sec. A6.3) and their possible concomitant links (Sec. A9.3.3).
2) The assembly rules imply general constraints for sustainability of purely trophic food webs:
k X l=1
nl (−1)k+l+1
≤ (−1)k
∀
k ∈ [1, L] , if No − Ne = 1
≤ 0
∀
k ∈ [1, L] , if No − Ne = 0
11
(A22)
and No − Ne ∈ {0, 1} ,
(A23)
for the case of one common resource. Here No and Ne are again defined as the total species richness at all odd, respectively even, trophic levels. Eq. A22 does justice to the constraints imposed by the species richness at distinct trophic levels, as discussed in the main text. Violation of these constraints can e.g. by quantified by the difference Ne − No , which expresses to which extent the food web is in the consumer-dominated state. Ne − No can be used to estimate the minimal rank deficiency of a given set of data for a food web with sharp trophic levels and an excess number of consumers.
A6
1
Omnivory and parasitism
Omnivores (Fagan, 1997; Thompson et al., 2007) or some parasites (McCann, 2000; Bascompte et al., 2005; Lafferty et al., 2008; Dunne et al., 2013) or pathogens (Mordecai, 2011) that prey on multiple trophic levels, have been suggested to play a moderating role in food webs. When disregarding concomitant links (Secs A9.3.3, A10.4), parasites, despite their complex life-cycles (Huxham et al., 1996), can be seen to formally act as omnivores in our steady-state equations, when considering a given parasite species with its population as a whole. In our analysis, even the sign of interaction would not matter, i.e. also mutualistic interactions with species at distinct trophic levels, or a combination of different interaction types (e.g. antagonistic towards one species at one trophic level and mutualistic to another species at a distinct level) would be possible. In terms of the assembly rules, omnivores whose prey includes species on a given trophic level l can be interpreted as trophic species on the level l + 1. Specifically, consider an omnivore preying on all species at the Lg trophic levels {i1 , i2 , . . . , iLg }. This omnivore can then be interpreted as a trophic consumer residing on any of the trophic levels {i1 + 1, i2 + 1, . . . , iLg + 1}. Consider now the introduction of this omnivore into an existing L-level food web containing only nearest-level trophic interaction, i.e. omnivores are absent prior to the addition. The L diversities are then given by {n1 , . . . , nL }. The result of introducing the omnivore is, that the assembly rules now allow the redefinition of any of the existing diversities at the corresponding trophic levels, i.e. ni1 +1 can now be redefined to become n ˜ i∗ +1 ≡ ni∗ +1 + 1 ,
(A24)
where i∗ ∈ {i1 , i2 , . . . , iLg }. Note that for any single omnivore this replacement can only be made for a single selection of the Lg trophic diversities {i1 , i2 , . . . , iLg }. This means, once the omnivore has been associated with a given trophic level, it can no longer be re-interpreted as adding to species richness at 1 Note: For the case of excess resource species, minimal rank deficiency could analogously be expressed by the term No − Ne − ∆, with ∆ chosen as the positive integer less than or equal to the number of distinct nutrient sources nS , so that the term is minimized.
12
another. That said, the previous assembly rules now still hold when the diversities ni are replaced by diversities n ˜ i , for i ∈ {1, . . . , L}. Consider the introduction of an omnivore into an existing food web, which previously obeyed the assembly rules. Using the guideline defined by Eq. A24, the omnivore may or may not lead to violation of the resulting set of assembly rules given by the n ˜ i . In the resource-limited case, the requirement then becomes that interpretation of the omnivore as a consumer is possible, i.e. it must prey on one of the resource species levels (odd levels). Further, any of the inequalities in Eq. A22 must still hold when replacing with n ˜i. This would e.g. be impossible if the added species richness were to occur at a level already at the limit of the allowed number, e.g. when adding another top predator to a system where the number of top predators equals that of the bordering trophic level, nL = nL−1 .
A6.1
Example of omnivory
Generally, multiple omnivores may be present in a given food web. We exemplify the effect of four omnivores (ng = 4) on a possible configuration of diversities in a food web consisting of seven trophic levels (Fig. A3). For simplicity, in the figure we consider all omnivores to prey on the same three trophic levels, 1, 4, and 6. Note also, that under absence of the omnivores, the food web presented in the figure violates the assembly rules, since n2 ≥ n1 − 1 is not obeyed.
By interpreting one of the omnivores as a consumer of species at trophic level 1, the replacement n ˜ 2 ≡ n2 + 1 can be made. This modification allows the condition to be met, and we yield n ˜ 2 = n1 − 1. The subsequent condition 4 = n3 ≥ n ˜ 2 − n1 + 1 = 0 is trivially fulfilled and 7 = n4 ≥ n1 − 1 − n ˜ 2 + n3 = 4 is also fulfilled in this example. However, at the next level, 2 = n5 ≥ n4 − n3 + n2 − n1 + 1 = 3 is violated. Again, an omnivore can be interpreted to feed on a convenient trophic level, namely n4 , and thus allows the interpretation of this omnivore as a consumer at the level 5. This leads to the redefinition n ˜ 5 ≡ n5 + 1 and the assembly rule is now obeyed. The final two assembly rules involving n6 and n7 prove to be fulfilled. At this stage of the analysis, two of the omnivores have been re-interpreted as trophic species allowing all assembly rules to be obeyed. However, the two remaining omnivores must also be accommodated without violation of the assembly rules. In the given example, this is possible by interpreting again one to reside at level 2, the other at level 5, all inequalities are still obeyed. Note that the possibility of accommodating multiple omnivores into an existing food web that is compatible with the assembly rules, is only possible when they allow interpretation as species on both even and odd trophic levels. This is the case for the example in Fig. A3, where the omnivores prey on levels one and four.
13
Ng
n7
Trophic Levels
n6 n5
n4
n3 n2 n1 Ng
n7
n6
n5
n4
n3
n2
n1
Trophic Levels
Figure A3: Inclusion of omnivory. Interaction matrix for seven trophic levels with respective species diversities n1 , . . . n7 . Ng specifies the number of omnivore species preying on more than one trophic level, here n6 , n4 , and n1 . The schematic shows that n2 and n5 do not fulfill the standard assembly rules but that the species richness at these levels can be supplemented by the omnivore count, i.e. n˜2 ≡ n2 + 1, and n˜5 ≡ n5 + 1, yielding N˜g ≡ Ng − 2. The row and column of addition of these two omnivores is indicated by arrows. The two remaining omnivores can further be accommodated by interpreting one as further contributing to n˜2 , the other as contributing to n˜5 .
A6.2
Generalist omnivory
The previous discussion shows that the presence of an omnivore allows more flexibility in interpreting the diversities ni in the assembly rules (Eq. A22). The presence of an omnivore can hence allow for fulfillment of the assembly rules when a trophic species is added or removed, which would otherwise lead to a violation of the rules. The situation is most transparent for a generalist omnivore, which we define as a species that preys on all existing trophic levels. This type of omnivore allows greatest flexibility: (i) Incorporation into an existing food web is always compatible with the assembly rules as the generalist omnivore can either be interpreted as adding to No or Ne (which would then lead to an increase of No ˜o , respectively N ˜e ). (ii) Upon addition of or Ne by up to the number of omnivores, yielding a modified N a new species into a food web encompassing the generalist omnivore, a possible violation of the assembly rules by the new species can often be circumvented by re-interpreting the generalist omnivore as a species residing on a convenient trophic level. Specifically, the assembly rules under presence of NG generalist omnivores (subscript capital ”G“) in addition to the food web species richness described in the main text, are:
14
k X l=1
n ˜ l (−1)k+l+1
≤ (−1)k
∀
˜o − N ˜e = 1 k ∈ [1, L] , if N
≤ 0
∀
˜o − N ˜e = 0 , k ∈ [1, L] , if N
(A25)
PL ˜o − N ˜e ∈ {0, 1}, n where N ˜ l ≡ nl + Ωl , Ω ≡ l=1 Ωl ≤ NG and Ωl ≥ 0 for all l ∈ {0, . . . , L}. This means that either of the NG ≥ Ω generalist omnivores can — but need not — be interpreted as an additional trophic species residing on any of the existing trophic levels.
A6.3
Pairing requirements for food-webs with omnivores
In terms of the pairing presented in Sec. A5, omnivores can allow additional pairing options as they prey on multiple trophic levels, making it easier to fulfill our basic assembly rule, det(R) 6= 0 (compare Fig. 3c, main text). It is however necessary to also consider an extension of the perfect matching criterion (Sec. A5), as omnivores in addition can open for nonzero determinants by the use of loops (or equivalently cycles), see Fig. A4. To discuss this we introduce the concept of directed pairing. Each pairing of species that are connected, may influence the population sizes of each other (so far marked by a single solid arrow, Figs 1 and 3, main text). To more explicitly highlight that the interaction influences both species involved in the pairing, we now use a more specific notation, where the single arrow is replaced by two dashed arrows (shown in Fig. A4a). Thereby, the interaction that allows species i to influence species j is marked with a directed dashed link from i to j. This may for example be a predator i that reduces the population of prey j, corresponding to a nonzero matrix element Rji . If the predator also gains in population by the interaction, there is a link from j to i, resulting in a nonzero matrix element Rij . A directed pairing is now defined as an interaction involving two species with only one direct link between them, i.e. either, but not both, of the two matrix elements Rji and Rij . Pairings hence always involve two directed pairings in opposite directions (both matrix elements Rji and Rij ). Complete directed pairing: Consider a sequence of nonzero matrix elements Rij , Rjk , Rkl , . . . , Rmi . We call this type of sequence a closed loop of directed pairings, i.e. a chain of directed pairings where the direction is maintained and the last element connects to the first. When the entire food web is covered by non-overlapping closed loops of directed pairings, we refer to this as complete directed pairing. Each nutrient source can, but need not, participate as a species (and in this case each nutrient source participates in a bi-directional directed pairing with each basal species). det(R) can only be nonzero if the requirement of complete directed pairing is fulfilled. Note that a food web that can be perfectly matched will automatically allow a complete directed pairing. Fig. A4a shows a food web that is not completely paired. The orange species needs to be paired with the resource, making it impossible to pair the remaining three species. Fig. A4b shows that the same food web can indeed be covered by complete directed pairings. Further, there are two permutations that
15
can cover the food-web, the one shown and its mirror image for the closed loop of length three. For future use, Fig. A4c shows that even a network with uni-directional relations may be covered by a complete directed pairing. This is of relevance for later discussions of links from predators to the parasites of their prey (so-called concomitant links, see Sec. A9.3.3).
a ~ no complete pairing
b
x x x x Det x x
≠0
1 1 two complete directed pairings
c
x x Det x x x
≠0
1 1 one complete directed pairing
Figure A4: Pairing requirements for food webs involving omnivore species. Example of a simple food web that can allow for coexistence, even if it is not completely paired. Yellow circles/ovals show interactions (links) that can be used to make a non-zero contribution to the determinant of R.
A7
Basal species with separate nutrients
So far we have restricted the discussion to the case of a single fundamental energy source that was shared by all primary producers. In practice, there may be several fundamental energy sources, such as different chemical compounds, in addition to sun light. Further, nutrient partitioning can also result from e.g. spatial subdivision or temporally varying availability (Tilman, 1994; McPeek, 1996; Tilman, 2000, 2011). As a consequence, a partitioning of the nutrient sources between the primary producers will occur. For simplicity, we take the n1 primary producers to each depend on one of nS ≤ n1 nutrients. In Eq. A1 this corresponds to setting only those pij equal to unity where i and j share a nutrient. In that case, A0 breaks down into a block structure with nS blocks. Each of these blocks is a square matrix of unit entries and dimension equal to the number of primary producers sharing the corresponding nutrient. Each of these blocks can then be manipulated to yield a single row of ones and otherwise zeros. This means that nS rows of unit entries persist. In the assembly rules the availability of the nS nutrients again allows for
16
more flexibility: k X
nl (−1)k+l+1 ≤ ∆(−1)k
∀
k ∈ [1, L] ,
(A26)
l=1
where ∆ ≡ No −Ne ∈ [0, nS ]. This set of conditions is analogous to the one for a single nutrient (Eq. A22) except that now, ∆ can take any value between zero and the number of available nutrient types, a choice that then carries over to additional sets of inequalities.
A8
Species link requirement to include predator saturation
In consumer-resource relationships, type-II functional responses are characterized by a less-than linear increase of consumption as resource concentration grows. This distinguishes type-II from type-I responses where consumption is linear in resource concentration, i.e. c(R) ∼ R, where c is the consumption rate per consumer, R is the concentration of the resource species and K is a constant. In the following we set the non-linear type-II response to c(R) ∼ R/(K + R)
(A27)
and discuss the resulting modifications of the food web assembly rules.
A8.1
Model equation with type-II response
Predator saturation, i.e. the situation where the predator growth rate saturates at a high prey density, can be taken into account in the assembly rules by replacing the type-I by the type-II functional response. For simplicity, let us again first consider the case of a completely shared nutrients for basal species and absence of omnivores. The population dynamics for the basal, respectively higher trophic levels, now obeys the following modified equations (compare Eqs 3 and 4, main text): (1) dSi
dt
(1)
1 −
(1) Sj −
j
(l)
dSi dt
(1)
= ki Si
X
(l)
= Si
X
X
(1)
(2,1)
η˜ki
Si
(2)
Sk
(1) Si
k (l−1)
(l,l−1) (l,l−1) η˜ji
βji
j
Sj (l−1)
Sj
(l,l−1)
+ Kji
−
(1)
(2,1) Kki
+
X
(1)
− αi Si , respectively
(A28)
(l)
(l+1,l)
η˜ki
Si
(l+1)
Sk
k
(l)
(l)
(l+1,l)
Si + Kki
(l)
− αi Si .
Here, the term (l)
(l+1,l)
η˜ki
Si
(l+1)
Sk
(l)
(A29)
(l+1,l)
Si + Kki
denotes the predation rate of the species k at the trophic level l + 1 with respect to the species i at the (l+1,l)
trophic level l, with a half saturation density Kki
(l)
(l+1,l)
. When Si Kki
, the function will be reduced
to the type-I response function as (l+1,l)
ηki
(l+1)
Sk
17
(l)
· Si
(A30)
(l+1,l)
(l+1,l)
≡ η˜ki
with ηki
(l+1,l)
/Kki
(l)
(l+1,l)
, while when Sk Kki (l+1,l)
η˜ki
(l+1)
· Sk
, the function will be reduced to the term
,
(A31) (l+1)
which implies that the species at level l is not fully exposed to the species population of Sk
A8.2
.
Constraints from steady state condition and extended species pairing
The non-overlapping species pairing of nodes required for coexistence (described in the main text) can be extended to the type-II functional response case. In order to derive it, we first consider top predators, middle species, and basal species separately, and then summarize the extended rules.
A8.2.1
Top predators
Consider the top predators at the level L. Their population dynamics is given by (L)
1 dSi (L) dt S
=
(L)
(L−1)
(l,l−1) (L,L−1) η˜ji
βji
j
i
Assuming Si
X
Sj (L−1) Sj
(l)
+
(L,L−1) Kji
− αi .
(A32)
6= 0, the steady state condition gives X
(L−1)
(l,l−1) (L,L−1) βji η˜ji
j
Sj (L−1) Sj
(l)
+
(L,L−1) Kji
− αi = 0 .
(A33)
(L−1)
Eq. A33 amounts to one constraint among the populations’ of the all of its prey species Sj
. For
nL top predator species in steady state, it must be ensured that Eq. A33 does not over-constrain the densities of prey populations. There should hence be nL prey species whose populations are determined by these equations. In other words, it should be possible to pair each top predator with one of its direct prey species in a non-overlapping way. Hence, for top predators and their prey, the pairing requirement remains unchanged, irrespective of whether the type-I or type-II response is of interest.
A8.2.2
Intermediate species
Now consider a species i at a trophic level l, that is neither a top predator nor a basal species. (l)
Type-I response of all predators with respect to Si enough to satisfy
(l) Si
(l+1,l) Kki
(l)
— Suppose the population Si
for all of its predator k; for convenience, we term such species ”type-I (l)
response species”. The dynamical equation for the population density of Si (l)
1 dSi (l) dt S i
=
X j
is small
(l−1)
(l,l−1) (l,l−1) η˜ji
βji
Sj (l−1) Sj
+
18
(l,l−1) Kji
−
X k
(l+1,l)
η˜ki
will become
(l+1)
Sk
(l)
− αi .
(A34)
(l)
In this case, with assuming Si 6= 0 the steady state of Eq. A34 gives one constraint among its predators’ (l+1)
population Sk
(l−1)
and its preys’ population Sj
. In order that this condition does not over-constrain
the populations, it should be possible to pair all the type-I response species either to one of its predators or one of its prey species in a non-overlapping way. This pairing requirement for the top predators and their prey species hence remains unchanged as compared to purely type-I response consumer-resource relationships (compare main text).
n4
n3
n1
x
n4 n3
n2
x x x
x x
x x
n2
x x x x x
x
x x x x x 1 1
n1
Figure A5: Simple food web containing a type-II response species. The figure is an extension of Fig. 3a, main text, where an additional species was included at trophic level 2, i.e. now n2 = 4. This species is taken to be in the saturation phase w.r.t. the response to predation by a predator on trophic level 3 (the type-II response is indicated by a dotted arrow). Under these circumstances, the species can be considered to pair with itself in the species pairing, indicated by a shaded circle. When viewing the adjacency matrix as a table of relations between a consumer and a resource species, the self-pairing could be interpreted as diagonal matrix element (shown as a pink square), i.e. a dependency of the change in species density on itself.
(l)
Type II response of some predators with respect to Si (l+1,l)
than at least one of the constants Kki
(l)
— Suppose Si is comparable or larger
; we term such interactions ”type-II response”. The equation
will then take the form (l)
1 dSi (l) dt S i
=
X
(l−1)
(l,l−1) (l,l−1) η˜ji
βji
Sj (l−1)
Sj
(l,l−1)
+ Kji X (l+1,l) (l+1) X (l+1,l) (l+1) − ηk0 i Sk0 − η˜ki Sk j
k0
k
1 (l) Si
(l)
(l+1,l)
+ Kki
− αi .
(A35)
(l)
In this case, given Si 6= 0, the steady state of Eq. A34 yields one constraint among its own population (l)
(l+1)
Si , its predators’ populations Sk
(l−1)
, and its prey species’ populations Sj
. For this condition not to
over-constrain the populations, it should now be possible to link each of the type-II response species to either itself, one of its predators, or one of its prey species in a non-overlapping way. An example of a food web involving a type-II response species is shown in Fig. A5, demonstrating the possible self-pairing for the added species given that the saturation regime is reached. Note that 19
the ability of a species’ self-pairing comes with the condition that the saturation level K in Eq. A27 is reached by the species population density. In the example given in Fig. A5 this may impose relatively strict requirements on K since its resource species (shown in blue) is already in a consumer-limited state (small circle radius). Recent work has explored different functional responses as well as different types of interaction (competitive vs. mutualistic). It was found that combining a range of responses can have a stabilizing effect on communities (Mougi and Kondoh, 2012; Brose et al., 2006).
A8.2.3
Basal species (1)
(1)
Type-I response of all predators with respect to Si
— If Si
(2,1)
Kki
for all its predator k,
a basal species obeys the equation
(1)
1 dSi (1) dt S
=
(1) ki
1 −
(1)
The steady state condition with Si its predator population
(1) Sj
−
X
j
i
(2) Sk ’s.
X
(2,1)
ηki
(2)
(1)
Sk − αi .
(A36)
k
6= 0 gives a constraint among the sum of basal species
P j
(1)
Sj
and
Hence, for this condition not to over-constrain the populations, it should
be possible to pair each of the type-I response basal species to one of its predators in a non-overlapping way, allowing additionally one of the basal species to pair with the nutrient source. (1)
Type-II response of some predators with respect to Si (1)
1 dSi (1) dt S
(1) = ki 1 −
i
X
(1) Sj −
X
(2,1)
(2)
ηk0 i Sk −
k0
j
X k
— The equation now becomes
(2,1)
η˜ki
1
(2)
Sk
(1) Si
(1)
(1)
(2,1)
+ Kki
− αi .
(A37)
(1)
The steady state condition with Si 6= 0 gives a constraint among its own population Si , the sum of P (1) (2) basal species j Sj and its predators’ populations Sk . In order for this condition not to over-constrain the populations, it should be possible to pair each of the type-II response basal species to either itself or one of its predators in a non-overlapping way, allowing again one of the basal species to pair with the nutrient source. The generalization to multiple nutrient sources is analogous to the discussion in Sec. A7, i.e. each nutrient source may, but need not, be used for pairing a a basal species.
A8.2.4
Summary of the extended species pairing requirement
Summarizing the conditions above, the species pairing requirements with type-II response read as follows: • It must be possible to produce a pairing of each species and one of its “partners” in a non-overlapping way.
20
• This partner can be any of its predator or prey species if the species’ response to all its predators is in the type-I regime (sufficiently low species density). • If a species’ response to one or more of its predators are not in the regime of type-I response (sufficiently high species density), it can be paired to itself. This self-pairing corresponds to a self-limit on growth which prevents it from fully controlling its prey. • If the species is a basal species, each nutrient source is also a possible partner.
A9
Data Analysis
This section describes details of the data analysis performed. In particular, we describe the characteristics of the food web data considered (Sec. A9.1), an analysis of the sub webs formed by free-living species only (Sec. A9.2) as well as the analysis of the sub webs including parasite species (Sec. A9.3). The latter also involves the discussion of concomitant predation (Sec. A9.3.3). Several of these sections involve specific comments on the individual food webs, we have noted this by “details”, to facilitate reading. We have also moved more detailed figures (included for completeness) to the end of the text. We further examine the possible existence of secondary extinctions by removal of species in the different empirical food webs (Sec. A9.4).
A9.1
Data considerations
A9.1.1
Food web data
We use high-resolution data on seven food webs including free-living and parasite species: The North American Pacific Coast webs Carpinteria Salt Marsh, Estero de Punta Banda, Bahia Falsa (Hechinger et al., 2011; Lafferty et al., 2006); the coastal webs Flensburg Fjord (Zander et al., 2011), Sylt Tidal Basin (Thieltges et al., 2011), and Otago Harbor, New Zealand (Mouritsen et al., 2011), as well as the Ythan Estuary, Scotland (Huxham et al., 1996). These food webs describe consumer-resource interactions between basal, predatory and parasite species. A compilation of all seven food webs has recently been provided (Dunne et al., 2013). Specifically, the data distinguish three types of links: (i) links between free-living species only (“Free”), (ii) additional links between parasites and other species (“Par”) and (iii) links from free-living consumers to the parasites of their resources (“ParCon”), i.e. so-called concomitant links.
A9.1.2
Technical considerations on parasites
Similar to predators, parasites feed on their hosts. Parasite-host interaction can be distinguished from predator-prey interactions in that parasites undergo complex life cycles (Huxham et al., 1996), where a
21
single parasite at one stage may benefit from one host but a different host at another stage. In addition, a parasite often lives in or on its hosts, and concomitant links from the hosts’ predators’ to the parasite may result (Sec. A9.3.3 and Sec. A10.4). On the population level, the total population of a given parasite species can however be regarded as feeding on a number of different hosts simultaneously. We adopt this view in the following as far as “Par” is concerned, as the details do not impact on our mathematical description in terms of food web matrices. However, these simplifications should be kept in mind when considering, in particular, removal of species and the question of potential sustainability for the resulting webs (Sec. A9.4). When considering concomitant links, we make additional distinctions (Sec. A9.3.3). For example, one aspect, which we further consider, is that of the host potentially also being impacted upon by the parasite, e.g. by reduction or modification of activity, thereby affecting metabolism or reproductive success (Lafferty and Morris, 1996; Johnson et al., 2010).
A9.1.3
Matrix rank, basal species and nutrient sources
For our general coexistence condition of det(R) 6= 0 we do not require details of the numerical values of matrix coefficients. Instead, we first symmetrize the interaction matrix to mimic the bi-directional effect in predator-prey, respectively parasite-host pairs. Further, we allow basal species to feed on individual (pij = δij ). We then assign random numbers to all nonzero matrix elements to numerically compute the determinant. To allow a more refined view on food webs exhibiting det(R) = 0 we have computed the rank r of R (Otto and Day, 2007). r describes the number of linearly independent rows (or columns) of the matrix. Ecologically, the rank amounts to the number of uniquely occupied niches in the ecology. We also repeatedly use the rank deficiency (i.e. nullity) d ≡ S − r, with S the total number of species. We distinguish df , df +p , dcon,asym , and dcon,sym , the deficiencies of the sub webs of free-living, free-living and parasite species, as well as additional asymmetric and symmetric concomitant links (Sec. A9.3.3).
A9.1.4
Treatment of detritus and particulate organic matter (POM) in the data
Several of the empirical food web data sets contain detritus or POM as basal species (Wilson and Wolkovich, 2011; Gaedke et al., 1996; Pauly, 1996). The equation describing the concentration of such non-alive “basal species” Di should have a source term that is independent of Di , namely 2 X 1 dDi k (D) (2,1) (2) = i − αi − ηki Sk . Di dt Di
n
(D)
(A38)
k=1
The first term on the RHS is not linear in Di , therefore the equation for the steady state is not expressed by the matrix form. However, as summarized in section A8, the species pairing can be extended to the nonlinear equation as a condition not to over-constrain the variables. In this regard, the steady state of (2)
Eq. A38 constrains its own “population” Di and its consumers’ populations Sk . Thus, the Di can be
22
paired with either themselves or one of their consumers, as in the case of a basal species with its own nutrient. We therefore treat such resources as basal species with their own nutrient in the data analysis.
A9.1.5
Approximate trophic levels
For all empirical food webs, we first organize species according to their approximate trophic level. In accordance with the literature (Williams and Martinez, 2004; Thompson et al., 2007), we here aim to define the trophic level of a given species as its average food chain length. Basal species, which do not consume other species, are thereby assigned trophic level 1. Within the prey-averaged definition (Williams and Martinez, 2004), the trophic level of a given species is then the average trophic level of all its resource species plus one, e.g. a species that only consumes basal species has trophic level 2. In practice, we have determined chain lengths by following paths from a given species to one of the basal species, choosing a random resource at equal probability at each intersection. As choices are random, we have repeated the procedure to ensure that fluctuations are small and the general conclusions do not depend on the sampling. For two species with very similar approximate trophic levels the order could occasionally switch, an effect that we were able to reduce by using larger sample size. We compared results using between 2 × 103 and 2 × 104 food chain samples per species and detected very little variation. Some food webs contain loops, where the same species recurs within a single food chain. These loops were dropped from the analysis on trophic levels. In the following, we use the computed average and standard deviation of a species’ food chain lengths as a proxy its trophic position and degree of omnivory, respectively. A similar consideration has been made in the literature (Lafferty and Morris, 1996). We will, for simplified syntax, in the following refer repeatedly to “trophic level” when discussing species, by which we mean the approximate average food chain length of the given species.
A9.2
Analysis of food web matrices of free-living species
A9.2.1
Trophic levels and omnivory
For all food webs, we compute the frequency distributions of average food chain lengths. We also aggregate the data for all food webs and produce the corresponding frequency distribution (Fig. A6). Notably, the distribution of aggregated empirical data (Fig. A6a) has pronounced, sharp peaks at chain length 1 and 2, corresponding to basal species and their consumers. Approximately 25 percent of species are sharply located at trophic level 2 (dashed red line in the plot). The absolute highest average chain length lies near 4.5 (compare also to the literature, e.g. (Thompson et al., 2007)). Comparing with the individual food webs (Fig. A21), this overall pattern is generally preserved. In both figures (Figs A6 and A21), we compare to data produced using the niche and cascade models, with total species number S and total number of links L for each food web as an input. These models generally produce fewer species at the level 2 and overall considerably broader distribution functions ranging up to chain lengths of 5.5. 23
a
c
b
F raction of Species
Empirical Data
Niche Mode
Cascade Model
0.2
0. 1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
Average Food Chain Length Figure A6: Average food chain length distribution (aggregated for all webs). a, Histograms of species’ average food chain lengths when aggregating species of all seven food webs. b, c, comparison with niche and cascade model, respectively. Red arrow highlights the spike formed by level 2 species in the empirical data. dashed horizontal lines are guides to the eye, illustrating the respective concentrations of level 2 species in the three cases. To quantify the extent to which species can be associated with sharp trophic levels, we compute also the standard deviation of food chain length for each species, and show the aggregated distribution functions for all resulting values (Fig. A13a—c). We hereby define a species to have a “sharp trophic level”, if the species shows no variation, i.e. zero standard deviation, in food chain length. In the plots (Fig. A13a—c), a value of zero correspondingly means that all chain lengths associated with a given species are identical. The figure demonstrates that in the empirical data (Fig. A13a), all food webs have a considerable fraction (approximately 40 percent on average) of species with sharp trophic levels. The remaining species have chain length standard deviations of generally less than one. A value of ∼ 1/2 would e.g. be obtained by a species that consumes at two neighboring trophic levels equally. In comparison, the niche and cascade models (Fig. A13b,c) predict substantially fewer sharp trophic levels. Further, they produce a continuum of values ranging up to 2, which could e.g. be obtained by a species consuming equally on trophic levels 1 and 5. These aggregated results should also be compared to the distributions for the individuals webs (Fig. A22), where the general pattern is preserved, albeit with some variation (which we will return to in Sec. A10).
A9.2.2
Trophic structure of food web matrices
We now use the obtained mean chain length (CL) of each species to assign an order to species from shortest to longest CL. Using the Bahia Falsa food web as an example, we describe the resulting depiction of food webs (Fig. A7, caption). We will subsequently employ this presentation style repeatedly in the remaining text. We also use integer-rounded CL to broadly categorize species (the CL is thereby simply rounded to the nearest integer value). To simplify the discussion, in the following, we refer to interactions between species with integer-rounded chain length l1 and l2 as l1 −l2 interactions. Consider now the corresponding depiction of all empirical food web matrices of free-living species (Fig. A19). 24
b 150
100
50
1
a 100
50
150
150
100
100
1
100
100
n4 n3
50
50
n3 50
50
n2
n2
n1
n1 1
1 100
n4
50
n3
1
1
1
n2
n4
150
n1
100
n4
50
n3
1
n2
n1
Figure A7: Depiction of Bahia Falsa food web matrix using ordering by approximate trophic levels. a, Free-living web consisting of 119 species. b, Web of free-living and 52 parasite species. Orange and blue colors of matrix elements represent the direction of interaction between species. Color-coding shown at top and left edge of matrix indicates the ordering of species in terms of their average food chain length (red=short, blue=long). This color coding is used in (b) to indicate the new positions of free-living species once parasites are introduced (there is little re-ordering for the lowest 60 species, but more reordering above). Gray and black “measuring bars” at the lower and right edge of the food web matrices indicates the number of integer-rounded approximate trophic levels and corresponding diversities ni . These measuring bars can be used to compare the structure of the matrices in (a) and (b). Basal species have again been assigned individual nutrient sources, yielding finite matrix elements on the diagonal in the lower right block matrix. Figure is a reproduction of main text Fig. 4a,d. Additional detail. We here focus first on the sub web of free-living species only. All food webs show a clear structure of blocks of high link density and blocks where links are generally absent. We have in Sec. A9.2.1 noted that species of integer-rounded CL 2 have little variation in chain length. This is also evident from the figure, as the positions of the white blocks in the matrices indicate that species of CL 2 predominantly consume basal species and are in turn mainly consumed by species of CL 3. We further find that the structure of the matrices becomes less defined for species of CL 3 and above. There, different food webs show different behavior. Generically, more links can be observed between species of similar chain lengths, e.g. in the block with chain lengths ∼ 3 considerable interaction is present, while this is e.g. absent for species with CL 2 (white spaces). These matrices show additional detail on the trophic organization of species: In addition, a clear hierarchical structure is evident, illustrated by blue and red colors (corresponding to the direction of the consumer-resource links). Species located at higher trophic levels tend to act as consumers of species below them.
A9.2.3
Discussion of individual matrices (details)
Using Fig. 1, main text, we attempt to construct a path through the matrices. For the different matrices, we make the following observations: • Bahia Falsa: Constructing a path is not possible as a large number of well-defined chain length 2
25
species has formed a white block that is impossible to circumvent using the available 2-3 interactions (interactions between species of integer-rounded CL 2 and 3). This could be remedied by the presence of 4-2 interactions, but also these are mainly absent. In this particular food web, the interactions 4-4 are also generally missing (there is only a thin band of omnivores feeding on levels 3 and 4). Hence, the Bahia Falsa food web constitutes a relatively well-structured food web and the high degree of organization, together with a large number of CL 2 species leads to failure to comply with the assembly rules (the rank difference — i.e. nullity — is df = 16, compare the table in Fig. A12d). • There are two other food webs with clear structure: Flensburg Fjord and Otago Harbor. Both have nearly complete lack of 2-2 interaction and little 2-4 interaction. Both have large 2-2 blocks. Also in these food webs, the assembly rules can not be fulfilled, the respective rank differences are 15 and 19. • A further food web with large rank difference is Ythan Estuary (df = 17). Here, however, the overall link density is very low, compared to the other food webs, leading to an additional difficulty in reaching agreement with the assembly rules. • The remaining food webs Carpinteria, Punta Banda, and the Sylt Tidal Basin are less clearly structured and the 2-2 block is somewhat smaller. Correspondingly, a path through the matrix can be constructed for Carpinteria and Punta Banda (df = 0). Only a small rank difference results for Sylt Tidal Basin (df = 6). Again, we compare with the simulations obtained from the niche and cascade models. • Both models describe the hierarchical organization of species (clear organization of blue and red matrix elements), where those with larger CL prey on those with shorter CL. This is clear by the definition of the two models, where an ordering of species is the starting point. • Both models deviate from the empirical data in that they do not produce any obvious signature of trophic levels, i.e. no block structure is apparent in either of the two models. • Correspondingly, for all instances of the models considered (with the empirical numbers of species and links), full rank was achieved. Indeed, for random matrices containing exclusively values −1, 0, and 1, with relatively mild conditions, one is very likely to achieve full rank (Tao and Vu, 2006). • We have further reduced the number of links considerably for the two models: Demanding only that each species be connected to at least one other species, essentially all (> 99 percent) of samples achieved full rank, even when using only 2/3 of the links. In the remaining few cases, a rank difference of 1 resulted.
26
A9.3
Analysis of food web matrices including parasite species
A9.3.1
Trophic positions with parasites
We now consider the empirical data also for food webs including parasites (Fig. A20). We aim to characterize the apparent re-organization that takes place once parasites are included in the food web. We again make use of approximate trophic positions, but it should be kept in mind that for parasites trophic positions are a matter of debate, as they undergo complex life-stages and localizing them at any specific trophic level (or assigning a specific food chain length) is questionable. Here, we use the concept of trophic levels, respectively food chain length, solely as a means to organize parasite species relative to free-living species and discuss the respective host range. To this end, we use a color-coding (displayed at the edge of the food web matrices in the figure and explained in Fig. A7). For the free-living food webs, the color coding simply indicates the sequential organization in terms of the chain length computed for the free-living species alone. When parasites are included, the sequence of the color coding becomes disrupted. However, the disruption is again far from random: Nearly all species with CL 1 and 2 have unchanged positions in the extended food webs. For species with CL 3, some disruption takes place, i.e. some parasites now have chain lengths ∼ 3. However, most of the parasites enter near the top end of the existing chain lengths and top-predators of the free-living web often no longer have largest chain length in the combined system (Lafferty et al., 2006). This feature of disruption is observed generally for all food webs studied. We also show distribution functions of the respective chain lengths in Fig. A26. We note more specific features of the matrix structure: Parasites have re-organized parts of the structure at higher trophic positions, leading to substantial randomization there. However, much of the structure in the lower part of the matrix is left intact, in particular, the white block formed by 2-2 interactions is nearly unchanged in most of the food webs. Another noteworthy observation is that the hierarchical structure, highlighted earlier for the free-living webs, is strongly disrupted for the higher trophic levels, when parasites are considered (blue and red matrix elements are now more mixed). A9.3.2
Discussion of individual food webs (details)
Here it is important to consider the different food webs separately (Fig. A20): Carpinteria and Punta Banda show a strong mix-up of hierarchy for the higher levels, while this effect is less clear for some of the other food webs. In particular, Ythan Estuary shows nearly no reduction of the hierarchical structure. Mixing of hierarchical structure is important as it can lead to an increase in chain length (e.g. free-living species of formerly short chain length can sometimes feed on parasites that then feed on others, thereby increasing chain length). This particular feature is most evident when again considering the distributions of chain length standard deviation (Fig. A25) or the coefficient of variation (Fig. A18). Carpinteria and Punta Banda have now acquired much stronger variation of food chain length for some species, values that now even exceed those produced by the corresponding Niche and Cascade model simulations. In 27
Niche Mode
F raction of Species
Empirical Data
Cascade Model
0.2
0. 1
2
3 4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
Average Food Chain Length Figure A8: Aggregated chain length distributions for food webs including parasite species. Similar to Fig. A6 but for all data including parasite species. the other extreme, the Ythan Estuary food web, the unbroken hierarchical organization leads to nearly unchanged chain length variation distribution. For completeness, we present also the aggregated data of chain length standard deviation for the case of free-living and parasite species (Fig. A24). The above findings on chain length standard deviation and mixing of hierarchical structure are also consistent with the changes in chain lengths (Fig. A23, compare Lafferty et al. (2006)). Here, it is found that a continuum of chain lengths beyond 3 ranging up to 5 or 6, in some cases, is now present. However, the pronounced concentration of level 2 species is conserved (most evident in the aggregated data, Fig. A6 and Fig. A8). This means, that the disruption, caused by the presence of parasites, affects mainly the upper part of the food web and introduces re-organization there. The lower part of the food web remains nearly unchanged.
Returning to Fig. A20, we can discuss the relation to the assembly rules. For the Bahia Falsa web, some parasites prey on level 2, granting better chances to find a path through the food web matrix as demanded by the assembly rules (Fig. 1, main text). Indeed, in the Bahia Falsa food web, presence of parasites reduced the nullity from df = 16 to df +p = 3. Similar behavior is observed for Otago Harbor, where substantially more species now prey on level 2 species. On the other hand, in the case of the Ythan Estuary, not much structure is added and overall low link density makes finding a path less likely. This may be one reason why the nullity in fact increases when including parasites (others are discussed in Sec. A10).
A9.3.3
Concomitant predation
We further consider concomitant links: Concomitant links are links from a free-living consumer to the parasites of its resources (Lafferty et al., 2008, 2006), e.g. by consuming the resource, also the parasite will be consumed, causing an additional death rate to the parasite. Such links have e.g. been suggested to contribute considerably to parasite mortality (Johnson et al., 2010). Specifically, in the following we
28
distinguish asymmetric and symmetric concomitant links. Asymmetric concomitant effects arise, when a predator causes death to its prey’s parasite(s), but no effect of the consumption of the parasite is felt by the predator. Generally, this is a type of 3-species interaction, which can be a function of all three species involved (which is beyond the simple products of the generalized Lotka-Volterra equations). In principle, one could e.g. include terms of the type: d[parasite] ∼ . . . . − η ∗ · [predator] · [host] · [parasite] + . . . , dt
(A39)
where η ∗ is a modified interaction probability. We here make the point that such interactions can lead to additional niches, as they impose new relations between parasites and the hosts’ predators. To consider the presence of such interaction, it is here therefore sufficient to consider only the case where the coupling between the parasite and the host’s predator does not vary with host density: d[parasite] ∼ . . . . − η ∗∗ · [predator] · [parasite] + . . . , dt
(A40)
implying a linear truncation. (Non-linear interactions could also be considered in analogy to Sec. A8.) In terms of the interaction matrix, this is a directed link where the parasite feels the negative effect of the host’s predator. In such cases, the interaction matrix has a nonzero entry at some position (i, j) (Fig. A4c). The entry at the symmetric position (j, i) might be zero, if the predator’s gain resulting from consumption of its prey does not change when the prey is infected by the parasite. For food webs with some uni-directional links, the basic condition det(R) 6= 0, i.e. “finding a path through the matrix”, amounts to the requirement of possible selection of an out-link and an in-link for each species in the interaction matrix. Fig. A4c illustrates how directed links can provide sustainability: In the case depicted, the parasite (green) is exposed to concomitant predation by the predator (red). Without the concomitant link, this food-web would violate the competitive exclusion principle. In fact the rank deficiency of the interaction matrix would be d = 1 because the parasite and the predator compete. However, when one includes one directed concomitant link between predator and parasite, d = 0. This reduction in nullity comes about through inclusion of the cyclic interactions
predator → parasite → prey → predator
(A41)
corresponding to the entries marked with yellow circles in the matrix on the right side of Fig. A4c. Fig. A4 emphasizes that directed concomitant links can decrease the nullity of the interaction matrix. However, they will not always do so. In practice, when assuming that all concomitant links are directed, the nullity is only improved in two out of the seven food webs (Ythan Estuary and Otago Harbor), see
29
a) Bahia Falsa
Free-living
Free-living + parasites
Free-living + parasites + concomitant
(df=16)
(dc,asym=df+p=3)
(dc,sym=0)
(df=0)
(dc,asym=df+p=1)
(dc,asym=dc,sym=0)
1000
0
b) Carpinteria Salt Marsh
Data set
1000
0
c) Estero de Punta Banda
(df=0)
(dc,asym=df+p=5)
(dc,asym=dc,sym=0)
(df=15)
(dc,asym=df+p=10)
(dc,asym=dc,sym=9)
d) Flensburg Fjord
e) Otago Harbor
Number of triangles
1000
0
0
(df=19)
(df+p=6 dc,asym=4)
(dc,sym=2)
(df=6)
(dc,asym=df+p=3)
(dc,sym=0)
(df=17)
(df+p=21, dc,asym=12)
(dc,sym=7)
Bahia Falsa Carpinteria Punta Banda Flensburg Fjord Otago Harbor Sylt Tidal Basin Ythan Estuary
41 23 40 33 45 42 45
Data set
odd levels no para para
Bahia Falsa Carpinteria Punta Banda Flensburg Fjord Otago Harbor Sylt Tidal Basin Ythan Estuary
1000
all free-living no para para
9 8 11 12 18 7 26
78 84 98 44 78 84 46
42 39 48 17 26 40 15
Sf
Sp
119 107 138 77 123 126 91
52 58 76 46 19 35 42
even levels no para para 32 15 29 21 27 35 19
36 45 50 27 52 44 31
1000
0
f ) Sylt Tidal Basin 1000
0
g) Ythan Estuary 1000
0
0
100
200
0
100
200
0
100
200
Species ordered by chain length
Figure A9: Characterization of concomitant links. Numbers of loops of length three (triangles) of each species for all food webs (as labeled), for the sub-webs of free-living, free-living and parasite links, as well as additional symmetric concomitant links. Loops were computed by first removing diagonal elements from R and collecting the diagonal elements of the cube of the resulting matrix. Note that we count each existing closed loop only once, irrespective of whether the loop may be directed or bidirectional. Panels mark the rank deficiencies for all cases, red labels highlight where changes occur for asymmetric concomitant links. Plot also shows the frequency distribution for numbers of free-living species with parasites, distinguishing additionally the set of high level (i.e. at least CL 3) free-living species. The table shows the numbers of free-living species that interact with parasites (i.e. either act as parasite hosts or feed on parasites, “para”), and do not interact (“no para”). Also shown: numbers of free-living and parasite species, as well as the numbers for interaction with free-living species at odd and even levels, respectively. the labels in the panels of Fig. A9. In the remaining five food webs the nullity remains unchanged. Although directed links may help, one should realize that the use of loops constrains the pairing of other species. Thus, when the pairing involves one directed interaction, complete directed pairing requires a closed cyclic loop of reactions, involving species that then cannot participate in another pairing.
Fig. A9 examines the number of loops of length three (“triangles”) for the seven food webs, with and without parasites. The panels also show the nullity of the corresponding food web. The figure illustrates that absence of rank gain from directed concomitant links is associated to food webs with many loops. In contrast, food webs with few loops (Ythan Estuary and Otago Harbor) open for a decrease in nullity by inclusion of these directed concomitant links. Symmetric concomitant effects. The presence of a parasite may in some cases have effective impact on the consumption rate (Johnson et al., 2010). For example, by altering host behavior, the 30
x x x Det x x x x
x x ≠0 x 1
Figure A10: Possible pairing for symmetric concomitant links. Food web consisting of host (blue), two predators of the host (orange, red) as well as a parasite (green). Consumer-resource (concomitant) links are shows as black (gray) arrows. Matrix shows the existing interactions and a possible path through the matrix (circles). The food web is sustainable with symmetric concomitant links, but not without. parasite may facilitate consumption of its host by the consumer, as was shown for killifish that become more susceptible to their natural enemies (birds) when they were infected by trematodes (Lafferty and Morris, 1996). In such cases, the fraction of hosts infected by a parasite would have a larger probability to be consumed by the predator. This again constitutes a non-linear 3-species interaction as in Eq. A39. Again making a linear approximation, where the predator gain now linearly increases with parasite density, we then consider symmetric concomitant links. These links involve a bi-directional interaction between the host’s parasite and predator (see Fig. A10 for a simple example where coexistence can be obtained). However, in all empirical cases, the full food webs with parasites and bi-directional concomitant links have lower nullity than the food webs without parasites. We have explored in the empirical data, whether only a small subset of bi-directional concomitant links is sufficient, finding that usually 20 percent yield already substantial improvement of the rank (compare Fig. 3g, main text). Four out of seven full food webs fulfill our assembly rule. Concomitant links are further explored in the simulations in Sec. A10.
A9.4
Removal of species from food web
The lack of niches d can be used to address the potential effect of removal of one species from a food web. If removal increases the lack of niches further, it can potentially trigger secondary extinctions, especially if d = 0 before the removal of the species. If the removal of a species causes d to decrease (which is possible only if d > 0 before the removal), then the removal of the species could potentially make the food web more compatible with coexistence. It should however be noted that — especially in food webs containing parasites — removal of species, even when decreasing the nullity, can lead to strong disruptions. E.g., when parasites use different hosts at distinct life stages (Lafferty et al., 2008), removal of only one of the hosts could be sufficient to cause extinction of the parasite. We tested how removal of each species affects d in the seven food web data sets. We again distinguish 31
food webs with free-living species only, with parasites, and with parasites and concomitant links. For simplicity, symmetric entry to the matrix is assumed for concomitant links. In Fig. A11, species are labeled based on their average chain length. Species for which removal increases d are marked by red symbols (“+”), those for which removal decreases d are marked by green symbols (“×”). For the free-living species webs with d > 0 (Bahia Falsa, Flensburg Fjord, Otago Harbor, Sylt Tidal Basin, and Ythan Estuary), there is a clear tendency that removal of species with chain lengths near one or three increases d, while the removal of species near the chain length two or four decreases d. This is consistent with the observation that the excess species at the even trophic levels cause the lack of niches (cf. Fig. A12). a) Bahia Falsa
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Figure A11: Effect of removal of a species on the lack of niches d. The species are labeled based on their average chain length. If the removal of a species results in increase of d, the species is marked by a red + (potentially triggering the secondary extinction), while if the removal of a species results in the decrease of d, the species is marked a green ×. If d does not change, no symbol is given. In the figure, the value of d before removal is given. Number of species, m, that increase d is also given in each figure.
For the webs with d = 0 before removal (Estero de Punta Banda, Carpinteria Salt Marsh), only an increase of d can occur, and removal of such species will cause secondary extinction. The Estero de Punta Banda free-living web is quite sensitive to the removal of one species, while CA is not. When considering food webs including parasites, it must again be noted that substantial disruption (and extinction) could be caused, when only one of the parasites’ hosts’ is removed. Given this limitation, we state for Bahia Falsa, Flensburg Fjord, and Sylt Tidal Basin, that adding parasites could in principle
32
reduce the sensitivity of the food web to the removal, as seen by reduced values of the number of species m that increase d by removal. Carpinteria Salt Marsh and Punta Banda appear to be less stable when only adding parasites since d = 0 was already realized in the free-living species web, but concomitant links might have the potential to stabilize these webs. Ythan Estuary also shows such a tendency, though it starts with a rather high value of d in the free-living species web. Otago Harbor remains rather sensitive to the removal of species even with addition of parasites. This may be due to the rather large excess number of species in the trophic level 2 (cf. Figure A12). The addition of parasites decreases d but not enough to completely compensate the excess species, which make it sensitive to removal of species in other levels and also removal of species from level 2 helping to reduce d for all the three cases.
A10
Modeling
This section provides modeling of an idealized food web structure. Sec. A10.1 states the basic assumptions made to obtain an idealized, food web with sharp trophic levels (no omnivores). Originating from this “free-living” food web, we discuss several perturbations: Modification of the free-living food web to include omnivores (Sec. A10.2), addition of different types of parasites to the free-living food web (Sec. A10.3), as well as a discussion of asymmetric and symmetric concomitant links (Sec. A10.4).
A10.1
Idealized food web structure
To obtain additional insight into the generic features of the data, we now produce an idealized free-living food web. To this end, we consider all seven empirical food webs and determine the species’ approximate trophic levels (Fig. A12a and b). For simplicity, for any species we assign the closest integer trophic level as its default trophic position, e.g. a species with mean chain length 3.2 would be assigned a value of 3. Proceeding in this way for all food webs, indeed some degree of generality is found (compare Fig. A12a and b): Basal species constitute the smallest fraction of trophic levels, levels 2 and 3 are most diverse and level 4 often constitutes the top level (for details, see also the table in Fig. A12c). Four webs have a small number of species (between 1 and 3) at level 5, which we ignore for simplicity. We thereby arrive at a generic distribution of species richness in terms of trophic levels. This particular distribution is characterized by an inherent lack of niches: a default nullity of 22 is predicted since No − Ne = −22 (compare Sec. A5). In terms of resource vs. consumer limitation (Sec. A3) the system would be in a state where far too many consumers were added to be consistent with the assembly rules (we call this consumer-dominated in the following). We note that a compensation by nutrient sources is thereby not possible, since additional nutrient sources could only be used to compensate for an “overweight” in terms of unpaired basal species. In the current situation, however, compensation would be required for species
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Figure A12: Idealized food web data for simulations. a, Approximate species richnesses ni at trophic levels: Approximate trophic positions were determined by rounding the species chain lengths to the nearest integer value. For similar colors, bars from left to right correspond to the sequence of food webs given in the table (c). b, Generic web used in the simulations: Values of ni were obtained by averaging the corresponding trophic richnesses of species in the empirical food webs and rounding to the nearest integer value. c, Table of food web data for all empirical and the idealized food web. ni are the numbers of species at trophic levels i obtained by the rounded mean food chain lengths for all species in the respective webs. No − Ne is the difference of the numbers of species at odd and even levels, respectively. Sf and Sp are the numbers of free-living and parasite species, respectively. d, Table containing information on links in the different food webs: Lf are the numbers of links in the free-living web, Lp are the numbers involving at least one parasite. Lp−p are the numbers of links between parasites. df and df +p denote the nullities in the webs of free-living, respectively free-living and parasite species webs. dc,asym and dc,sym denote the nullities when including asymmetric and symmetric concomitant links, respectively. The table also indicates where values vary in the simulations. e, An example of a sample of the idealized free-living food web matrix Rsim f ree , used as a starting point for the simulations: All species are placed at integer-valued trophic positions without food chain variation. {n1 ,n2 ,n3 ,n4 }={9, 42, 35, 24}. Basal species are all assigned individual nutrient sources (yielding a sequence of diagonal elements in the lower right block matrix). Panels (a), (b) and (e) are a reproduction of main text Fig. 5a,b. at levels 2 or 4. By inspecting the table in Fig. A12c, we further find that in each of the empirical food webs, the difference No − Ne is negative, indicating that all webs are in a consumer-dominated state, an inherent nullity, or rank deficiency, hence results for all food webs, ranging from −9 for the Ythan Estuary web to −37 for Otago Harbor. This means, if the food webs were organized strictly in terms of these integer trophic positions (no variation of food chain length for a given species), the assembly rules could not be fulfilled for any of them. One can speculate as to the origin of the imbalance in No − Ne . One 34
explanation could be that basal species are poorly resolved. Another is that parasites can be seen as adding a counterweight at higher trophic levels and cause additional mixing. Using the idealized structure as a starting point, we initially assume strict trophic positions of all species and assign random interactions within the allowed sub-block of the matrix until the empirical average of links is reached (compare table Fig. A12d). The strict trophic positions require that the distribution of chain length variation would be a spike at zero. In the following, we refer to this type sim of matrix as Rsim f ree , a sample is shown in Fig. A12e. In sections A10.2 and A10.3 we use Rf ree as a
starting point for the discussion of two specific types of modifications: additions of omnivores to Rsim f ree by re-assignment of links (no link addition); addition of parasites to Rsim f ree (increasing total number of species to Sf + Sp , see Fig. A12c).
A10.2
Simulations of omnivore addition within the free-living web
We apply two types of perturbations to Rsim f ree without changing the number of species or links: • Some of the links described by Rsim f ree are reassigned to become omnivorous links (Fig. A13d). Even for a small fraction of omnivorous links, the variation of chain lengths widens and at 65 links already becomes similar to that of the aggregated empirical data (compare Fig. A13a). Addition of omnivorous links is further associated with a rapid reduction of the nullity (Fig. A13f, dashed line). • We consider specific omnivory, namely only that of level 3 species (which initially feed only on level 2) being able to feed on other level 3 species (3-3 interaction in addition to the existing 3-2 interaction). This type of interaction appears to be quite abundant in the empirical data (Fig. A20). In this case, the distribution of chain length variation changes more slowly, even at 125 omnivorous links the empirical distribution has not yet been attained. Considering again the change of nullity as function of added links, there is no change. This is clear as the addition of 3-3 interactions does not help in relaxing the consumer-dominated state of the systems towards a more balanced configuration. In terms of the species pairing, 3-3 interaction would allow for pairing of level 3 species with level 3 species. However, all existing level 3 species were previously already paired with either level 2 or 4. Conversely, more 4-2 or 2-2 interactions would be beneficial, as in that case e.g. the abundant level 2 species could be paired with other species at that level or with some of those at level 4. Real food webs do have high density of 3-3 interactions but some also show substantial 4-2 interaction. Actual food webs therefore likely lie in between the two extremes (solid and dashed lines in Fig. A13f). However, nearly no 2-2 interactions are present in empirical webs, which makes pairing between such species rare.
35
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Figure A13: Chain length variation for free living species. Normalized histograms of the standard deviation of food chain length for free living species food webs. a, Aggregated empirical data for all seven empirical food web data sets. b, Niche model results for corresponding values of S and L for the seven food webs. c, Similar to (b) but for cascade model results. d, Simulation data for the idealized free-living food web containing the average numbers of species at approximate trophic levels 1, 2, 3, and 4. The corresponding node richness equals {9, 42, 35, 24}, respectively, leading to a theoretical lack of niches of 22. For Lomni = 0 each species in the constructed food web only consumes species belonging to the trophic level below, i.e. all species have zero food chain length variation. Panels show results of simulation using varying numbers of omnivorous links (Lomni , as labeled) to perturb a system of species with well-defined trophic levels. Omnivorous links produce consumer-resource pairs where the difference in trophic levels between the respective partners is larger than 1. e, Similar to (d) but for omnivorous links only allowed between species formerly located at trophic level three and constraining the omnivorous links to include interaction with another species at level three. f, Dependence of rank difference (“lack of niches”) on Lomni for the two cases shown in d and e as dashed and solid curves, respectively. Percentages on top horizontal axis specify the fraction Lomni /L, i.e. the fraction of omnivorous links.
36
A10.3
Simulations of parasite addition
We now perform a similar analysis for webs including parasites (Fig. A14). Starting again from the idealized free-living food web Rsim f ree (Fig. A14a), we perform four distinct simulations: 1. Addition of the empirical number of parasite species that interact with any existing levels at random, to ensure that each parasite will be connected. Initially, each parasite is granted one link to a random existing species. All subsequent links are completely random (Fig. A14b). 2. Addition of parasites that are specialized at consuming species of a given trophic level. All parasites are thereby first assigned a random host, this defines the specific trophic level of hosts for the given parasite. Subsequent links are then added by ensuring that the given parasite will only consume hosts with identical chain lengths to the first. Link additions are otherwise random (Fig. A14c). 3. Initially, each parasite is given one link to an existing free-living species at level 3 or 4. Subsequent links are assigned randomly between any parasite and any level 3 or 4 species (Fig. A14d). 4. Similar to (d) but parasites consume hosts at levels 3 and 4 and also previously added parasites at random (Fig. A14e). Approximately a 5 percent fraction of parasite links are parasite-parasite links.
Simulations show qualitatively different results for the four cases: (1) Random addition and consumption behavior of parasites leads to rapid degradation of the nullity, addition of less than 5 links per parasite generally suffices to remove the nullity, i.e. allow full rank. (2) Completely level-specific addition of parasites leads to a slight initial reduction of rank. This is due to a compensating effect of adding parasites to random hosts: Dominant trophic levels will acquire more parasites. Those parasites are then counted towards the node richness at the neighboring (higher) trophic level. Apart from this (statistical) effect, the addition of level-specific parasites can not remove the nullity. (compare the barplot in Fig. A14a,(ii) where we also indicate the expected nullity when adding parasites). (3) A similar effect is initially seen for parasites that consume at two trophic levels. There, an initial increase in nullity is found due to the larger likelihood of consuming at level 3, thus adding more species to level 4 and thereby increasing the consumer-dominated state. The nullity however declines when further links are added and saturates at a value of df +p = 21, which is obtained by pairing parasites with all level-4 species and the remaining unpaired parasites with level 3 (compare Fig. A14a,(iii)). Another case arises, when parasites are viewed simply as adding an additional trophic level, i.e. all parasites consume exclusively level-4 species. This case in fact yields again a completely structured food web, where simply a level 5 with n5 = 47 species is attached to the existing free-living web. The assembly rules then predict that only n4 = 24 parasites can be paired, leaving n5 − n4 = 23 parasite species unpaired. This sets the nullity of the resulting matrix (Fig. A14a,(i)). This also implies that adding parasites can in fact increase the 37
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Figure A14: Simulations of different food web matrices. a, The lack of niches d as a function of the number of links per parasite for the different cases b—e (as labeled). Vertical lines indicate the standard deviation of results when repeating for many different samples of food web matrices. Barplot indicates the resulting distributions of node richness on the respective trophic levels when considering parasites as an additional trophic level (i), in the case described in c, (ii), as well as that described in d, (iii). b, Addition of parasites that form random links to existing free-living species. c, Addition of parasites that are confined to consumption on a specific trophic level. d, Similar to (b) but with the restriction of parasites consuming only free-living species at levels 3 and 4. e, Similar to (d) but with additional parasite-parasite interactions (hyperparasitism), approximately 5 percent of parasite links are from parasite to parasite. Panels are reproduced from main text Fig. 5c—g.
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nullity, if hosts are dominantly located at a specific, single trophic level. (4) When parasites can also consume other parasites (in the empirical data typically more than five percent of parasite links involve parasite-parasite interaction) the nullity can again be removed and approximately 10 links per parasite are sufficient for the moderate concentration of parasite-parasite links used here. Within the limitations of the idealization made, reality is best described by the curve corresponding to (Fig. A14e), when also the order of interaction is random, i.e. in some cases parasites consume freeliving species, but in others the order is reversed. This best mimics the overall features seen in the data (Fig. A20), also regarding the shuffling of average chain lengths (indicated at the matrix boundaries in color shades). Note also, that in real food webs there are also interactions between parasites and level-1 and level-2 species (Figs A20 and A9), which could also be included in the model. Specific food webs (details). We again compare with several empirical cases: • The Bahia Falsa web (Fig. A20a) might be reasonably well described by Fig. A14e with substantial interaction between parasites and level-3 and level-4 free-living species but also some interaction within the parasite-parasite sub web. This may explain the substantial reduction of nullity for that web (df = 16 → df +p = 3, compare Fig. A12c,d). • The Ythan Estuary food web (Fig. A20g) does not include any parasite-parasite interaction and shows relatively strict hierarchical organization, even for interactions involving parasites. Hence, that particular web might be best described by Fig. A14d, leading to a sizable nullity, even with parasites (df = 17 → df +p = 22). We have tested, whether adding several random parasite-parasite interaction to that food web could change the nullity. We find that when adding on average one parasite-parasite link to each existing parasite (the approximate value of the other food webs), the nullity decreased to values as low as 7 or 8. • Ythan Estuary further has comparably low numbers of links per parasite (Lp /Sp = 4.2 compared to an overall average of more than 15), which further makes it difficult to achieve full rank. • Another food web, Flensburg Fjord, maintains relatively high nullity (df +p = 10). There, adding further random parasite-parasite interactions can even reduce the nullity to zero. • Otago Harbor stands out with a very small fraction of parasite species (Sp /Sf = 0.15, compared to an average of 0.43), but an overweight of even-level species of 37. This extreme configuration in the measured data might possibly be a hint of an under-sampling of parasite species in addition to the possible under-sampling of basal species.
A10.4
Simulations of the effect of concomitant predation
We also perform simulations to describe the effect of concomitant predation. There are two qualitatively different effects present in the empirical data (Fig. A13d and Fig. 3g, main text): 39
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Figure A15: Schematic of trophic levels with concomitant predation. a, Free-living food web (reproduction from Fig. A12b). b, Inclusion of parasites feeding on a subset of hosts at trophic levels 3 and 4. After pairing, there are 17 remaining “unpairable” parasites, n3 and n4 have been reduced to 17 and 12 unpaired species, respectively. c, Inclusion of asymmetric concomitant links in the web shown in (b). Use of free-living species for pairing indicated by blue boxes on negative vertical axis. Use of concomitant links for directed pairing indicated by dashed blue boxes on negative vertical axis. d, Including also symmetric concomitant links. e, Schematic showing the type of concomitant links present in the simulated food web. “con” indicates the concomitant link. Numbers indicate the trophic levels of the parasite’s (blue) free-living host (yellow) and host’s predator (green). Panels also show the respective niche deficiencies df , df +p , dcon,asym , and dcon,sym . 1. Symmetric concomitant links lead to an overall improvement of rank, i. e. a reduction in rank deficiency. 2. Asymmetric concomitant links (only affecting the parasite) lead to an improvement in rank in only few cases, most food webs do not show any change in rank. To start the discussion, it is important to note several empirical observations: Parasites predominantly interact with species at more than one trophic level, e.g. as implied by the variation in their chain lengths (Fig. A26). A substantial fraction of free-living species do not have parasite interactions (table in Fig. A9), e.g. approximately 50 percent for the Ythan Estuary web. In many webs, parasites also interact with other parasites (Fig. A12c,d), but there are exceptions (Ythan Estuary).
Parasite-free-living links only. To allow for a simple model, we again start from the idealized food web structure (Fig. A12b) and the matrix Rsim f ree . We then include a typical number of parasites (set to 47), and allow these to prey on a fraction (we choose 50 percent for simplicity) of species at levels 3 and 4 with otherwise random links (Fig. A15a). In practice, parasites do prey on all trophic levels (albeit with some bias towards the higher levels). Our theory can easily be extended to more levels.
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Figure A16: Numbers of loops of length three for simulations and empirical data. a, Food web formed by free-living and parasite links. Schematic in (a) indicates the types of triangles that can be formed (bi-directional between level 3, level 4 and parasite species, shown in yellow, green and blue, respectively); b, Food web including additional concomitant links. Schematic in (b) indicates the additional type of triangle that can be formed when including concomitant links. c, as in (a) but for the empirical Ythan Estuary data (reproduction of Fig. A9g, middle panel). d, as in (c) but for the empirical Ythan Estuary data (reproduction of part of Fig. A9g, right panel). Given a sufficient number of links, the usual pairing will allow those free-living species which interact with parasites to pair with these, i.e. half the species at levels 3 and 4 can pair with parasites, leaving a reduced number of unpaired (and “unpairable”) parasites (see panel b). Collecting the remaining unpaired free-living and parasite species, the lowest possible nullity is df +p = 45.
Asymmetric concomitant links. We now examine how inclusion of asymmetric concomitant links can improve the situation. Consider the directed loop shown in Fig. A15e and let all level-4 species, previously lacking connections to parasites, participate in such loops of length three (this requires sufficient numbers of links, but as we will see in the simulations, this is usually possible). The loops engage n4 /2 triples in directed pairings. The remaining n4 /2 free-living species are open to bi-directional interactions with parasites and can hence also be paired. We arrive at the situation shown in Fig. A15c, where the types of pairings are indicated. Notably, the nullity has now been reduced to dcon,asym = 33. Symmetric concomitant links. Consider now symmetric concomitant links (Fig. A15d). In this case, level-4 free-living species are likely to acquire bi-directional links to parasites and can all be paired. Level-3 species however do not acquire additional links to parasites, only half of these could be paired. In the example at hand, there is already an overweight of even level species, level-3 species can therefore be 41
equally well paired with level-2 species. In any case, the nullity can now be decreased to ncon,sym = 21. ree We have simulated all the above cases using the idealized matrix Rfsim (Fig.A17a—c). In particular,
we have also checked if indeed more loops of length three are produced when adding concomitant links. Fig. A16 shows the number of triangles when simulating free-living and parasite species alone (shown in panel a), as well as additional concomitant links (panel b). In the panels, we distinguish three groups of species: those without parasites (levels 1 and 2), those at level 3 (some have parasites) and those at level 4 or parasites. Notably, in Fig. A16a, loops of length three can only be formed when a parasite interacts both with a species at level 3 and level 4 and additionally, the level 4 species preys on the level 3 species (shown in the schematic). This means, that any free-living species, that does not directly interact with a parasite, can not be involved in a loop of length three (there are correspondingly a number of species without loops, even in levels 3 and above). Conversely, when adding concomitant links, additional triangles can be formed (see the schematic in Fig. A16b). Notably, level 4 species that were not involved in any triangles in Fig. A16a now often have substantial numbers of triangles. These idealized simulation results should be compared to Fig. A9g, central and right panels (Ythan Estuary web, reproduced also in Fig. A16c,d). Many of the features are similar, in particular, the increase of the number of triangles when adding concomitant links, the large number of free-living species without triangles in the case without concomitant links, as well as the reasonable quantitative agreement. The plots differ in that the actual data does have some triangles for low-level species, which is due to the fact that parasites do interact with some species at those levels as well, and omnivory does exist for free-living species. Additional parasite-parasite links. When a small fraction of links between parasites are included, the system will acquire the ability to pair many parasites with other parasites, which can sometimes allow all parasites to be paired with others. However, a subset of parasites might be needed to pair free-living species, in the example, a number of level-4 species should be paired to reduce the overweight of the even levels. As only n4 /2 of these have parasite links, only these can be paired. The nullity can then be reduced to df +p = 10. Adding additional asymmetric concomitant links will not help, since those always involve both a level-3 and a level-4 species. In this case, there will be no benefit in such 3-species pairs, since they will not reduce the overweight of the even levels. The situation is different for symmetric concomitant links, where additional pairings between level-4 species and parasites become possible, thereby reducing the overweight of the even levels. Simulating the case where approximately 15 percent of parasite links were parasite-parasite links, overall reduction of the nullity is achieved (Fig.A17d), even without inclusion of concomitant links. However, considering now the inclusion of asymmetric concomitant links (Fig.A17e), the nullity is not decreased further (df +p = fcon,asym = 10). When including symmetric concomitant links, full rank is achieved. This type of simulation (where some parasite-parasite links are present) might be a reasonable abstraction of what is seen in the webs in Fig. A20a,b,c,d,f, where the nullity changes with symmetric, but not with asymmetric, concomitant links.
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Figure A20: Empirical food web matrices. a—g, Food webs as labeled. Empirical data for freeliving webs (left column), free-living and parasite species webs (central column), as well as those including additional concomitant links (right column). Matrix indices organized by approximate food chain length (basal species: bottom right). Blue and red matrix elements indicate direction of links. Black and gray stripes at boundary of empirical food web matrices indicate ranges of approximate trophic levels. Additional color-coding along the matrix boundaries indicates the respective location of species in freeliving webs in the parasite webs, e.g. most basal and level-2 species remain at approximately unchanged locations.
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Acknowledgments The authors acknowledge financial support by the Danish National Research Foundation through the Center for Models of Life.
Conflict of Interest Statement The authors have no conflict of interest to declare.
References Bascompte, J., Meli´an, C. J., and Sala, E. (2005). Interaction strength combinations and the overfishing of a marine food web. Proceedings of the National Academy of Sciences of the United States of America, 102(15):5443–5447. Brose, U., Williams, R. J., and Martinez, N. D. (2006). Allometric scaling enhances stability in complex food webs. Ecology Letters, 9(11):1228–1236. Dunne, J. A., Lafferty, K. D., Dobson, A. P., Hechinger, R. F., Kuris, A. M., Martinez, N. D., McLaughlin, J. P., Mouritsen, K. N., Poulin, R., Reise, K., et al. (2013). Parasites affect food web structure primarily through increased diversity and complexity. PLoS biology, 11(6):e1001579. Fagan, W. F. (1997). Omnivory as a stabilizing feature of natural communities. The American Naturalist, 150(5):554–567. Gaedke, U., Straile, D., and Pahl-Wostl, C. (1996). Trophic structure and carbon flow dynamics in the pelagic community of a large lake. In Food webs, pages 60–71. Springer. Gross, T., Rudolf, L., Levin, S. A., and Dieckmann, U. (2009). Generalized models reveal stabilizing factors in food webs. Science, 325(5941):747–750. Haerter, J. O. and Sneppen, K. (2012). Spatial structure and lamarckian adaptation explain extreme genetic diversity at crispr locus. mBio, 3:4. ¨ Hafner, L. (1978). Okologie. Schroedel-Verlag. Heath, M. R., Speirs, D. C., and Steele, J. H. (2014). Understanding patterns and processes in models of trophic cascades. Ecology letters, 17(1):101–114.
51
Hechinger, R. F., Lafferty, K. D., McLaughlin, J. P., Fredensborg, B. L., Huspeni, T. C., Lorda, J., Sandhu, P. K., Shaw, J. C., Torchin, M. E., Whitney, K. L., et al. (2011). Food webs including parasites, biomass, body sizes, and life stages for three california/baja california estuaries: Ecological archives e092-066. Ecology, 92(3):791–791. Huxham, M., Beaney, S., and Raffaelli, D. (1996). Do parasites reduce the chances of triangulation in a real food web? Oikos, pages 284–300. Johnson, P. T., Dobson, A., Lafferty, K. D., Marcogliese, D. J., Memmott, J., Orlofske, S. A., Poulin, R., and Thieltges, D. W. (2010). When parasites become prey: ecological and epidemiological significance of eating parasites. Trends in Ecology & Evolution, 25(6):362–371. Lafferty, K. D., Allesina, S., Arim, M., Briggs, C. J., De Leo, G., Dobson, A. P., Dunne, J. A., Johnson, P. T., Kuris, A. M., Marcogliese, D. J., et al. (2008). Parasites in food webs: the ultimate missing links. Ecology letters, 11(6):533–546. Lafferty, K. D., Dobson, A. P., and Kuris, A. M. (2006). Parasites dominate food web links. Proceedings of the National Academy of Sciences, 103(30):11211–11216. Lafferty, K. D. and Morris, A. K. (1996). Altered behavior of parasitized killifish increases susceptibility to predation by bird final hosts. Ecology, 77(5):1390–1397. Lov´asz, L. and Plummer, M. D. (1986). Matching theory. New York. MacArthur, R. H. (1967). The theory of island biogeography, volume 1. Princeton University Press. McCann, K. S. (2000). The diversity–stability debate. Nature, 405(6783):228–233. McPeek, M. A. (1996). Trade-offs, food web structure, and the coexistence of habitat specialists and generalists. American Naturalist, pages S124–S138. Montoya, J. M., Pimm, S. L., and Sol´e, R. V. (2006). Ecological networks and their fragility. Nature, 442(7100):259–264. Mordecai, E. A. (2011). Pathogen impacts on plant communities: unifying theory, concepts, and empirical work. Ecological Monographs, 81(3):429–441. Mougi, A. and Kondoh, M. (2012). Diversity of interaction types and ecological community stability. Science, 337(6092):349–351. Mouritsen, K. N., Poulin, R., McLaughlin, J. P., and Thieltges, D. W. (2011). Food web including metazoan parasites for an intertidal ecosystem in new zealand: Ecological archives e092-173. Ecology, 92(10):2006–2006.
52
Otto, S. P. and Day, T. (2007). A biologist’s guide to mathematical modeling in ecology and evolution, volume 13. Princeton University Press. Pauly, D. (1996). Fishing down marine food webs. Science, 273:1706. Smith, J. M. and Slatkin, M. (1973). The stability of predator-prey systems. Ecology, 54(2):384–391. Tao, T. and Vu, V. (2006). On random±1 matrices: singularity and determinant. Random Structures & Algorithms, 28(1):1–23. Thieltges, D. W., Reise, K., Mouritsen, K. N., McLaughlin, J. P., and Poulin, R. (2011). Food web including metazoan parasites for a tidal basin in germany and denmark: Ecological archives e092-172. Ecology, 92(10):2005–2005. Thingstad, T. F. (2000). Elements of a theory for the mechanisms controlling abundance, diversity, and biogeochemical role of lytic bacterial viruses in aquatic systems. Limnology and Oceanography, 45(6):1320–1328. Thompson, R. M., Hemberg, M., Starzomski, B. M., and Shurin, J. B. (2007). Trophic levels and trophic tangles: the prevalence of omnivory in real food webs. Ecology, 88(3):612–617. Tilman, D. (1990). Constraints and tradeoffs: toward a predictive theory of competition and succession. Oikos, pages 3–15. Tilman, D. (1994). Competition and biodiversity in spatially structured habitats. Ecology, 75(1):2–16. Tilman, D. (2000). Causes, consequences and ethics of biodiversity. Nature, 405(6783):208–211. Tilman, D. (2011). Diversification, biotic interchange, and the universal trade-off hypothesis. The American Naturalist, 178(3):355–371. Williams, R. J. and Martinez, N. D. (2004). Limits to trophic levels and omnivory in complex food webs: theory and data. The American Naturalist, 163(3):458–468. Wilson, E. E. and Wolkovich, E. M. (2011). Scavenging: how carnivores and carrion structure communities. Trends in ecology & evolution, 26(3):129–135. Zander, C. D., Josten, N., Detloff, K. C., Poulin, R., McLaughlin, J. P., and Thieltges, D. W. (2011). Food web including metazoan parasites for a brackish shallow water ecosystem in germany and denmark: Ecological archives e092-174. Ecology, 92(10):2007–2007.
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