Network architecture of food webs Q: How do food webs tend to differ from other types of networks that have been studied? smaller, also energy flow rather than info processing It is difficult to figure out the genesis of a network simply by looking at its end state. Therefore, new topological measures are needed that might provide additional insight into network genesis (Guimarães et al., 2005), especially for networks such as food webs whose genesis cannot be directly observed. Food webs have very few “motifs” in common with other networks (Milo et al., 2005). (Give examples?) None in common with WWW, only one in common with neuronal networks. That’s because... Most networks that have been studied are large, with thousands or even millions of nodes (Barabási and Albert, 1999), whereas most food webs are described with 200 nodes at most. This in itself makes it difficult to compare food webs with other networks. Despite recent efforts to come up with general rules, networks in the real world show much variety in their topological properties (Aldana et al., 2000, and other REFs). Garlaschelli et al. (2003) suggested that food Figure X (Figure 1 from Garlaschelli et al., 2003). Transformation of a food web (a) into a minimal webs be thought of as transportation spanning tree (b). Species are arranged in trophic networks, such as watersheds or vascular levels; the black node represents the abiotic systems. (Food webs are transporting energy environment. See the authors’ paper for details. across trophic levels). There exists welldeveloped theory to determine the maximal transport efficiency in these systems; however, the theory does not allow loops, so Garlaschelli et al. reduced the food web to a “minimal spanning tree” based on shortest path lengths (Figure X). In so doing, they found that food webs can be described as self-similar trees with an efficiency exponent of η = 1.13. (1 is maximally efficient, and 2 is maximally inefficient.) They found that this exponent η was remarkably consistent across different webs, not related to web size or connectance. Presumably this is due to an optimization process in ecological communities (the only reason trophic levels exist at all is b/c of competition; they’re necessarily inefficient). But, van Veen and Murrell (2005) refuted this seemingly interesting result, showing that, given a realistic number of trophic levels, Garlaschelli et al’s method of constructing a minimum spanning tree would inevitably lead to η ≈ 1.13. Allesina and Bodini (2005) also published a refutation, saying that minimal spanning trees are an unacceptable simplification, as they reduce everything to its lowest trophic level. They considered food webs as directed acyclic graphs (DAGs), the only modification from a real food web is the removal of loops. Their calculated values of η indicated systems that were far from optimal transport efficiency. That is to be expected, due to the presence of various biological interactions such as competition. Q: Can food webs be thought of as information-processing systems, like neural networks? ZM: It would be neat if a gestalt shift could be made, and the food web could be seen as computing some function, in the same way that a gas released in an evacuated cylinder tries to fill the cylinder, even though the molecules don't care. So far I haven’t come across any research that compares trophic networks to computational networks. To the contrary, Milo et al. (2005) postulate that because food webs represent energy flows, not information processing, it is reasonable to expect their network structure to be fundamentally different from neural networks or the World Wide Web. (REF) studied various kinds of networks ranging from electric power grids to friendship circles. They found that.... Q: Do food webs show scale-free architecture, that is, does the link density decay as a power law? Not exactly. It seems that food webs are not, as a rule, scale-free (Dunne et al. 2002; Garlaschelli et al., Fig. 3 (Fig. 2 from Dunne et al., 2002). Linear-log plots of the cumulative distributions of links per species (both predator and prey links) in 16 food webs. Webs are ordered by increasing connectance. Lines and r2 values show the fit to the data of the best of three simple models: power–law distribution (upward curved line), exponential decay (straight line), or uniform distribution (downward curved line). No food web is well fit by a Poissonian or Gaussian distribution. 2003; Williams et al., 2004; Guimarães et al. 2005; Stouffer et al., 2005). Most food webs are more homogenous in their link structure than the scale-free model would predict (i.e., food webs don’t have as many highly-linked nodes as predicted). Instead, food webs tend to fall into the category of “broad-scale” networks (usually, the number of connections per node initially decays as a power law, but the tail is truncated by an exponential decay; Guimarães et al., 2005). This may be due to the small size of food webs relative to the scale-free networks that have been studied; it may be due to their mechanisms of network formation (namely, a lack of preferential attachment, and restrictions on forming new links); or it may be due to the presence of an initial “core” of links that follows different rules. Note that this usage of the term “scale-free” is not synonymous with “scale-invariant.” The latter, as applied to food webs, means “does not vary across food webs as a function of S (the number of species in the food web).” For some examples of which food web parameters are scale-invariant, see the list beginning on page 12. Early assertions of scale-freeness are largely a result of limited data (Dunne et al., 2002). Of the 16 high-resolution food webs examined by Dunne et al. (2002), only one showed a scale-free degree distribution, whereas eight showed an exponential distribution and six showed a uniform distribution. Stouffer et al. (2005) agreed that the exponential distribution seems to be the most common type of distribution for food webs. (Montoya and Solé, 2002) studied three food webs (Ythan Estuary, Silwood grassland, and Little Rock Lake) and reported that the the first two followed a power-law decay of link density distribution, with γ = 0.94 and γ = 1.13. They explained the lack of fit for the third food web, and other food webs in general, as an artifact of poor taxonomic resolution. Dunne et al. (2002) criticize Montoya and Solé (2002) for looking at only three food webs, using absolute rather than cumulative degree distributions, and dismissing a contradictory result as an artifact. No reason to think that food webs would have the same assembly processes, and therefore the same architecture, as other types of networks (Dunne et al., 2002). The tail of the link distribution may contain information about how the network formed (Guimarães et al., 2005). There are three types of network: single-scale (link density distribution decay is Gaussian), broad-scale (decay is exponential), and scale-free (decay is power-law). (Amaral et al., 2000). (and other REFs). Scale-free networks can be created by preferential attachment. But this doesn’t apply when nodes “age” or have a “cost” of adding links. (Amaral et al., 2000); that can discuss . Some FWs are scale free, but most are not if they exceed relatively low levels of connectance. (Dunne et al., 2002). It is agreed, however, that the degree distributions of food webs certainly are not random (Dunne et al., 2002). Smaller food webs seem to be closer to random (Montoya and Solé, 2003), but even those do not have homogenous degree distribution. Important to use cumulative, rather than absolute, degree distributions for small networks, because otherwise the “bins” will be too small and the data will be too noisy (Dunne et al., 2002). From an e-mail to ZM, 7/23/06: “So as a small food web accumulates more species, and thus grows bigger, and thus (as described above) each individual species accumulates a larger number of links on average... that doesn't necessarily disrupt the scale-freeness, it just changes the exponent thereof. Right?” Umm... actually, that doesn’t sound right to me. Q: Do food webs show small-world architecture, that is, do neighbors of a given node also tend to be connected to each other? Some say yes, some say no. Again, early assertions of small-worldness are largely a result of limited data (Dunne et al., 2002). Food webs are not scale-free (Garlaschelli et al., 2003). Small-world networks (as defined by Strogatz, 2001) show an exponential decay of degree distribution (Barabási and Albert, 1999). Scale-free networks are necessarily small-world networks, but not vice-versa (Amaral et al., 2000). Small-world structure can contribute to network homeostasis (Montoya and Solé, 2002). Q: Is there evidence of compartmentalization of food webs? Disagreement abounds on this issue. Melián and Bascompte (2004) studied 12 empirical food webs and found subwebs in all of them: that is, the “densest subweb” their algorithm could find was always significantly denser in the real food web than in its random equivalent. Similarly, Krause et al. (2003) presented a detailed study of a Chesapeake Bay food web and found that it contained two clearly defined subwebs: benthic organisms and pelagic organisms. Other authors disagree that compartmentalization is a characteristic of food webs (REF). If there is significant compartmentalization, that would imply that C (connectance) is not an adequate measure of food web structure. Melián and Bascompte found that C of the densest subweb does not depend on the size of the subweb, whereas C of the whole web does depend on the size of the web (note that this conflicts with Martinez, 1992). This can’t be adequately explained by any null model except for the local rewiring algorithm, which isn’t exactly a null model. How does this relate to the definition of food webs as “small-world” networks? Are they small-world networks? (REF) Various methods exist for determining and quantifying compartmentalization (contrast the ksubweb approach of Melián and Bascompte with the KliqueFinder algorithm of Krause et al. with the “uplinked mutual dyad” motif of Milo et al. (2002).) (well uh... maybe not that last one) In small-world networks the average path length increases as log(n), where n is the number of nodes (in food webs, S). Clustering coefficients are much higher than expected in smallworld networks. (Amaral et al., 2000). Scale-free networks are necessarily small-world, but small-world networks are not necessarily scale-free (Amaral et al., 2000). One unanswered compartmentalization problem is: do the “loner” species (those not very closely connected; on the periphery) tend to fall into any particular category? Stability of food webs Q: How is stability defined with respect to food webs? Does it mean resistance to change, resilience from change, or both? General note: Both dynamic stability and topologic stability are important considerations for food webs (Melián and Bascompte (2004) . I don’t understand what “topologic stability” means, though. -------“Population persistence in large and complex natural food webs remains a long-standing dilemma in ecology.” (Brose et al., 2003) This is a matter of some confusion. There is no single agreed-upon measure of stability. Brose et al. (2003) define “community robustness” as the fraction of species within a network that persist (presumably as t goes to infinity). The concept of stability may or may not include the idea of an outside perturbation. Montoya and Solé (2003) define stability as the speed with which the system recovers after a disturbance. In the general ecology literature, “stability” usually has connotations of both resistance and resilience. The mathematical definition we learned of “stability” in ER 202 was something along the lines of “the system will, eventually, return to its initial state after a perturbation of arbitrary size.” (Virtually all systems are stable within certain limits, that is, if the perturbation does not exceed ε.) This mathematical definition only includes the concept of resilience (returning to an initial state), and ignores resistance. It sets no limit on how wildly the state of the system can fluctuate before it returns to the initial state, nor how long the system can take to regain its initial state (the answer is usually ∞, of course.) I’m not sure if stability for food webs is, or should be, defined similarly... ZM: Other notions come to mind. For instance, meta-stable states. Maybe the system is stable in terms of always being in some region of phase space - perturb it and it moves, and possibly doesn't ever come back to its original position, but stays in its phase space region. Dunne et al. (2004) define “robustness” as the fraction of species that must be removed to cause the loss of 50% of the total species. Maximum robustness, therefore, is 0.5 (there are no “secondary extinctions”). (There’s much more to say on this... it’s not at all complete.) “Stability” of ecosystem processes (for example, nutrient cycling) is not necessarily the same as stability of any particular species or even group of species (Tilman, 1996 – don’t have REF). Therefore, a food web could be “unstable” in that its species are fluctuating or disappearing, but the ecosystem could still work just fine overall. The most strictly mathematical definition, but not a very useful one, is the “Lyapunov pointset stability.” A more general definition is “tendency to persist over time,” or “tendency to recover from disturbance.” (Vandermeer, 2006) Aldana and Cluzel (2003) define “robustness” as the “ability... to perform a sequence of biological tasks in the presence of perturbations.” They were referring to intracellular processing networks, not food webs, but the principle is perhaps the same. Neubert et al. (2004) define a system as stable if “small perturbations to the equilibrium eventually decay,” and define resilience as the asymptotic rate of decay. He also introduced a term called “reactivity,” which means “how much worse things get before they get better” – i.e., after the initial disturbance, the system tends to continue to move away from equilibrium for awhile. Mathematically, they show that most dynamic food web models with densityindependent mortality rates tend to be reactive. Density-dependence, on the other hand, reduces reactivity. This is an example of a potentially important non-trophic effect that is usually omitted from food web models. Q: What types of food webs tend to be most stable in real life? Does this align with what is predicted and/or observed for other types of networks? The ongoing complexity / stability debate is still unresolved. (For a mediocre review, see Worm and Duffy, 2003). There haven’t been many empirical studies of food web stability; it’s been mostly model simulations. The main question everyone seems to want to answer is “How can (all) food webs be so stable when we have so much trouble making our models turn out stable?” The niche model and the cascade model generally predict that, all other things being equal, increasing connectance should decrease robustness. (Brose et al., 2003). Here, robustntess is defined as “the fraction of species that persist” after a long time period. One can see why that would be so: disturbances could more easily propagate through the web. But, on the other hand, being more connected may give you more “options” if one of your prey species goes extinct. (I really don’t understand “topological stability,” that is, the stability of static food web models, at all... how can you model extinctions without dynamics?) Compartmentalization in food webs should, theoretically, make them more resistant to disturbance, as the effects of disturbance should be mainly limited to one compartment (Krause et al., 2003). (Compare this to theory on small-world networks). Neutel et al. (2002) propose that weak interactions in long “trophic loops” are crucial for the stability of food webs. They attribute these weak interactions to both the pyramidal structure of biomass at different trophic levels, and to the presence of omnivores. I found their paper tricky to understand... perhaps I should read it again. Neutel et al. (2002) introduced a new idea called “loop weight,” the geometric mean of the absolute values of the interaction strengths within a closed chain of trophic links. (They used directed links, considering a Jacobian matrix of interaction strengths for each species pair. Hence any two or more species would form a loop: A  B.) They demonstrated mathematically that stability was enhanced by long trophic loops that were made up mostly of weak links and a few strong links. In accordance with this, they found that maximum loop weight was much lower in real community matrices than in randomized ones. They attributed this to (1) omnivores, who feed at many different levels and thus “spread their weights around,” and (2) pyramidal biomass structures, which encourage feeding at lower trophic levels simply because that’s “easier” (hence weakening the links to higher trophic levels). Community tends to be stable when the intra-specific interactions tend to be strong compared to the inter-specific interactions (Neutel et al., 2002). As far as empirical studies go – that is, resistance of real ecosystems to disturbance – attention has focused mostly on the basal species of ecosystems (Worm and Duffy, 2002) and not given enough consideration to predator-prey interactions. Need to say something about deleting heavily connected nodes versus random nodes. (REF) Omnivory can be either stabilizing or destabilizing depending on the conditions (Vandermeer, 2006). In this simulation, a stable system was sometimes destabilized by an omnivore, and vice versa. Intermediate values of omnivory seemed to be the most stable. He didn’t use parameters that had much to do with biological reality, though. Dunne et al. (2004) found that, in marine food webs, higher connectance (L/S2) seemed to confer robustness to species loss, even though theory might have predicted the opposite. Food webs with skewed degree distributions (ie power-law or exponential decay) tend to be robust to the removal of most species, but fragile when the highly-connected species are removed. By contrast, food webs with Poissonian degree distributions are fragile to the removal of any species. (Montoya and Solé, 2003). Q: In empirical and/or theoretical food webs, what generalizations can be made about (1) nestedness / fractal behavior; (2) density of connections; (3) interaction strength? (Of course this may end up being several separate questions if I get enough material.) These answers may end up overlapping with the answer to “What makes a food web stable”? ZM: I don't know what a theoretical food web is. Well, any graph I want to call a food web, I suppose, but hard to say anything sensible about those. Q: Is there any a priori reason to expect scale-freeness, or complexity, to enhance stability? Uhhh... Scale-freeness, yes. See Aldana and Cluzel (2003). Food webs may be scale-free enough to see some of those benefits, but it seems unlikely, since (as far as I know) there are few published examples of food webs that show scale-free behavior at all, and (as far as I know) none that calculate a γ greater than 2. Complexity... well, the jury’s still out (Worm and Duffy, 2003). The debate is between “portfolio theory,” which (as applied to ecology) suggests that diverse food sources make a species resistant to environmental fluctuations, versus May (1974) who created food web models in which increasing trophic complexity increased the probability of destabilizing positive feedbacks (Martinez and Dunne, 1998). Kondoh (2003a, 2006) published the controversial idea that a complexity/stability relationship emerges if one assumes adaptive foraging. That is, if a predator focuses its foraging efforts on “easy prey” in order to maximize energy gain and minimize effort, it will tend to shift away from prey species when their population densities get low, thus making extinctions much less likely. Yay! I explained that better than they did! :-) Dunne et al. (2004) did not find that robustness increased with S. High connectance means (usually) high robustness to species loss, because everything has alternate prey sources. Short path length would tend to decrease stability, because everything effects everything else. But small-world networks get around that problem by compartmentalization. (REF?) Evolution of food webs Q: How do current food web models account for change of food webs over time? They don’t, usually. Most of the community assembly literature bypasses the issue of trophic structure (Montoya and Solé, 2003) and focuses instead on competition. Loeuille and Loreau (2005) were able to create a simple evolutionary food web model, based on body size, that recreated many of the results found in the community assembly literature. E.g. after enough time has passed, basic structure remains even though species composition may change; and invasability decreases over time. (Montoya and Solé, 2003) created a simple dynamic assembly model but failed to reproduce key properties of food webs. (Need to look at their Methods.) based on Pimm, 1987. Q: What sorts of processes can lead to a change in food web structure? - adjacent food webs interacting? (ZM) (yes, they do interact – examples?) - Evolution of new species from within - Immigration of species from outside - Extinction of species, either through biotic or abiotic factors (I would say “immigration” just to be parallel, but that’s the same as local extinction) Q: What empirical evidence do we have (direct or indirect) of changes in food webs over time? There is very little empirical data on this question. Three different possible approaches come to mind: - Paleontological evidence (e.g. Crame, 1990) - Recent introductions of species to existing food webs - Comparison of similar food webs that have diverged allopatrically5 (assuming we can’t just directly observe changes) 5 Thanks to Rex Kerr for this suggestion. It would help to understand and explain the network structure of food webs if we could actually “see” how the networks evolved, that is, how new nodes (species) attached themselves. This point is made by Montoya and Solé (2002) although they don’t say anything about how to do it. Presence of scale-freeness or other topological properties may tell us how networks evolve (Barabási and Albert, 1999). The best we’ve been able to do for paleo food webs is to estimate different prevalences of functional groups (e.g. predators, suspension feeders, and detritus feeders), roughly estimate stock and flux between each of these compartments, and fill in for the organisms we know must not have fossilized. (Crame, 1990). The biased incompleteness of the fossil record is a big problem here, though we can try to compensate. Estimating energy flow, as opposed to just biomass, is particularly difficult. It’s difficult to guess what the prey of extinct predators must have been. Q: When a new species enters a food web, what attachment rules does it follow? I think the only direct evidence we have for that is for recently human-introduced species, and it’s not necessarily safe to assume that those are representative of species in general. But, the following studies have been done... And indirect assessments have also been made, for example... (take an abstract model, give it some rules, try to replicate real FW structure, and then conclude, “Well, I guess real FWs must follow these attachment rules as well.) Did a reasonably thorough literature search on this and found essentially nothing, altohough maybe my search strategies were not ideal. Q: What tends to happen when exotic species are introduced into a food web? (And is this at all analogous to a new species evolving?) I know there is quite a bit of empirical work on this [oh yeah?], but I don’t know about the theory... Note: Do not confuse introduced species (that is, any species that comes in from the outside) with invasive species (that is, introduced species with certain characteristics that allow them to proliferate aggressively). Note also: Some authors do use the above two terms interchangeably. See Strauss et al. (2006). Follow-up Q: Do invasive species tend to have different characteristics in a food web than do native, non-invasive species? I.e. do invasives tend to have more links or fewer links? Stronger links or weaker links? Perhaps this is straying too far from the general into the particular. ZM: Oh, I don't think so at all. In so far as the immediate goal is to find a good theory explaining the scale-free topology of food-webs. If invasive species have unusual connective properties, that would be of great interest. (see Strauss et al. 2006) Researchers seem to agree that this is an important question (REFs) but the only specific things I’ve read about it is that it’s hard to answer. Takes place on a time scale too slow to measure. Cite Strauss (2006). Wish I had a better ref for that. Q: Is a food web itself a unit of evolution? I am really not sure. No one else seems to be, either. Aldana and Cluzel (2003) mention natural selection as a possibility for networks in general, but they do not mention food webs. Q: How do food webs come into existence – that is, how are they able to assemble themselves from nothing? Does the order of introduction of species matter? Yes, it most certainly does! This is the place to review the community assembly literature. ZM: Take a given food web. Have all the species involved been present and co-evolving since time out of mind? Have some been plopped down recently, while others have been hanging around and co-evolving for a long time? Is there any regularity in this? Is there a relationship between how recently a species entered a food web and its connectivity? (This seems a promising approach to scale freeness.) (I haven’t read anything about this yet.) Q: Is a newly emerging island, or some other type of primary succession, a good empirical example of food web assembly? Or, in those cases, would we simply expect neighboring food webs to reassemble themselves in the newly available habitat? AK: Should look into work on paleo food webs. This is one of Jennifer Dunne’s current areas of research. Guimaraes et al. (2005) describe how the initial core of a network may differ from its more recently added nodes. But they seemed to imply that it was not possible to examine these dynamics for food webs, because of the time scales involved. Need more refs on this. Q: What happens when two food webs collide? We could consider either a dynamic situation (i.e. one ecosystem literally crashing into another) or a static situation (e.g. the intermediate type of food web that might be found at the boundary between a forest and a grassland). Certain aspects of food web collision might be generalizable, whether it’s an old dead tree falling down on the forest floor, or the subcontinent of India ramming into Asia. In any case, this question is meant to elucidate the previous question, about the genesis of food webs. Didn’t read any papers on this... Q: Is there any sense in which food webs (or communities, or ecosystems) can be thought of as individuals undergoing natural selection? I.e., there are multiple types of species assemblages that could persist in a given abiotic environment, and it’s a sort of competition that determines which one actually will persist? And is the analogy of identical islands joined by intermittent land bridges a useful one? Aldana and Cluzel (2003) suggest that it does make sense to consider “natural selection” as a force shaping network structure; specifically, they postulate that the prevalence of scale-free networks in nature can be explained by the fact that those networks tend to be robust to perturbations. This is the only reference in which I have encountered such a postulation. But follow-up is called for. AK: This is starting to sound rather like the Gaia hypothesis. ZM: [dismisses the Gaia hypothesis], but I’m in favor of finding higher level descriptions, if they can be correctly grounded out in the physics of their substrate.