Number of predators per prey after extinction I: A start

Tags

extinction, food webs, link distribution, Network theory, networks

This series of posts are inspired by two questions that Jarrett Byrnes asked:

1. Given the extinction of E predators out of N, what is the probability that a prey species will still have at least one predator remaining?
2. Given E out of N, what then is the probability that all prey species will have at least one predator remaining?

As Jarrett and I have been discovering, these are actually quite difficult questions to answer in a general manner, i.e. for all topologies of a certain size!

Say we have a two trophic level food web with N predators, what is the probability that a prey species has at least one predator remaining after the extinction of E predators? The solution provided here depends on having the out-degree of prey species, and finding the probability that all predators of a prey species become extinct as a result of E. Say that the out-degree of the prey species is , then that probability is a hypergeometric solution

which reduces to

The probability then of a prey species having at least one prey is 1 minus the above

that is, the sum of the probabilities of having 1 predator, 2 predators, etc.

Example

Let the adjacency matrix of a food web be

where predators are rows and prey are columns. Our prey out-degree set is therefore {3, 2}. For E=1, both prey will have at least one predator since their out-degrees both exceed 1. For E=2, the possible resulting topologies are

For s=2

This is correct since our prey species of out-degree 2 (second column of A) has at least one predator in two of our three post-extinction topologies. The probability should be zero for s=3 (since E<s). If we add a third prey species, with s=1, making

then for E=1, the post-extinction topologies are

The probability that this third species has at least one prey is also 2/3.

A further example

So far so good, right? Well, Jarrett posed this example,

Notice that we now have two prey of out-degree 2. For E=1, the post-extinction topologies are

Applying the above formula yields

which is correct, since two of the three topologies maintain at least one predator for each prey. When E=2, the topologies become

Obviously, . But the formula gives

What went wrong?! The answer points to just how devilish the questions are, and how deceptive! There are two species of out-degree 2 (s=2) in the food web, hence the second term in the formula is squared (see above). BUT, the predator-prey topologies of the species are different, meaning that simple hypergeometric counting cannot work. We literally must list and examine all the post-extinction topologies, but this is prohibitively impractical for food webs and networks of even modest size (a dozen species). So there we stand. We currently have a partial solution, and I will explore the difficulty and the partial solution in the next post.

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A species’s tragedy of the commons

Tags

biodiversity, carrying capacity, cascades, competition, extinction, food webs, interaction strength, link distribution, link strength, modeling, networks, paleo-food web, paleontology, Robustness, Scientific models, simulations, Tipping point, trophic guild

My colleague Ken Angielczyk and I have a new paper out in the Royal Society‘s Biology Letters, entitled “The evolutionary palaeoecology of species and the tragedy of the commons“. If you have never read Garrett Hardin’s original paper on the tragedy of the commons, I strongly suggest that you do. It is a principle that I believe has broad application, and would well be worth a re-visit (first visit?!) by today’s leaders and economists. Our paper can be found here or here (first page only). And here is the abstract, as a little teaser!

Abstract

The fossil record presents palaeoecological pat-
terns of rise and fall on multiple scales of time
and biological organization. Here, we argue that
the rise and fall of species can result from a tragedy
of the commons, wherein the pursuit of self-inter-
ests by individual agents in a larger interactive
system is detrimental to the overall performance
or condition of the system. Species evolving
within particular communities may conform to
this situation, affecting the ecological robustness
of their communities. Results from a trophic
network model of Permian–Triassic terrestrial
communities suggest that community perform-
ance on geological timescales may in turn
constrain the evolutionary opportunities and
histories of the species within them.

Jamaican reefs lack specialized foragers

Tags

connectance, coral reef, food webs, generalists, link distribution, marine communities, specialists

Results from an earlier post, based on REEF data only, suggested that the Jamaican reef food web is more connected than either the Caymans or Cuba. Our augmented data set (see previous post) now confirms this. Jamaica, of intermediate vertebrate species richness (S=160) has a connectance, C=0.06032, while the Caymans are S=156, C=0.05949, and Cuba, S=176 and C=0.05972. Are these differences in any way significant? That’s a difficult question to answer, since we really don’t know if there is a distribution underlying food web connectance. We tested it, again as reported earlier, but asking the following question: “Given a regional species pool comprising all species observed on at least one of our islands, what is the expected C for a system of size S?” To answer the question, we generated numerous food webs with random draws from that regional species pool (1,000 food webs per S shown in the figure). There is an expected regular relationship between C and S (blue regression line), but Jamaica is well above the curve. Random draws for S equalling exactly the richnesses of the island food webs confirms that Jamaica is more significantly connected than would be expected if it was a random draw from the regional species pool (10,000 randomizations, p=0.0054; p=0.2296 and 0.0548 for the Caymans and Cuba respectively).

Recall that the formula for connectance is

Two questions now come to mind. First, why does C increase with S? And second, how can Jamaica be more connected than Cuba? The answer for the first question lies in the shape of the link distribution curves. They are right-skewed and long-tailed. This means that a random draw from one of those distributions is likely to yield a species of intermediate to low in-degree, that is, a species with a low density of links. But as the number of species drawn approaches the maximum number of species available, the probability of drawing species out on the long tail, that is, link dense species, increases. So as far as C is concerned, you’re getting more bang for you buck. Okay, then why is Jamaica’s connectance greater than Cuba’s? Because Jamaica is not an unbiased or random draw from the pool. Somehow, Jamaica has an unusually high component of link-dense or high degree species. It’s drawing preferentially from the long tail! A better way to view this, is that Jamaica is lacking in trophic specialists.

Comparison of in-degree distributions for reef foragers present in Jamaica (blue) vs. those missing (green)

A little data exploration shows this to indeed be the case. If we compare species present in one of our communities to species absent., but present in one or both of the other two communities, it is clear that Jamaica has a lower than expected number of specialized reef foragers. In other words, if you’re looking for species that forage only on the reef (and not seagrass beds), and that have a specialized diet, don’t expect to have as much luck finding them in Jamaica as you would in the Caymans or Cuba. Why? Well, that’s a very difficult question to answer. Perhaps specialists have a more difficult time establishing themselves as new colonists; but they don’t seem to have a problem other than in Jamaica. Maybe specialists, with their smaller number of resources, are more prone to extinction if the system is disturbed in some way. Or, maybe us Jamaicans find specialists to be tastier than generalists! We’ll explore these possibilities in future posts.

More resources for Cuban reef fish

Tags

coral reef, food webs, link distribution, marine communities

We’ve spent some time augmenting our REEF-based data in light of the uneven sampling intensities in the Caymans, Cuba and Jamaica. Major additional sources include GBIF and Fishbase. We now have both occurrence records and trophic data for 192 vertebrate species, with Cuba being the richest island (177 species), followed by Jamaica (163 species) and the Caymans (159 species). The trend coincides with reef area of the three systems, but we must be careful to not over-interpret these somewhat noisy data. So what have we learned so far?

Comparisons of the trophic in-degree distributions, or the number of prey species/resources of the vertebrate species reinforces our earlier result showing that these are modal, right-skewed and long-tailed distributions. They differ from “conventional” food web distributions which are almost never modal, with attention being focused on the long tail. Here, we see that on average, species in all three communities have a similar number of incoming links. There are more specialized species, but specialists are never most common. Also note the difference between Cuba and the other two systems. Density is essentially shifted toward the right tail, i.e. generalists, in the Caymans and Jamaica. What does this mean for the species? Pairwise comparisons of the in-degrees of species that are found in two systems indicate that species in Cuba (CU) have significantly more prey resources, in-links, than they do in the Caymans (CY) or Jamaica (JA) (Wilcoxon test of equality; CY vs. CU, n=153, p<0.0001; JA vs. CU, n=142,p<0.0001; JA vs. CY, n-142, p-0.5653; Bonferroni corrections applied). When broken down by the three major foraging habitats, seagrass beds, reefs, and both seagrass beds and reefs, the significance holds up for the latter two habitats. The meaning for populations in these systems is a bit difficult to predict, but it would at least imply a greater susceptibility to stochastic or exploitative perturbations of the Cayman and Jamaican communities.

Some of you might have picked up on an apparent paradox: the Cuban distribution has more specialists, yet species in the Caymans and Jamaica have fewer resources. There’s no paradox really; the difference lies in species that do not occur in two or more systems. I’ll show in the next post that not only are species in the spatially smaller communities of lesser degree, but that in at least one of those communities, there is a dearth of specialists.

New paper – Networks, extinction and paleocommunity food webs

Tags

connectance, extinction, food webs, graph, link distribution, metanetwork, Network theory, networks, nonlinear, paleo-food web, power law, probability, real world networks, Robustness, simulations, trophic guild

Roopnarine, P. D. 2010. Networks, extinction and paleocommunity food webs in J. Alroy and G. Hunt, eds., Quantitative Methods in Paleobiology, The Paleontological Society Papers, 16: 143-161. (available here).

The paper is part of a volume, Quantitative Methods in Paleobiology, sponsored by The Paleontological Society. Full details are available here. The volume is also available for sale. Purchase one and support the Society!

Species richness and connectance

Tags

connectance, coral reef, food webs, link distribution, marine communities, Network theory, power law, Robustness

Ever since Lord Robert May challenged Robert MacArthur’s assertion that there is a positive relationship between diversity and stability, the argument has raged as to whether there really is a relationship between the proxies, species richness and connectance. May demonstrated that, at least within randomly connected food webs (more properly graphs), diversity does not beget stability, and that there is a critical connectance above which the system becomes unstable. I say that richness and connectance are proxies because diversity is more than richness, and stability is more than a critical point of connectance. Many workers, stimulated by May’s contention, have since shown that the non-random connection topologies of food webs matter; that is, functional diversity and hierarchical arrangements of species interactions allow real food webs, apparently, to be far more complex than allowed in May’s framework. Is there then no limit, or indeed no relationship between species richness and food web connectance?

I showed in an earlier post that there is a positive relationship between node richness and the number of links spanning a broad array of food web types. The same has been demonstrated before, most recently by Ings et al. Indeed, workers such as Jennifer Dunne and others have hypothesized that increased connectance confers greater robustness on food webs, hence allowing increases in richness as long as complexity also increases. I, on the other hand, doubt that this relationship actually exists for several reasons. First, the data upon which these hypotheses are based are extremely heterogeneous, and it is unclear whether connectance as measured across the array of food webs is actually the same thing from one web to the next. Second, measures of robustness typically are incapable of assessing robustness against anything other than the bottom-up perturbation of unparameterized systems; that is, no link strengths, population sizes, etc. Additionally, there should hence be no expectation of similarity of connectance values among any food webs.

In continuing our work on Greater Antillean coral reef food webs, I wanted to examine this relationship for our three food webs, namely those of the Cayman Islands, Cuba and Jamaica. The food web models differ only in vertebrate richness, and are ordered as Caymans>Jamaica>Cuba. This ordination corresponds nicely with sampling events and efforts. Yet, Jamaica has by far the greatest connectance of the three. Is this unusual or unexpected? We assessed this by stochastically drawing food webs of varying vertebrate richness, ranging from 80 to 160 species, from the regional species pool, and calculating their connectances. We did this for about 9000 food webs, and discovered this very nice, linear relationship. Connectance clearly increases, linearly in this case, with increasing richness. Why? The explanation is rather simple. Recall that the in-degree distributions of the island food webs, and hence the regional pool, is modal, yet with a significant right long tail. As one increases the number of species in a randomly drawn food web, the probability of drawing species from the long tail, those of high in-degree, also increases. Think of those species as being more “link dense”. Connectance will therefore increase, and will not be a constant value.

A wonderful surprise, however, is that the real food webs do not necessarily conform to this. The Cayman and Cuban food webs are indeed indistinguishable from the random food webs, but look at Jamaica! It’s connectance is well above random expectation. We know why, but I won’t tell you yet.

We Have Met the Enemy and He Is PowerPoint

Tags

connectance, link distribution, Network theory, networks

Gen. Stanley A. McChrystal, the leader of American and NATO forces in Afghanistan, was shown a PowerPoint slide in Kabul…. It certainly would be interesting to apply some network analyses to this! Which components are central? Is the “strategy” highly modular and clustered, or diffuse? What does the degree distribution look like?

p.s. I always knew that PowerPoint was the Enemy.

Different numbers of interactions in Caribbean coral reefs

Tags

connectance, coral reef, food webs, link distribution, real world networks

In-degree cumulative distributions

The last post on this topic reported that alpha vertebrate diversity differs among reef communities in the Cayman Islands, Cuba and Jamaica, with the Caymans having the greatest species richness. I also showed that if we consider the reef communities to be random draws from the gamma-level (regional) species pool, we cannot reconstruct food webs with the observed Jamaican connectance. What’s causing this? At least a partial answer is the bias in the degrees of trophic specialization in the communities. Jamaica has greater than expected connectance because it has a relatively greater proportion of generalist species, i.e. those with high in-degrees (incoming links). Interestingly, if we compare the in-degree distributions of the communities, we find no significant differences (Kolmogorov-Smirnov tests). This first figure illustrates the cumulative frequency functions of each community. While the K-S test says no difference, however, we note that there are asymmetries in the distributions, and it could well be worth decomposing these distributions into pre- and post-modal portions. They are right-skewed and relatively long-tailed.

Another way to compare the interaction or link distributions is to look at the properties of those species, present in the regional pool, that are missing from each community. The box plots at left plot the in-link distributions of “missing” vertebrates from each community. There is a clear trend, suggesting again that Jamaica is relatively poor in trophic specialists, while the Caymans are relatively rich, with Cuba in between. K-S tests fails to confirm any significance here, but sample sizes are pretty small, and there is likely a problem with test power.

A final interesting observation. Given that there are species common to two or more of the communities, we can compare the in-degree distributions of those species only. A series of paired t-tests confirm that species in Jamaica have significantly more incoming interactions than conspecifics in the Caymans and Cuba (Pr(T>t)=0.0004 and 0.0001 respectively). Can this be reconciled with the above observations? This result is telling us that if a species exists in Jamaica and elsewhere, it will have more prey resources in Jamaica! Given that we are recording only vertebrate differences among the communities, then it means that they have more vertebrate prey resources. I find this to be very odd, and I’m going to have to wrap my brain around it a bit to explain it. Might be time to decompose those distributions.

The Probability of Extinction

Tags

connectance, extinction, link distribution, metanetwork, paleo-food web, probability, Robustness

p(extinction) when number of prey species=10

The CEG model asserts that food web structure plays a role in extinction. The intricate patterns of relationships among species in a community distribute the effects of changes in one species to others in its community. Therefore, while the ultimate causes of increased extinction in an interval of time may be abiotic, and might affect only some species directly, the effects could be felt more broadly.

Topographic secondary extinction.–The narrow definition of secondary extinction, where a species becomes extinct because it has lost all its resources, is termed topological secondary extinction (Roopnarine, 2009). Topological refers to the dependence of extinction solely upon the topology (pattern) of the network. Note that topological secondary extinction affects the network only in a bottom-up fashion, that is, in the direction of energy flow from producers to consumers of increasing trophic level. Measuring or estimating topological secondary extinction in a food web, in response to a particular perturbation, can be approached in three ways, again depending on whether one assumes accuracy of a higher-level representation of the food web (e.g. a metanetwork) or precision of a species-level network. Here I will outline a probabilistic approach using metanetworks.

Let a perturbation of magnitude be equal to the number of species removed randomly from the network. The probability that a species will become secondarily extinct is the probability that all its links are to species that are a subset of the set. This is determined from a hypergeometric distribution, where we can first ask: Given an in-degree of , what is the probability that of them will be lost?

where there are S-1 other species in the network and . The probability of becoming extinct occurs when ,

The formula can be re-stated interestingly as

where trophic breadth is the number of species consumed by species , out of a pool of potential prey (species richness). For example, in a network of 10 species, and , and the probability of extinction increases as , and decreases as (see figure). Species with more resources are thus more resistant to topological secondary extinction. This is the same as saying that the more trophically generalized a species, the greater its resistance to extinction.

This explains in part the suggestion that food webs of greater connectance (C) are more resistant to topological secondary extinction (Dunne et al., 2002). Overall food web or network resistance to this type of extinction has been termed structural robustness (Dunne and Williams, 2009). Food web connectance does not increase, however, because of uniform increases in the in-degrees of all species in the network, but increases instead because of the presence of highly linked species. The skewed, long-tailed in-link distributions discussed earlier indicate the non-uniformity of in-degrees within real food webs. The above formula for extinction shows that $latex p(e_{i}^{x})

r_{i}^{y}$, that is, is of greater in-degree than . This will also be the case if the species on which preys are more resistant to extinction, even if . The presence of generalist consumers therefore enhances robustness both because of their own greater resistance, and the resistance which they confer upon their consumers.

Species-level networks

Tags

food webs, graph, link distribution, metanetwork, networks, paleo-food web, paleontology, trophic species

The final step in food web construction is the generation of species-level networks (SLNs). A SLN is considered here a single potential pattern of community interactions in any given place at an instant of time, and may be constructed in two distinctly different ways. First, using empirical observations, one could construct the SLN of a community. This is typically the fashion in which SLNs are reconstructed for modern communities; workers observe and record the community’s trophic interactions. SLNs of this type are precise and without error, though usually taxonomically incomplete, and we cannot have similar confidence in their accuracies because of the sources of uncertainty described above. Capturing their variability requires repeated observations, which is possible under some circumstances. For example, there has been documentation of seasonal variation in food webs. Repeat observations are impossible for paleo-food webs. The best that can be done is to measure spatial or temporal variation in taxonomic composition. The latter of course could describe variability on only the longest of ecological timescales. Dunne et al. (2008) (see previous post) compiled SLNs of two Cambrian food webs derived from the Burgess Shale and Chenjiang lagerstatten, comprising 142 and 85 taxa respectively. The taxa in both these networks were subsequently aggregated into trophic species, 48 and 33 respectively, on the basis that species within the trophic species have identical consumers and resources. As argued above, it is impossible to validate this claim for fossil taxa. Trophic species-level links were ranked according to uncertainty in these networks, but there was no explicit attention paid to uncertainty at the level of species within the trophic species.

SLN derived from metanetwork in previous post

The CEG model takes an alternative approach to SLN reconstruction, generating multiple plausible SLNs from the metanetwork and hypothetical or underlying principles of food web networks as gleaned from modern food webs. This type of SLN generation requires a trophic in-link distribution for each guild. Recall that a trophic in-link distribution describes the number of prey per species within a guild. This number ranges from 1 (a heterotrophic species must prey upon at least 1 other species) up to the total species-richness of all guilds that are specified as prey of the guild in the metanetwork. SLN-generation requires initially that species within a guild be treated neutrally, that is, they have no distinguishing trophic properties. Stochastic draws from specific guild trophic link distributions then determine the number of prey to be assigned to each species. The prey species themselves are drawn randomly from the pool of prey guilds of the predatory species. The result is a directed graph or network in which each species in the community has been assigned prey, and many therefore also have predators (see figure). SLNs capture the uncertainty associated with the reconstruction of fossil food webs, and in fact any food web, in a manner in which static or unvarying trophic link determinations cannot. Repeated stochastic generation of SLNs accounts for the sources of uncertainty discussed earlier, namely uncertainty of the particular interactions of a species, and the temporal and spatial variability of a community type. Also, even though any two SLNs derived from any moderately complex metanetwork are unlikely to share the same exact topology (isomorphic), they are drawn from the same ensemble, as discussed earlier for Erdӧs-Renyi random graphs. Whether the argument can then be extended to claim that they will also have the same behavior on average, as with random graphs, is an interesting question, because the ensemble is the range of variation possible for a paleocommunity’s food web based on paleontological uncertainty. The next few posts will therefore deal with a description of the ensemble, and the ecological dynamics of the SLNs in an ensemble.