Fig. 1 - Species-level trophic link distribution for entire coral reef.

We’ve examined records of fish occurrences on Jamaican reefs for the past 10 years, and compared it to our “master” food web. Of the 196 species in our food web, 136 have records in Jamaica. Many of these species are present in very low numbers, and some reefs are noticeably depauperate, recording less than 60 species. Nevertheless, to be conservative, we assume that we can integrate over all the reefs, thereby counting all 136 species as being present. We next expanded our metanetwork, or guild-level food web (in this case almost exactly the same as a trophic species-based web) to the species level, therefore accounting for all expected links in the food web. For the master or pristine web, this yields an overall connectance of 0.059. The trophic link distribution is shown in Fig. 1. Interestingly, this is clearly not a decay distribution (e.g. power law), but has a definite modality of about 25 links. One needs to question the extent to which under-sampling of natural food webs, and aggregation into trophic species, affects interpretation of link distributions.

The next step of course is to assess the state of the Jamaican reef system. Our initial analysis has been to simply remove the “missing” species (extirpated) from the web, and to re-calculate the statistics. Connectance declines to 0.055. Is this significant? Probably impossible to answer that question for network connectance. Also, it should be noted that hundreds of invertebrate species are included here, and they will dampen the impact of any fish removals or additions. Perhaps the next question regards the link properties of the extirpated species.

Roopnarine, P. D. 2009. Ecological modeling of paleocommunity food webs. in G. Dietl and K. Flessa, eds., Conservation Paleobiology, The Paleontological Society Papers, 15: 195-220.

Find the paper here:
http://zeus.calacademy.org/roopnarine/Selected_Publications/Roopnarine_09.pdf
or here
http://zeus.calacademy.org/publications/

I started to match our Caribbean coral reef food web data to assessments of Jamaican reefs today. I used 10 years of careful observations. I basically just sat there and watched my dataset fall apart as species after species failed to appear on the Jamaica list. Where have all the species gone? I’ve worked on extinction for quite some time now, I know many of the people who work on these systems well and we talk, and I talk a lot with relatives (mom included) who are Jamaican and remember the way that it was. It is depressing, it’s sobering, and it’s humbling. I don’t think that I’ve ever had a more depressing day of science. Our species is both remarkable, and remarkably stupid. Sigh.

Guild-level trophic link distribution

Spent a great week at the Annual Meeting of the Geological Society of America. The Paleontology Society session on Conservation Paleobiology was a lot of fun, and my students also presented great posters. Now back to the coral reef.

I’ve been cleaning up the data, because with some much data, errors are bound to creep in. I believe that the current data are now accurate, and the metanetwork statistics are 265 guilds (including primary producers) and 4,651 links. That yields a metanetwork connectance of 0.066. The link distribution should therefore also be different, and indeed it is. The figure shows the no. of links per guild, and the regression plot demonstrates that the distribution is still a power law distribution. The exponent is smaller than previously calculated, (), but this is the guild-level network and does not reflect species richnesses (yet).

Trophic level vs. no. of links

The next question that I’m looking at is the distribution of trophic levels among guilds and species. I therefore calculated trophic level for all guilds. The first figure (scatter plot) plots trophic level against the number of prey or incoming links to each guild. There are two things to notice: First, the variance of trophic levels decreases as the number of links, or diet generality of the guild increases. Second, the decrease in the variance is asymmetric, in that there is a bias against being a generalist of low trophic level. This is obvious if you look at all the empty space being vacated below the data points as no. of links increases. I can think of two non-exclusive explanations for this. If you think about a food chain, consumers toward the top of the chain simply have more prey to select from (on an evolutionary timescale), and therefore there should be a natural increase in the number of generalists as trophic level increases. Also, note that there are also many specialists of high trophic level. Perhaps the ability to exert power over other species, as a predator, combined with the previous statement, explains this observation. Finally, what is the distribution of trophic levels within the community? The second figure is a simple histogram plot of all non-primary consumer guilds (i.e. omnivores and carnivores). The distribution is approximately normal, with a definite central tendency. On average, most guilds in the reef are of similar trophic level! That’s very interesting. And referring to the previous scatter plot, we know that there is a biased composition in the tails of the distribution, in that the upper tail (higher trophic level) is a mixed composition of specialist to generalist guilds, but the lower tail is basically restricted to low trophic level specialists.

Guild trophic level distribution

Some of you may have noticed that our trophic levels are non-integer numbers. Primary producers all occupy trophic level 1, and primary consumers are trophic level 2. “Above” that, trophic level is calculated on the basis of the trophic levels of your prey. Exactly how we do that will remain a secret for now.

Species-level trophic link distribution.

The data presented in the previous post examined in-link or in-degree distribution at the guild level, i.e. species are aggregated into ecological guilds. A comment on the previous post asked whether we’ve used any grouping algorithms for guild recognition, and the answer is no, at least not yet (and thanks again for the comment). The current guilds are based primarily on trophic habits and habitat, and other features such as the presence of photo- or chemosymbionts. Guild derived algorithmically would be based on species-level network topology, and ideally, the two would be very similar. Anyway, I noticed the comment when I logged on to post the current results. What I’ve done is to expand the guild-level network (metanetwork) to the species-level, and then re-examine the trophic link distribution. There is no guarantee that the two distributions should agree. For example, it is quite possible that guilds of high in-degree (lots of prey), though few in number, are very species rich, and hence one would lose the decay distribution at the species level. Conversely, guilds of low in-degree could be tremendously more species rich, and would expand disproportionately, when compared to high in-degree guilds, when expanded into member species. Nevertheless, for this dataset, when guilds are actually expanded from 255 consumer guilds to 704 consumer species, the scale-free nature of the distribution is reinforced. The new function is y=11158x^-1.981, implying a power law exponent very close to 2. Neat.

Trophic link distribution

What sort of network is the coral reef food web? In other words, how are the links or interactions between nodes in a food web distributed? Food webs have been modelled variously as everything from random (Poisson) networks to networks based on exponential, power law or mixed distributions, with or without hierarchical structure. Empirical measures suggest that link distributions in real world food webs follow exponential or power law distributions, perhaps a mixture of both (differentiated by scale). One of my worries with those measures is that they are based on food webs of varying sizes, and more importantly, levels of taxonomic and ecological resolution. So, for example, how much does it matter if your food web covers only a small part of the community’s taxonomic diversity, or only part of the trophic diversity? What about the level of aggregation of species into more inclusive groups? The high resolution of the coral food web presents an opportunity to address some of these questions, and here’s the first one: How are trophic in-links distributed at the guild level? Recall that guilds here are groups of species with potentially the same prey and predators. I say potentially, for while we have very specific trophic data for some species, e.g. heavily studied fish, data are less certain for many smaller or less well known species. Still, there are 265 guilds in this dataset, and 4,756 links (see previous post). The histogram is a basic frequency histogram of the number of links per guild. As predicted on the basis of previously studied food webs, the distribution is a (right-skewed) decay distribution, with a greater number of species possessing fewer prey, i.e. being relative specialists, and a few species having a broad repetoire of prey, i.e. relative generalists. The extreme generalists (to the right or tail of the distribution) are all large sharks, the most extreme being the tiger shark, Galeocerdo cuvier. These species range from microscopic, single-celled dinoflagellates to large carcharhinid sharks!

What type of distribution is this? A simple logarithmic transform of the data is shown in the second figure, and regression of the data yields the following function: y = 17238x^-1.9496 (r-squared=0.95). The significant and extremely good fit of a linear function to the transformed data suggests that the underlying link distribution is a power law distribution of the form , where is the link probability, is the number of prey available, and is the power law exponent. An exponent of ~1.95 is tantalizingly close to other empirical measures. Even more exciting, for me at least, is the fact that we have predicted on the basis of previous work that power law exponents that promote resistance or robustness to secondary extinctions should lie in the range 2-2.5. That work was based on terrestrial food webs from the Late Permian, 250+ million years ago!

Caribbean coral reef food web

I’ve been compiling data for a Caribbean coral reef food web. This is intended to be a “typical” coral reef of the Greater Antilles region, focusing on Jamaica. Data are drawn, however, from a more general region encompassing the Cayman Islands, Jamaica, Cuba, Hispaniola, Puerto Rico and the U.S. Virgin Islands. The U.S.V.I. were included because of the large amount of data available for the reefs there, particularly fish. It has been a rather large task to assemble species lists for this region because of the tremendous species richness of the reefs, as well as the scattered nature of the literature. Most major animal groups have been included, with notable exceptions being barnacles, sea stars, and some minor but probably important groups, such as sipuncula, echiura, crinoids and brachiopods. All major producer groups are also specified at the species level, including nannoplankton, diatoms, macroalgae, etc. The community comprises the reef habitat and adjoining seagrass beds.

The current compilation includes a total of 905 species, for which trophic data are available for 761 (84%). The 761 species are further collapsed into 265 guilds. Guilds range in size from 1 species up to 54 species (symbiont-bearing scleractinian corals). There are 4756 links among guilds, yielding a guild-based connectance of 0.068, well within the range of connectances for published, lower resolution communities. That’s a lot of stuff happening on the reef! It is surprising to realize, though, how little we know about many familiar species, or perhaps how poorly documented that knowledge is. The situation would be far worse if I accounted to the true diversity of the reef, which must range into several thousand species. One can only imagine the difficulties in attempting this with an Indo-Pacific reef or a tropical rain forest. Sadly, the current condition of many Caribbean reefs means that my compilation is an overestimate, being based on accounts dating back to the 1950’s, when the reefs were still in reasonably good shape, by 20th century standards anyway.

Jennifer Dunne, in a recent paper (Dunne et al., 2009), defines the structural robustness of a food web as a minimum level of secondary extinction that occurs in response to a particular perturbation (species removal). This is roughly what I’ve termed “resistance”, but I think that structural robustness will be a very useful and more precise definition. The paper is part of a recent issue of the Philosophical Transactions of the Royal Society on food webs. Several papers in that volume point to the relationship between diversity and “robustness” (often used less specifically than defined by Dunne), but the nature of this relationship, if any, remains problematic.

Given our (the CEG group) growing collection of ancient and modern data sets, plus the array of CEG programs that we now have, I’ve decided to examine this question a bit more closely using a number of different communities. The main questions are:

1. Is there a straightforward relationship between species richness and food web robustness?
2. Does the relationship differ between marine and terrestrial communities?
3. Does it differ on the basis of geological age?
4. Does it differ between fossil and modern communities, given differences in data completeness?

Both topological or structural perturbations, as well as CEG dynamic perturbations are being performed on each data set. Answers to the above questions most likely differ dependent on whether species interactions are purely topological, or are dynamic! Perturbations are bottom-up disruptions of primary productivity, removal of top predators (top-down cascades), removal of most connected, and removal of least connected species.

The figure shows results from the Early Permian Waurika locality of Oklahoma. These data were compiled by Ken Angielczyk. The treatment is a bottom-up disruption of primary productivity at three different levels of species diversity: 1x, 2x and 3x observed (higher) taxon diversity. Treatments are also repeated for three different models for trophic link distributions, exponential , mixed power law-exponential where , and power law . Two conclusions: First, the effect of increasing total species diversity is to reduce the variance of the results, and perhaps reduce the overall mean (i.e. increase robustness), but the significance of this change has to be tested. Second, there is a striking difference among the trophic link distributions. The transition from Level I to Level II secondary extinction is discontinuous for the exponential and mixed distributions, but continuous for the power law (though the interval of transition is represented by acceleration of secondary extinction). What causes this?

There continues to be a lack of clarity of the role of interaction strengths in stabilizing ecological communities. Most of the empirical and theoretical work done suggests a predominance of weak links. Strongly coupled species tend to have oscillatory or pseudo-oscillatory interactions, but weak links to stable species may tend to dampen, or reduce the amplitude, of the oscillations. The extent to which this is true, given a large and complex network of a species-rich system, remains unknown. Perhaps one way to explore this is to examine network robustness, CEG-style, while manipulating interaction strengths in the following way:

1. Topological extinction with no link strengths, i.e. all links are of equal and static strength.
2. Current CEG-style link strengths, where in-link strengths for a species are all equal. Strengths would be static.
3. Same as above, but strengths are now dynamic, reflecting compensation for lost links.
4. Same as previous two options, but now repeat with -distributed link strengths, both static and dynamic.