Coral reef fish food webs

Reef fish food web, Greater Antilles

Here are a couple of renderings of the vertebrate-only component of the coral reef food web. Reminder: the food web is what we expect to see for a reef in the Greater Antilles of the Caribbean, based on data collected around the mid-20th century. The vertebrate component comprises all fish and sea turtle species. The upper figure is the expected food web, and includes 196 species and 995 trophic interactions. Species are arranged on the periphery of the diagram, with interaction represented by the lines crossing the interior. The very busy, or hub species are higher trophic level predators, mostly carcharhinid sharks.

Jamaica coral reef fish food web

The lower figure is what we observe today in Jamaica. (Note: Jamaica is of particular interest for me as a starting comparison, both because of the excellent documentation of those reefs, and my Jamaican heritage; not picking on Jamaica). The number of species, out of 196, observed there over the past 10 years is dramatically smaller. Perhaps more obvious is the loss of interactions. I won’t present the actual data yet, since we will eventually prepare a paper to report all this, but the differences between the two food webs are obvious. We are currently rendering the complete food web, including primary producers and invertebrates, which will be an update of the figures presented in earlier posts. But there are a lot of species in there, and the computers have been churning now for about 17 hours!

New paper: Ecological modeling of paleocommunity food webs

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/

Power law confirmed

Species-level trophic link distribution

Okay, this post just disappeared, so let’s try again. The updated and correct coral reef food web comprises 759 species. The incoming trophic link distribution, when expanded to the species level (compared to the guild level in the previous post), is a definite power law distribution. The log-transformed data (see figure) yield a function of , i.e. . See the earlier coral reef posts to understand why this is significant.

Coral reef trophic levels, & update

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.

Coral reef species link distribution

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.

Coral reef food web II

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!

Interaction (edge) strength and compensation

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.

Corals, algae and space

A new project involves working with the CEG model and coral reef communities. The main goal is an interactive and instructive module for education, but there’s no reason why the data could not be used for some research also. The exercises are to model the impacts of coral bleaching, and reduction/removal of higher trophic-level fish from the system. Now CEG specifically models potential secondary extinction of species, but it occurs to me that one of the major impacts that we observe on reefs is the decline of corals as dominant or co-dominant benthic cover. This is usually accompanied by an expansion of macroalgae with which the corals compete for space. So the model is being modified to examine the impact of the manipulation of trophic networks (food webs) on the spatial state of the reef (along with secondary extinctions, of course). You can read a bit more about this here.

Assume that the community begins in equlibrium (with regard to spatial competition) at time 0 (). If the relative population size of species i is , then equlibrium is expressed as

where is a competition coefficient (not a constant), and there are n competing species. Therefore,

Because population sizes are changing in response to non-competitive factors (trophic), we expect changes to relative population sizes, and hence the coefficient is dynamic. Hence the difference equation governing relative population size during a CEG cascade becomes

where

is the same for all competitors, and need be computed only once per cascade step.

Network equivalence

The objective here is to establish the degree of similarity between successive communities, in spite of changing taxonomic compositions and diversities, and ecological/guild diversity.

Let represent an ecosystem over time (e.g. the Karoo series) comprising a series of chronologically successive communities, The metanetwork representation of is identical for all if guild diversities are omitted. That is, the unparameterized metanetworks are automorphisms Species-level networks (slns) are generated from the parameterized metanetworks, a finite set for each one. The question being addressed here is, how many of those slns are isomorphic between metanetworks? In other words, how many of a community’s networks are identical to networks in the preceding and succeeding communities? This is a very important question from the perspective of CEG dynamics, because in the model equivalence of ecological dynamics can transcend taxonomic composition and identity. Consider the two slns in the first figure. Specifically they are different, but dynamically they will respond identically to perturbation. This isomorphism extends to the guild level also. Imagine for a moment that the species in the figure are actually different guilds, and that guilds 2 and 3 are very different organisms. IF guild diversities permit the generation of slns with the same numbers of links, then isomorphic slns will be generated. Furthermore, we can remove any guild and species identity completely from the networks and maintain an equivalence of CEG dynamics under the following condition: If the perturbed species/nodes are part of a connected subgraph, and the connected graph is isomorphic with another subgraph, then CEG dynamics of the two networks will be identical! A minimum measure, therefore, of the continuity of ecosystem dynamics between successive communities, is the number of slns that are isomorphic between the sln sets.

Unfortunately, determining whether two graphs or networks are isomorphic is an NP-complete problem, and slns are very complex graphs. Given the size of the sln sets for each community, it would be impossible to determine the power of their intersection. Given that CEG dynamics are drawn however from the likeliest region of the sln space, that is, stochastic draws are made from defined trophic link distributions, we are really interested in establishing isomorphism of those subsets, not the entire sln space. So, one procedure would be to generate a set of high probability slns from each community, and then test those for isomorphism. The test would have to be one of elimination, i.e. whittling down the number that could be isomorphic, without ever actually arriving at the number that are isomorphic. But that would give us an upper limit of the number that could be isomorphic, and an upper limit on the measure of similarity between the two communities. In the next post I will outline an MCMC approach to generate sets of slns with high likelihood.

It is entirely possible that the result will be that no networks are isomorphic between communities. In that case, it would be obvious that, given similarity of CEG dynamics, the networks are “close enough”. We would then be faced with the even more difficult task of measuring minimum distances between networks, but this is not at all an impossible task. We’d just need some reasonably powerful computers, and acknowledge the fact that our carbon footprints would far outweigh any benefits to be gained from our work.

Trophic network probability

Given any species level topology where represents edges or links, and represents vertices or species, the probability of that specific topology is the probability of the set of interspecific links specified or composing the topology. Therefore, the probability of the network, given metanetwork is

where the right product is the probability of any species having a particular pattern or topology of links to other species, and the left product is the product of those probabilities for all species in the community. We can see immediately how the probability of any particular species-level food web is built from the probabilities of individual species networks.

The number of species-level networks that can be derived from a metanetwork depends upon permutations of all possible combinations of link topologies of species in the community, and is generally an astronomical number for even a modest number of species and guilds. The network with the greatest probability, or maximum likelihood of occurrence, is one where the probabilities in the formula above are maximized. This can be approximated if one considers that the probability of being linked to any is equal, regardless of the in-degree of One can therefore consider the probability of linking to any to simply be the proportion of the predator’s set of prey species that is represented by These are simply maximum likelihood estimates of the metanetwork link probabilities.

The in-links of each species are assigned randomly to species in other guilds for which a metanetwork in-link exists, i.e. those other guilds comprise potential prey of the species, and where the species belongs to guild The resulting network may be represented as a adjacency matrix, where is the total number of species in the community. Binary entries indicate whether species is prey to species Furthermore, row sums, equal the in-degree of species

The probability of a species link topology, or the binary pattern of the row is the product of the probabilities of each link. This can be calculated efficiently as the multinomial probability of a pattern of links spread among the guilds. Say that the probability of a link between and a species in guild is equal to the fraction of the diversity of prey that is represented by the species richness of then the probability of is

where is the number of links to species in guild is the species richness of and is the total number of prey potentially available to according to metanetwork , where It is important to note that each term in the formula exists iff i.e. a metanetwork link exists This prevents the inclusion of zero probabilities. Finally, the overall probability of the network is calculated as the product of all species probabilities, which in turn are the products of all link probabilities.