Resource overlap in Caribbean reef fish

Tags

competition, food webs, interaction strength, marine communities, nature, science, trophic level

I introduced a weighted index of interspecific resource overlap in the previous post. The overlap is measured as the number of prey resources shared by two species, as indicated in a food web network (or more properly, its adjacency matrix). The index is the ratio of the squared overlap to the product of the in-degree of the two species:

where C is the index for species m and n, I is the resource overlap, and k is the in-degree of a species. The index is symmetric for the species, equals 1 for a species compared to itself, and will also equal 1 if two species share identical prey and are of the same in-degree. Note that C falls short of a measure of interspecific competition in the absence of crucial demographic data about both species, as well as the strengths of interaction with prey.

So what can you do with this? Lots I think, but here’s something that we’ve been looking at. The first figure plots the ranked C values for the Caribbean reef shark, Carcharhinus perezi, versus all other species (or guilds) in the Cayman Islands food web, including invertebrate taxa. C is zero, or near zero, for many of those comparisons, because most nodes in the web share no or few prey resources with the shark. Note that the shape of the plot reflects this with its very long, flat tail. C rises sharply for highly ranked comparisons (left end of plot), indicating that the shark’s resource use overlaps with very few species, but when there is overlap, it is distinctly greater than most of the other comparisons. The red symbol is the species with which there is greatest overlap, the Yellowfin grouper, Mycteroperca venenosa. Does this indicate potentially significant competition between these two species? That’s difficult to tell from a single set of C values, so we’ll turn to the comparative method.

The second plot is also of ranked C values, but this time for the large Nassau grouper, Epinephelus striatus. Note two things right away. First, highly ranked C values are much larger than they are for the shark, indicating greater resource overlap between the grouper and a number of other species than there is for the shark. Second, the shapes of the plots are quite different! Whereas for the shark there are a few strong overlaps and a majority of weak ones, the grouper has strong overlap with a large number of species. In fact, the overlap between the shark and the Yellowfin grouper would only rank around 60 for the Nassau grouper! Things are certainly busier for the Nassau grouper. By the way, the most highly ranked overlapping species with the Nassau grouper is the gray snapper, Lutjanus griseus.

I find it fascinating that two large, and high trophic level predators on the reef exist under such different conditions of overlapping resource use. One very important thing to keep in mind, however, is that our food web reflects the (current) rarity of other large sharks on the Cayman reefs, and the situation could well be quite different where some of those species are present. Furthermore, as explained before, the reef food webs omit a fair number of species because the available trophic data are simply insufficient. And finally, I have to plug my invertebrate friends here, stating that I look forward to doing this sort of analysis on some of the very rich and functionally diverse molluscan and crustacean clades!

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Competition in food webs and other complex networks

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competition, coral reef, food webs, interaction strength, link strength, Network theory, networks, science

Competition is considered by many ecologists to be a major structuring factor in communities. It is a notoriously difficult thing to identify, classify and measure in the field and has been, in my opinion, an inspiration for some of the more elegant field studies. There is no doubt that species compete for resources in nature, but more elusive are answers to how much that competition matters to the stability of a species population, and the community as a whole, and what role competition might play on longer, evolutionary timescales. Typically, when we wish to measure competition, we require a few pieces of basic data, such as population sizes, interaction strengths and frequencies with the resource(s) being competed for, age structuring and so on. How can we go about doing this with complex food webs lacking these data? As usual, my answer is that you cannot, simply because of a lack of data. Nevertheless, I think that complex food webs do have something to say about competition, as long as one realizes that there is a trade-off between details of microscopic interspecific interactions and grabbing a macroscopic view of the community. Recently I’ve been mulling over appropriate ways to do this, and here are some ideas. I will preface them by saying that the interest stems from examining the potential impact of an invasive species as a competing consumer.

Let us begin with a (asymmetric) binary adjacency matrix, A, whose elements indicate whether species i preys on species j. The question is, what is the interaction between two consumer species, i and m. My first step is to simply count the number of prey shared between i and m, measured as the Hamming distance between the and rows; let’s designate that (). We can refine our view a bit by asking what fraction of a species’ prey is represented by that overlap, which is simply

where is the in-degree, or number of prey for species i in the food web network. You can think of this as the potential impact of species m on i. This is not quite satisfactory though, because and may be vastly different. For example, in our Caribbean coral reef food webs, many reef foraging piscivores (fish eaters) are specialists, preying mostly on maybe six other species, with those prey also being part of the repertoire of more generalist piscivores such as carcharhinid sharks who also forage on the reef and have k in the range of 70-80. It would be difficult to conceive of two such consumers as being strong competitors if the interactions of the generalist are distributed broadly over its prey. I therefore assume, in the absence of data on population densities, interaction strengths and functional responses of predators to prey, that this network measure of competitive interaction will be a function of both prey overlap (H) and consumer dietary breadth (k). There will be a trend of increasing pairwise strength of competitive interaction from generalist-generalist to generalist-specialist to specialist-specialist.

We can now extend our formulation in the following manner. First, count the number of prey shared between the consumers, . Then weight the interaction strength between m and its prey uniformly according to (ala CEG). The total interaction strength is

which is also the fraction of i’s prey that is being affected by m’s predation. The unaffected fraction, standardized to i’s dietary breadth is

yielding a standardized impact of

Note that this index is symmetric for i and m, i.e., it is the SAME for both species.

As a worked example, consider four species, A, B, C and D, with k’s of 60, 70, 2 and 2 respectively. The overlap of resources are: AB-35, AC-2, CD-1. The competitive indices are


and

I use in keeping with a conventional symbol for competitive interaction, but again point out that this is a very unparameterized measure compared to what is normally considered for use in Lotka-Volterra-type models or as measured empirically. You’ll notice that the values increase as the specialization of the interactors increases. It would be nice to scale these to a unit maximum to facilitate comparison, but I haven’t done that yet.

In a follow-up post I’ll provide some worked examples of all the above using real species from a real coral reef food web!

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.

Food webs as networks

Tags

connectance, food webs, graph, interaction strength, Network theory, networks, real world networks, Robustness

Perhaps the most obvious structural elements of real food webs that distinguishes them from the graphs presented earlier is directionality of the links. Links are trophic interactions, that is, predator-prey relationships, and describe the passage of energy from prey species to predators. They can also be used to describe the impact of predation on a prey species, recognizing that the relationship is an asymmetrical one between nodes. The “traditional” manner in which to depict this graphically is with arrows between nodes (Fig. A). Whereas the graphs illustrated so far have been undirected graphs, a food web is defined properly as a directed graph, or digraph. The asymmetry is also reflected by the adjacency matrix, which is no longer symmetric about the diagonal.

The most straightforward applications of Graph Theory to food web biology are analyses of the structure or topology of digraphs. Digraphs are often referred to as networks in modern usage, and the study of digraphs, especially those describing real-world networks such as the Internet or social networks, is described as Network Theory. The reader should be aware, however, that networks are technically graphs that are digraphs having weighted or parameterized links. A network therefore depicts a food web when it contains species interactions, the direction of those interactions, and some measure of the interactions, such as interaction strength. A digraph without measures or weights on the links is in reality a special case of a food web digraph, one in which all links are considered equivalent.

A very simple three species food web is illustrated in Fig. A. Species 1 (S1) is prey only (perhaps a primary producer), S2 is both a predator or consumer of S1 while being prey to S3, and S3 is the top consumer in the network. Alternative arrangements for three species are illustrated in Fig. B-D, including a simple food chain (Fig. B), a web where the top consumer is also cannibalistic (Fig. C), and a cycle among the three species (Fig. D). These networks bear only information about the existence and direction of interactions among species, but this information is important because structure always affects function (Strogatz, 2001). The basic network approach has proven useful as a means of capturing the complexity of food webs, deriving basic comparative properties such as connectance and link distributions, and assessing one type of robustness against perturbation.

New paper: Ecological modeling of paleocommunity food webs

Tags

cascades, competition, connectance, edge strength, extinction, food webs, interaction strength, link strength, modeling, Network theory, networks, nonlinear, paleontology, power law, probability, real world networks, Robustness, Scientific models, simulations, small world networks, Tipping point, top-down cascade

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/

Interaction (edge) strength and compensation

Tags

beta distribution, edge strength, interaction strength, link strength, networks

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.