Modern and paleocommunity analogues

Tags

connectance, coral reef, food webs, marine communities, modeling, Network theory, networks, paleo-food web, paleontology, real world networks, Scientific models, trophic guild, trophic level

Last week I gave a keynote presentation at the annual conference of the Geological Society of America in Vancouver. Here is the abstract, and a link to the presentation (pdf file).

ANCIENT AND MODERN COMMUNITIES AS RECIPROCAL ANALOGUES OF PERSISTENCE AND STABILITY

ROOPNARINE, Peter, Invertebrate Zoology and Geology, California Academy of Sciences, 55 Music Concourse Dr, Golden Gate Park, San Francisco, CA 94118, proopnarine@calacademy.org
Paleocommunities are spatio-temporally averaged communities structured by biotic interactions and abiotic factors. The best data on paleocommunity structures are estimates of species richness, number of biotic interactions and the topology of interactions. These provide insights into paleoecological dynamics if modern communities are used as analogs; e.g., the recent lionfish invasion of the western Atlantic is the first modern invasion of a marine ecosystem by a high trophic-level predator and serves as an analog for the invasion of paleocommunities by new predators during the Mesozoic Marine Revolution. Despite the invader’s broad diet, it targets very specific parts of the invaded food web. This will lead to non-uniform escalation on evolutionary timescales.

Theoretical ecology provides a rich framework for exploring dynamics of community persistence. Persistence–the stability of species richness and composition on geological timescales–is central to paleoecology. Ecological stability, a community’s return to stability after perturbation, is not necessary for geological persistence. However, it does dictate a community’s response to perturbation, and thus a species’ persistence or extinction. What then is the relationship between paleoecological richness/composition and ecological stability? How do communities respond to losses of species richness or ecological function? Questions of stability and diversity loss are addressed with an examination of transient responses and species deletion stability analyses of end-Permian terrestrial paleocommunities of the Karoo Basin. Transience is measured as the degree to which a perturbation is amplified over ecological time, even as a community returns asymptotically to stability. Transience during times of frequent perturbation, as during times of environmental crises, decreases the likelihood of a persistently stable community. Species deletion stability measures the dynamic response of a community to the loss of single species. It is an open question whether communities become more vulnerable or more resistant during environmental crises. That process, which has occurred repeatedly in the geological past, is important to the fate of threatened modern communities.

Experimental Space

Tags

coral reef, food webs, Network theory, networks

Cayman Islands coral reef food web

Hi everyone, if any of you will be in the San Francisco Bay Area in the coming month, there is an exhibition at the Aggregate Space Art Gallery, featuring scientific visualizations. A couple of pieces there are from my food web work! So please stop by. Here is the gallery’s announcement:

“In their search for evidence of theories that better explain our physical reality, scientists often discover unexpected and beautiful phenomena. The researchers who created the images and videos included in “Experimental Space” did not have an art gallery in mind while they worked. Nevertheless, the images, figures, and data on view are aesthetically compelling and seductive. Through this exhibition, Aggregate Space Gallery and BAASICS bring scientific images and perspectives from the laboratory and the academic journal to the realm of art, where subjectivity trumps objectivity and ambiguity is more celebrated than demystification.

Featuring Evidence by: Erin Jarvis Alberstat, PhD candidate; Roger Anguera, Multimedia Engineer; Daniel J. Cohen, PhD; Sara M. Freeman, PhD; Luke Gilbert, PhD; Angela Kaczmarczyk, PhD candidate; Arnaud Martin, PhD; Brian Null, PhD, and Dr. Peter D. Roopnarine, PhD.”

Links–
http://aggregatespace.com/
http://www.baasics.com/

Competition in food webs and other complex networks

Tags

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!

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.

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!

Unbalanced food webs

Tags

extinction, food webs, modeling, Network theory, networks, paleo-food web, paleontology, real world networks, simulations

Perturbation simulations of three Karoo communities. 100 sims. per community.

A number of earlier posts have discussed food webs of the Permian–Triassic of the Karoo Basin in South Africa. This terrestrial ecosystem was subjected to the devastating end Permian mass extinction. The community which emerged in the aftermath of the extinction, the Lystrosaurus Assemblage Zone (LAZ), has been identified as having very unusual food web dynamics. This first figure compares the CEG dynamics of the end Permian Dicynodon Assemblage Zone (DAZ), LAZ, and the successive Cynognathus Assemblage Zone (CAZ). The implication is that there was a breakdown of perturbation dynamics during and/or right after the extinction episode. LAZ differs from the other communities (and in fact from every other community that we’ve studied so far!) in two ways:

1. Levels of secondary extinction can be extremely high at low peturbation levels, implying food webs of very low resistance.
2. Many species level networks or food webs (SLNs) of LAZ are nevertheless quite resistant, and resemble SLNs from the other communities. So the SLNs, or at least their dynamics, are highly variable in LAZ.

So what causes all this?

Distribution of log(low pert. sd) for random networks. Karoo communities are marked in green. LAZ occupies the extreme right.

The first question we asked ourselves was, is LAZ an unusually bad community or metanetwork, or are the other Karoo communities just exceptionally good? Our approach to addressing this was to generate 1,000 random metanetworks by randomly selecting observed guild richnesses from among our observed communities to fill the richness of a random community. A random community or metanetwork could therefore have guild richnesses that never occur together in any of the observed communities, but every guild richness of a random community is observed in at least one real Karoo community. We then simulated perturbation of 100 SLNs for each random community, and collected data on the first observation above, i.e., the variability of resistance at low levels of perturbation. As we see in the second figure, LAZ really stands out, even among the random communities! Why?

Well, in order to address that, we’ve used a number of regression models to examine the dependence of that variability on proportional guild richness. Proportional guild richness, in contrast to absolute, is the fraction of a community’s total consumer richness encompassed by a particular guild. Several guilds consistently stand out: very large amphibians, very small herbivorous amniotes, very small carnivorous/insectivorous amniotes, small carnivorous/insectivorous amniotes, carnivorous insects, and herbivorous insects. Multiple regression models demonstrate that the herbivorous guilds affect resistance variability negatively, i.e., they dampen the variability, while carnivorous guilds affect it positively! Now here’s the neat part. If we examine the sub-metanetworks of DAZ, LAZ and CAZ comprising these guilds only (see figure), we can immediately see how the communities differed with respect to these crucial guilds. Guilds with a dampening effect are shown in blue, those in red have the opposite effect (producer guilds are brown). And if you think of LAZ as being somehow imbalanced or out of whack, the figures should suggest to you some ways to “restore the balance”. I’ll discuss those in the next post.

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.

The Network Structure of Baseball Blogs: Part 1

Tags

Network theory, networks, real world networks

Earlier in the week I read about the network structure of twitter employees’ accounts and that got me thinking about the network structure of baseball blogs….

This is very cool. Cute, interesting, of general interest, and it even mentions food webs! (Thanks to Steve W. for the link)

Dynamics and Control of Diseases in Networks with Community Structure

Tags

Network theory, networks, real world networks

The dynamics of infectious diseases spread via direct person-to-person transmission (such as influenza, smallpox, HIV/AIDS, etc.) depends on the underlying host contact network…

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.