Hot off the presses! The “tragedy of the commons” paper with Ken Angielczyk is featured here, along with a truly fantastic collection of other papers. A must-read for any of you modelling-nerds out there. Oh, and the papers are free to access!

Biology Letters Special Feature – Models in Palaeontology.

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.

In a recent paper in the Royal Society Proceedings B, Randy Irmis and Jessica Whiteside verify a prediction of the CEG model regarding earliest Triassic terrestrial communities of the Karoo Basin in South Africa. Ken Angielczyk and I were interviewed by Wired Science for an article about the paper. Read it all here!

We predicted that communities in the Lystrosaurus Assemblage Zone would exhibit intrinsic instability in the face of even mild disruptions of primary productivity. More recently (and here), we explained that the intrinsic instability stemmed from the rapid diversification of small to medium-sized synapsid carnivores in the aftermath of the end-Permian mass extinction, coupled with very low species richness of herbivorous tetrapod prey, and the resulting intensity of competitive interactions among the carnivores. The recent Proceedings B paper seems to support our prediction on the basis of relative abundances of species of different trophic ecologies, characterizing those species as “boom and bust”. It’s always great to have model verification!

I think that there are some unresolved questions though:

1. We also suggested that one way out of the conundrum would be the increased specialization of the carnivores. Contrary to Irmis and Whiteside, I don’t agree that uneven relative abundances necessarily lead to demographic boom and bust cycles. Community dynamics are more nuanced and flexible than that.
2. The authors also point to probable environmental instability based on carbon cycles (measured as carbon stable isotope signatures). They valiantly overlap the short Karoo signature with the much longer and highly resolved marine signature. We simply have no good correlation of these signatures, and this is at least a nice attempt to highlight this ongoing issue.

Whether you can observe a thing or not depends on the theory which you use. (Einstein)

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.

Mystery Fossils Link Fungi to Ancient Mass Extinction | Wired Science | Wired.com.

See also this older article: Fungus Feasted Off World’s Worst Extinction

Mindell DP, Fisher BL, Roopnarine P, Eisen J, Mace GM, et al. (2011) Aggregating, Tagging and Integrating Biodiversity Research. PLoS ONE 6(8): e19491. doi:10.1371/journal.pone.0019491

This is an extremely interesting new report by Boris Worm and Derek Tittensor. They compiled global range information for 13 species of large pelagic tuna and billfish, documenting significant range contractions in 9 of those species between 1960 and 2000. This implies that these top or near-apex predators have been extirpated over much of their range. Similar extirpations and extinctions of high trophic level terrestrial predators have resulted in now well-documented top down cascading effects, such as meso-predator release (no controls on smaller or less powerful predators) and subsequent declines of herbivore prey. It’s difficult for me to infer what the top down effects might look like in the ocean partly because of the tremendous ranges of some of these species and the likelihood of refuges, but also because, unlike most terrestrial predators, these species are exploited for food; as are the lower trophic level of meso-predators. We’re so busy knocking out the entire web that we could just be dampening many would-be cascades!

That is the opening quote from a recent news article in The Scientist. The full article may be found here. The original research paper is in the most recent issue of Ecology Letters. The article doesn’t quite get this right, does it? It has NOT been a long-standing assumption of Darwinian theory that phylogenetic relatedness on its own drives competition. Phylogenetic relatedness is correlated with the frequency and intensity of competition to the extent that it reflects phenotypic overlap or similarity, which in turn will be a major factor in the extent of overlap in resource utilization. Extinction risk, therefore, is not a direct function of phylogenetic relatedness, at least not from the standpoint of competition for prey and other resources. It could play a role in terms of, say, resistance to predation or disease. To emphasize, two distantly related species will face the same risk if their phenotypic resource utilization is comparable to that of two significantly more closely related species. The journal article tests the effects of both trait and phylogenetic similarity, and conclude that phylogenetic similarity is a better predictor of the outcome of coexistence. I don’t understand how this could be the case, unless unmeasured traits are underlying factors. Am I missing something here?

The first instalment of the Wellcome Trust’s “Rap Guide to Evolution”. This is good stuff. Maybe it raises more questions than it answers, but that’s science, right? It makes evolution relevant; it controls our lives, and yet has also given us the tools to make choices as individuals and societies. I’m looking forward to the rest of the videos.