PNAS: Late Cretaceous restructuring of terrestrial communities facilitated the End-Cretaceous mass extinction in North America

Tags

Cretaceous, dinosaurs, extinction, food webs, mass extinction, modeling, networks, paleo-food web, paleontology, Robustness, Scientific models, simulations, Tipping point, trophic guild

That’s the title of our new paper, hot off the PNAS press. This study was a lot of fun, because it combines my food web work with one of the best known events in the fossil record. The lead author is Jonathan Mitchell, a graduate student at the University of Chicago. Jon became familiar with the food web work via Ken Angielczyk at the Field Museum, also in Chicago, a former post-doctoral researcher in my lab and close collaborator.  Jon wondered what Late Cretaceous, dinosaur-bearing communities would look like when subjected to CEG perturbations (just search this blog for info. on CEG!), and presented his results two years ago at the Annual Meeting of the Geological Society of America. The results were so intriguing that we decided then to explore the question in much greater detail, and ask what sorts of community and ecosystem changes unfolded in the years before the Chicxulub impact, and what role they might have played in the subsequent extinctions. And here are the results! I will list the full reference below, and you can obtain a complete copy of the paper from PNAS (sorry, not open access). Also, here are links to some news websites that have covered the paper, as well as the paper’s abstract. Enjoy!

EurekAlert, Science Daily, Science Codex

Jonathan S. Mitchell, Peter D. Roopnarine, and Kenneth D. Angielczyk. Late Cretaceous restructuring of terrestrial communities facilitated the End-Cretaceous mass extinction in North America. PNAS, October 29, 2012

ABSTRACT

The sudden environmental catastrophe in the wake of the end-
Cretaceous asteroid impact had drastic effects that rippled through
animal communities. To explore how these effects may have been
exacerbated by prior ecological changes, we used a food-web
model to simulate the effects of primary productivity disruptions,
such as those predicted to result from an asteroid impact, on ten
Campanian and seven Maastrichtian terrestrial localities in North
America. Our analysis documents that a shift in trophic structure
between Campanian and Maastrichtian communities in North
America led Maastrichtian communities to experience more second-
ary extinction at lower levels of primary production shutdown and
possess a lower collapse threshold than Campanian communities.
Of particular note is the fact that changes in dinosaur richness had
a negative impact on the robustness of Maastrichtian ecosystems
against environmental perturbations. Therefore, earlier ecological
restructuring may have exacerbated the impact and severity of the
end-Cretaceous extinction, at least in North America.

Of cusps and folds

Tags

Catastrophe Theory, Tipping point

I am currently working on an essay (overdue!) for which I created this figure. It’s a cusp manifold and a very useful heuristic device for demonstrating some of the concepts and applications of Catastrophe Theory. Following is an excerpt from the current draft of the essay where the figure is introduced. And for those of you who are interested, following the excerpt is a very brief explanation and code for plotting the manifold.

Excerpt: Much of our theoretical understanding of tipping points is captured by Catastrophe Theory, a deep and somewhat ominously named mathematical theory. “Catastrophe” as used in the theory is generally understood to imply a dramatic change of state, with no necessary judgement as to whether the change is for the better or worse. Though mathematically complicated, the theory provides us with a very useful heuristic device, the catastrophe manifold, which can be used to visualize the manner in which a system will respond to external forces or controls. The manifold for a system controlled by two parameters is shown in Figure 1. The surface in the figure, known as a cusp catastrophe, illustrates the behaviour of a system, controlled by two factors, that is capable of a catastrophic state shift. For our case, the system is the global biosphere and the controlling factors are population size and resource consumption. The height of the surface is the condition of the biosphere’s state, with greater height corresponding to a healthier biosphere. It is easy to see that height, and hence biosphere condition, decreases as either population size or resource consumption increase.

The manifold was plotted using Mathematica, with code adopted from the notebook available here. The catastrophe is an unfolding of the singularity for the function

The controlling equation is

So the planar axes of the figure are parameters a and b, and the vertical axis is x. The surface of the manifold are the equilibrium points, or minima of the function, i.e. the points at which the first derivative

Those points are the real roots of the above equation. The Mathematica code is
F[x_, u_, v_] := x^4 + u*x^2 + v*x
y = ContourPlot3D[
Evaluate[D[F[x, u, v], x]], {u, -2.5, 3}, {v, -2.5, 3}, {x, -1.4,
1.4}, PlotPoints -> 7, ViewPoint -> {-1.25, 1.6, 1.2},
Axes -> False, Boxed -> False,
ContourStyle -> Directive[Red, Yellow, Opacity[0.5]], Mesh -> 0,
Contours -> 1, MaxRecursion -> 15, PlotPoints -> 50]

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.

FORA.tv – Ecosystem Impact of the Deepwater Horizon Disaster

Tags

california academy of sciences, Deepwater Horizon, marine communities, oil spill, pollution, Tipping point

Estimated tonnages of major oil spills

FORA.tv – Ecosystem Impact of the Deepwater Horizon Disaster.

My Pritzker Lecture at the California Academy of Sciences in November, 2010.

Critical threshold

Tags

giant component, networks, paleo-food web, paleontology, simulations, Threshold, Tipping point

LAZ with CAZ richnesses for g10 and g11

What causes the critical threshold increase of secondary extinction in a typical CEG simulation? It cannot be a simple result of increasing the perturbation magnitude, ; that is given by topololgical secondary extinction. CEG differs from the simple topological model in several ways. First, link strengths are not uniform but are instead a function of in-degree. Second, nodes are not equivalent, but have dynamic population sizes. And third, there are topdown effects. Somehow, top down feedback initiates catastrophic collapse at a very specific perturbation magnitude. Bel ow that threshold, responses are dampened. Why? I hypothesize that at the threshold, a giant component in the network is activated. Given that the component is present no matter the magnitude of perturbation, activation must depend on link strengths, which in CEG are dynamic.

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/

Corals, algae and space

Tags

competition, coral reef, corals, food webs, Tipping point

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.

Prey dynamics

Tags

extinction, Network theory, networks, nonlinear, quasi-periodic, simulations, Tipping point

The figure here is very similar to the one in the previous post, but these results are for the guild of shallow infaunal suspension feeders (primarily clams). The main difference is the more regular increase in the number of species that become extinct as the perturbation magnitude () increases. Another interesting note is that this guild is not the only driver, or any driver at all, of the behaviours exhibited by the guild of predators. Those predators may or may not prey on members of this guild, and also have an array of prey in other guilds. So the oscillatory behaviour seen at higher perturbation levels is probably system-wide. And it is system-wide because of indirect effects via network links. One wonders what a summary of the results would look like, and what the implications are for individual species population dynamics.

1. For example, even at a very low perturbation level, maximum sustainable population sizes oscillate wildly before settling down to a new stable state (which can in fact be the initial one, or zero, indicating extinction). One would assume that population sizes would follow this trend, if the timescales of the perturbation and population growth were sufficiently close. What if they are not? How does this affect what one would actually observe for a given species?
2. What is the distribution of stable states over the perturbation range? Are the oscillations observed at high perturbation level convergent, i.e. if run long enough they would also settle to a new stable state? Or are they asymptotic, but never settle down, or settle to two alternative states? One way to find out would be to simply run the series for many additional steps. Another would be model the oscillations themselves, and see if the convergence is linear or asymptotic. And what is the perturbation range of the bifurcations? At what point do we begin to observe oscillation/bifurcation, and is it synchronous throughout the community? Only one way to find out, but I’ll probably have to write some Sed/Awk or Perl scripts to handle these large datafiles.

Lighting up an ecosystem

Tags

bifurcation, chaos, extinction, Network theory, networks, quasi-periodic, simulations, Tipping point

One of the final pieces needed to explain the critical/threshold point in a bottom-up CEG perturbation is an understanding of which species become extinct, and what the species dynamics look like during the cascade. Therefore, what I’ve done is to modify the basic simulation to capture the demographic properties (technically, the carrying capacities) of each species; results in HUGE output files. Shown here in this figure are the species dynamics for the guild of shallow infaunal carnivores (e.g. naticid snails) at three different perturbation levels. Note that the levels correspond to a low secondary extinction response, and the two critical points identified earlier. The top row of figures plot the dynamics of surviving species, and lower show those of the species which become extinct. The community dynamics were recorded for 250 steps beyond the initial perturbation. The first thing to note is that species become extinct very quickly. Beginning K for each species is standardized at 1, and the species that become extinct have, on average, lower in-degrees, i.e. lower numbers of prey, than do surviving species (statistical tests to follow later). That result matches expectations of the CEG combinatoric model.

The other thing to note is that at the low and mid-perturbation levels ( 0.2 and 0.55), species’ K respond immediately to a perturbation of the producer guilds, oscillate for several steps, but eventually settle down to a new stable K. This is a transition to new stable states for the species populations. At the perturbation level which coincides with the major critical point of secondary extinction, however, there is no indication that the species ever settle to a new state. Instead, there seems to be bifurcation and subsequent alternation between two alternative stable states; the species are lit up by the disturbance (each species is given a different colour in the corresponding plot for easy distinction). The series is quasi-periodic though, in that the system never returns to quite the same point on alternating steps. It is possible that the series eventually converge to a single, or two, stable points, but the current data cannot address that. Therefore, I’ll next repeat this simulation, but extend the data capture to 1,000 steps (2Gb file). Hopefully that will give some indication of whether the series are converging, diverging, are stable, or perhaps chaotic. Also, the results for the other guilds need to be examined.

Tipping point II

Tags

cascades, extinction, food webs, Network theory, networks, nonlinear, simulations, Tipping point, top-down cascade

Red ellipses represent the two threshold points.

The story so far: We have a food web of a shallow coastal marine community from the Late Miocene of the Dominican republic. The metanetwork comprises 29 guilds, 139 guild-level links, and 130 species. A perturbation of the system, where all three primary producer guilds plus detritus were systematically and incrementally removed from derived species-level networks, results in the typical CEG result: that is, a relatively flat and low level of secondary extinction () over a broad range of perturbation magnitude (), succeeded by a rapid transition to a state of high secondary extinction. In fact, for this community, there are two transitions. The first occurs at , and represents a very minor but secular increase in . The second transition occurs at and represents a catastrophic increase in . Topological-only perturbation of the system makes it very clear that these transitions correspond exactly to two stages of the perturbation: First, the complete extinction or removal of benthic autotrophs and complete disruption of the particulate detritus supply. The second and greater transition occurs at the complete extinction of the benthic macroalgae and macrophytes. Accompanying the second transition is the complete extinction of the benthic herbivore guilds which specialize on the macroalgae and macrophytes (and derived detritus), comprising families such as the Phasianellidae, Cerithiidae, Vitrinellidae, Haminoeidae and Retusidae. This is accompanied by extinction of species in other more generalist guilds that include macroalgae in their diet.

Extinction of those heterotrophic taxa is not itself the cause of the major tipping point though. Simulations where the perturbation is specifically removal of these herbivores result in very low levels of secondary extinction, with no tipping point or threshold. The obvious question then is, why does extinction of the macroalgae drive the system to a new state? The qualitative answer is that the complete loss of this resource, and the bottom-up propagation to the herbivores, in turn cause intense top-down cascades of compensatory responses from higher level consumers. These cascades propagate throughout the network, even to the remaining source of production, the phytoplankton. The result is a tremendous loss of species. A very curious thing, however, is that phytoplankton productivity in the network is almost 3 times greater than macroalgal productivity, reflecting the much greater diversity of planktivores. So why does the collapse coincide with loss of the macroalgae?

Perturbation of macroalgae (top) and phytoplankton (bottom), and resulting loss of autotrophic productivity because of top-down effects (left), and secondary extinction of heterotrophic species (right).

I performed two separate perturbations to answer this question. First, I perturbed the system by removing macroalgae only, and second by removing phytoplankton only. The top row of the second figure shows the results of the first experiment. Secondary loss of autotrophic resources (left column) as a result of top-down effects is effectively zero. Secondary extinction of heterotrophs (right column) is significant but not dramatic. There is a mild increase in the region of , which represents the loss of the specialized herbivore guilds. Removing phytoplankton had a more dramatic impact, reflecting the greater overall dependence of the community on phyloplankton resources. There is a clear threshold, occurring at approximately . At this point, resource loss to the community is great enough to trigger the catastrophic top-down cascades and feedback within the network. Therefore, it seems that in the previous experiment, where all resource guilds were perturbed, the complete loss of macroalgae triggers the top-down cascades and compensatory feedback that in turn deplete phytoplankton resources to the point where the system transitions to a higher state of secondary extinction. This conclusion is supported by the fact that when all producer guilds are perturbed, the contribution or perturbation of phytoplankton at the tipping point is 38%, whereas when only phytoplankton are perturbedm the tipping point occurs at 50%.

Some closing observations:

1. Topological analyses of network vulnerabilities are likely to underestimate the severity of link losses when those links have variable interaction strengths, and the nodes have varying properties. In the case of a biological community, species could and are likely to alter interaction strengths to compensate for lost resources (i.e. links). Topological vulnerability analyses should be well suited for networks with static properties, perhaps such as power grids and the internet (though I’m no expert here!), but are ill-suited for dynamic networks, such as those describing transportation, metabolic/physiologic and ecologic systems.
2. An hierarchically structured, directed network such as an ecological community should be resistant to a broad array of random perturbations. This is a function of both the underlying link distributions (as already understood in the case of static networks or graphs), as well as the compensatory abilities of consumer species, and the variance of dietary breadth. The network is, however, vulnerable to the loss of highly linked nodes. Here I am referring specifically to basal, autotrophic nodes, and not necessarily keystone consumer species. Not all autotrophic nodes are equal, however, as shown in the above results. Nevertheless, because of the complexity of the species interactions and the hierarchical divisions of ecological functions, there should be strong nonlinearities in the network responses. This is borne out by the differences between the topological-only and fully dynamic simulation results. The nonlinearities are expressed as two or more alternative states of secondary extinction, separated by rather sharply defined thresholds of perturbation. I can think of no way in which to analytically predict the threshold points, but heuristically I would argue that they should exist in every ecological community.
3. Perturbation of top-level consumers are observed in nature to often result in top-down cascading effects, compatible with such notions as keystone predators. I will show in later results that the CEG model captures all this. The results will also show, however, that while top-down effects can be locally catastrophic, i.e. for individual species or groups of closely linked species, they are never globally catastrophic in the manner in which bottom-up perturbations are. This conclusion has implications for understanding the role of ecological collapse in large scale extinctions observed in the fossil record. It also has implications for the ongoing biodiversity crisis, where species far removed from the “tops” of food webs are increasingly threatened by climate change and habitat destruction.
4. An close examination of many of the results presented in this blog will show apparent “bifurcation” of the results, e.g. beyond the threshold point in the lower right graph above. These observations suggest that there is more than one type of species-level network that can be derived from the same metanetwork. So, while the higher-level organization of the community is the same, networks are being generated that vary enough in their interspecific link topologies to yield very different responses to the same level of perturbation. I believe that this is a statistical property of the underlying trophic link distributions and the resulting multinomial probabilities from which the species-level networks are drawn stochastically. In the case of the above results, where one set of networks is significantly more resistant than the other (i.e. they have a much higher tipping point), this mathematical feature of the model is not likely to be of great relevance ecologically. That is because the lower threshold is already so high, in this case, 50% shutdown of primary productivity. Those are catastrophic environmental conditions and would occur with very low frequency in nature on a large scale. There are cases, however, such as the Early Triassic Lystrosaurus zone community, where there seem to be multiple alternative states at very low perturbation levels. Those communities would very likely have experienced frequent low-level perturbations, and then one has to consider whether: (1) this feature of the model is a mathematical artifact, in which case one wonders about the constraints necessary to prohibit it in nature, or (2) the feature is real, and then one wonders how species within a community cope with such a situation.