Ecosystems, Epidemics, and Economies

Yesterday I gave a talk for the “Breakfast Club” series at the Academy (California Academy of Sciences). The club is a twice weekly series of online talks started by the Academy in response to the widespread shelter-in-place and shutdown orders. It’s intended to bring a bit of our science and other activities to those interested who, like so many of us, find ourselves mostly limited these days to online interaction.

My talk focused on some new work that we are doing in the lab, related to the COVID-19 pandemic, but inspired by and built partly on our paleoecological and modelling work. I hope that you find it interesting! Oh, and while you’re there, check out the other talks in the series (link above)!

Systems Paleoecology – States, Transitions, and Extinctions

Tags

alternative states, Complexkty, critical transition, Tipping point

WHAT IS ECOLOGICAL STABILITY ? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

1. Welcome Back Video
2. Introduction
3. Malthusian Populations
4. Logistic Populations
5. Logistic Populations II
6. Deviations from Equilibrium

7. r, R, and Bifurcations
8. Quasiperiodicity and Chaos
9. Chaotic Stability
10. Environmental Variation: Expectations and Averages
11. Nonlinearity and Inequality

The product of zero multiplied by zero is zero — Brahmagupta

The state of a population, as discussed to this point, is the result of intrinsic control exerted by internal variables (e.g. a life-history influenced trait such as R), the impacts of external parameters (e.g. water temperature), and often the response of internal variables to those parameters. These three factors, coupled with preservational conditions, underlie all the stratigraphic dynamics of an idealistically isolated fossil species. Even the dynamics of an isolated population will vary over time, though, because of evolutionary change and environmental variation and change. Thus the state of the population is expected to vary temporally. The states that we have so far considered have been either steady, or vary predictably with parameter changes (e.g. Fig. 1). It is now broadly recognized, however, that dynamic systems often behave or respond in non-smooth ways, where a system may transition discontinuously, and often unexpectedly, from one state to another. The surprises are twofold in nature: first, single systems may possess multiple states —multiple attractors. Second, the transitions between states are often abrupt. Such transitions bear various names that have entered into conventional ecological literature and everyday conversation, including tipping point, critical transition, and regime shift.

Two populations with different intrinsic rates responding to and recovering from a sudden loss of individuals. See here for an explanation.

Discussions of multiple states generally reference communities and ecosystems, e.g. clear vs. turbid lakes, forests vs. grasslands, and coral-dominated vs. algal-dominated tropical reefs. Transitions and multiple states in such multispecies systems are facilitated by nonlinear relationships among species, enhancing and balancing (positive and negative) feedback mechanisms among demographic variables and environmental parameters, and asynchronicity (or synchronicity) of driving and response processes. Can transitions and multiple states occur in the single species population systems on which we have focused so far? Hypothetically, it is possible, but we will have to re-examine and re-think some of the simpler models of environmental shifts and responses outlined in earlier posts. When the community to which a population belongs undergoes a transition between states, it is probable that the population will also change states, but not necessarily so. A species could persist within the multiple states of a community and yet maintain a stable population size or remain within a single attractor. Shifts and responses, however, may also yield a population with distinct stable states separated by a parameter threshold, or parameter range that is much shorter than the ranges within which the population would remain stable — an abrupt transition. “Abrupt” need not refer to time only, but instead more properly refers to the relatively narrow parameter range separating different system states. The state of the population within the transitional parameter range is transient, and we can therefore describe the dynamics of the population as comprising multiple stable states, separated by transient transitional conditions. And, whereas most work in this are has focused on communities and ecosystems, there are situations where transitions can be understood within the framework of single populations. Furthermore, such transitions often have implications for the persistence or extinction of the population. Those transitions and what they imply about population growth and extinction will be the focus of the remainder of this series.

However, before digging into the dirt that I love best, I will offer a rather random assortment of readings and other resources. State transitions, particularly those occurring within complex systems, are all the rage these days. This is the area, in my opinion, where systems science truly serves as a unifying concept across multiple parts of the real world, ranging from universal to microscopic scales, and across boundaries of the physical, biological, and human worlds. I wish that I could reach behind me right now and pull my favourite books off the shelves and list them for you, but, alas, I cannot. Why? Because here in the San Francisco Bay Area my institution remains closed (with most of my library) because of the awful intersection of complex little bundles of viral proteins and nucleic acids and complex human systems, including the biological, sociological, and economic. So, if you the reader is a fellow resident of the United States, I will leave you with a polite and humble request: Please wear your damned mask. Okay, now a few resources.

• Scheffer, Marten. Critical transitions in nature and society. Vol. 16. Princeton University Press, 2009. An absolutely outstanding and accessible presentation. Luckily, because I own multiple copies (not sure why), I actually have a copy at home.
• Complexity, a web resource at the University of Southampton.
• BioTip. This is a flagship program funded by the German government, and maaaybe the world would have a brighter future if this program was emulated elsewhere.
• Regime Shifts DataBase. An ambitious and super program from the Stockholm Resilience Centre.
• David Krakauer & Geoffrey West. The Damage We’re Not Attending To. A recent and very nice article of the great relevance of systems thinking.
• And finally, how about something hot of the presses? Lewis, K. M., G. L. van Dijken, and K. R. Arrigo. “Changes in phytoplankton concentration now drive increased Arctic Ocean primary production.” Science 369.6500 (2020): 198-202.

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.

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/

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.

Tipping point

Tags

extinction, food webs, Network theory, networks, Tipping point

Transition point refers to the perturbation magnitude at which there is a rapid or nearly discontinuous increase in the level of secondary extinction. One can think of it as a threshold (“tipping point”) separating two alternative states in which the food web can find itself after perturbation. The DR 525 results present an opportunity to understand the dynamics of this transition. I therefore ran a number of experiments and also examined in detail the network dynamics in this perturbation region.

The first experiment was to remove the two herbivore guilds whose topological secondary extinctions coincide with the dynamic threshold. Progressive perturbation or removal of species in these two guilds fails to reproduce the threshold. Secondary extinction among the other heterotroph guilds, and autotroph nodes, remains nearly constant and low ( and respectively) across the perturbation range. Therefore, removal of these herbivore species themselves is insufficient to generate the transition. Nevertheless, their topological extinction is a marker for the transition.

The transition can be explained by noting the points at which the autotroph guilds disappear. There are four such guilds which were perturbed: benthic microautotrophs, phytoplankton, benthic macroalgae and macrophytes, and detritus (this last one is not truly autotrophic, but is treated similarly in the metanetwork). Benthic microautotrophs and detritus are the smallest of these basal nodes because of the lower diversity of benthic micrograzers and detritivores in the community. Therefore, as we apply a uniform perturbation to all four guilds, these two disappear completely at a perturbation magnitude 0f 0.51. When we examine this point on the simulation output, we note immediately that there is a subtle, yet discrete and secular increase in secondary extinction at that point and above. The second such secular change, the tipping point, occurs at a perturbation magnitude of 0.67, which is the point at which the benthic macroautotrophs become completely extinct; there is yet planktonic production available to the community. While there are numerous phytoplankton consumers (e.g. suspension feeding bivalves) in the community, and only a minority of benthic macrograzers, a vast number of heterotrophs depend indirectly on the consumers of the benthic microautotrophs, benthic macroalgae and detritus, and the final loss of the benthic macroproducers therefore pushes the community to a state of very high secondary extinction. The transition is generated by the propagation of bottom-up losses of links that although topologically minor, result in significant top-down compensatory increases of interaction strengths, and the network collapses.

This makes perfect sense from an ecological standpoint. But we must remember that no recovery or replenishment is permitted by the model. Addition of these factors would increase model complexity significantly, and also require major assumptions of exactly how to parameterize the physical environment and metapopulation structures appropriately. Nevertheless, these do remain as potential escape routes for the community if the perturbation is localized. And a final word on the transition or tipping point. I can see no way, at this stage, in which one could easily predict the location of the tipping point solely from the network structure. That is, there is no elegant mathematical solution available because of the complexities of the network interaction. The computer simulations are algorithmic, not analytic. I do predict, however, that many if not most communities should have such a tipping point of secondary cascades and extinctions embedded in their structure. The extent to which this is true depends most likely on overall community richness, functional diversity at the higher guild level, and individual guild richnesses.