Systems Paleoecology – Regime Shifts I

Tags

alternative states, attractor, critical transition, regime shift, Tipping point, transience

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
12. States, Transitions and Extinction

Numerous terms, with roots across multiple disciplines that deal with dynamic complex systems, are used interchangeably in the study of transitions to some extent because they are related by process and implication. But they do not necessarily always refer to the same phenomena, and it is useful to be explicit in one’s usage (maybe at the risk of usage elsewhere). Regime shift, critical transition and tipping point are three of the more commonly applied terms in the ecological literature. They form a useful general framework within which to explore the concept of multiple states and transitions, and into which more detailed concepts can be introduced. Regime shift is defined here as an abrupt or rapid, and statistically significant change in the state of a system, such as a change of population size (Fig. 1A). Transient deviations or excursions from previous values, e.g. those illustrated in Fig. 1B}, are not regime shifts. “Regime” implies that the system has been observed to have remained at a stationary mean or within a range of variation over a period of time, and to then have shifted to another mean and range of variation. Regimes can be maintained by external or intrinsic processes, or sets of interacting external parameters and internal variables, but the ways in which the processes are organized can vary. Sets of processes can be dominant, reinforcing the regime; understanding this simply requires one to associate a regime with our previous discussions of system states and attractors. Regime shifts occur then when sets of processes are re-organized, and dominance or reinforcement shifts to other parameters and variables.

Fig. 1A – Hypothetical regime shift

Fig. 1B – Two populations of red-wing blackbirds. See here for an explanation.

Regime shifts may be distinguishable from variation within a state, or continuous variation across a parameter range, by the time interval during which the transition occurs, if the interval is notably shorter than the durations of the alternative states. This of course potentially limits the confirmation of regime shifts as we can never be certain that observation times were sufficient to classify the system as being in an alternative state. The interpretation though is that the duration of the transition was relatively short because the system entered into a transient phase, i.e. moving from one stable state to another. The transition itself may be precipitated in several different ways, dependent on the type of perturbation and the response of the system. The perturbation could be a short-term excursion of a controlling parameter that pushes the system into another state, with the transition being reversed if the threshold is crossed again. More complicated situations arise, however, if internal variables of the system respond to parameter change without a measurable response of the state variable itself, and if the system can exist in multiple states within the same parameter range. These various characteristics of regime shifts serve to distinguish important processes and types of shifts that are more complex than simple and reversible responses to external drivers, such as “critical transitions” and “tipping points”.

We have already discussed several model systems with multiple states, one of those being a trivial state of population extinction (), and the other being an attractor when . Zero population size was classified as an unstable state, because the addition of any individuals to the population — — leads immediately to an increase of population size, and the system converges to a non-zero attractor. This is true regardless of the nature of the attractor (e.g. static equilibrium, oscillatory, chaotic), and makes intuitive sense — sprinkle a few individuals into the environment and the population begins to grow. This is not always the case, however, and there are situations where zero population size, or extinction, can be a stable attractor, or where converges to different attractors, dependent either on population size itself, or forcing by extrinsic parameters. The system is then understood to have multiple alternative states. I reserve this definition for circumstances where does not vary smoothly or continuously in response to parameter change (e.g. Fig. 1), but will instead remain in a state, or at an attractor, within a parameter range, and where the states are separated by a parameter value or range within which the system cannot remain, but will instead transition to one of the alternative states. Thus, the multiple states are separated in parameter or phase space by transient conditions.

We will explore a real-life example in the next post, and here is a teaser.

Cod in the North Atlantic.

Vocabulary
Attractor – A compact subset of phase space to which system states will converge.
Regime shift – An abrupt or rapid, and statistically significant change in the state of a system.
System state – A non-transient set of biotic and abiotic conditions within which a system will remain unless acted upon by external forces.
Transient state – The temporary condition or trajectory of a population as it transitions from one system state to another.

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.

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.