Source: NYT

The flow of plastic bags into landfills is dramatically down from the 2005 rate. But the accumulation is up. This should be no surprise, because the structure of this system is:

The accumulation of bags in the landfill can only go up, because it has no outflow (though in reality there’s presumably some very slow rate of degradation). The integration in the stock renders intuitive pattern matching (flow down->stock down) incorrect.

Placing the flow and the stock on the same vertical scale, is also a bit misleading, because they’re apples and oranges – the flow of disposal has units of tons/year, while the accumulation has units of tons.

Also, initializing the stock to its 2005 value is a bit weird. If you integrate the disposal flow from 1980 (interpolating as needed), the accumulation is much more dramatic: about 36 million tons, by my eyeball.

LOG>CREATE|timing.txt
LOG>MESSAGE|timing.txt|"About to run."
LOG>TIMESTAMP|timing.txt
MENU>RUN|o
LOG>TIMESTAMP|timing.txt
LOG>MESSAGE|timing.txt|"Ran."

These commands were designed for logging user interaction, so they don’t offer millisecond resolution needed for small models. For that, another option is to use the .dll.

Generally, model execution time is close to proportional with equation count x time step count, with exceptions for iterative functions (FIND ZERO) and RK auto integration. You can use the .dll’s vensim_get_varattrib to count equations (expanding subscripts) if it’s helpful for planning to maximize simulation speed.

Vast stretches of Texas farmland lying over the aquifer no longer support irrigation. In west-central Kansas, up to a fifth of the irrigated farmland along a 100-mile swath of the aquifer has already gone dry. In many other places, there no longer is enough water to supply farmers’ peak needs during Kansas’ scorching summers.

And when the groundwater runs out, it is gone for good. Refilling the aquifer would require hundreds, if not thousands, of years of rains.

This is a widespread, generic problem, and instance of the tragedy of the commons archetype:

Alternately, as an economist might put it, depletion of groundwater is typically an externality, with zero rent paid from users to owners (us).

There are three facets to management of the problem: the commons problem, detail complexity from the complex geology of aquifers, and dynamic complexity. Allocation of property rights solves the commons problem, but not the others (see Erling Moxnes’ interesting work on this).

Sniffing around for material for this post, I found it easy to get lots of complicated information about aquifers, but hard to find any simple stock-flow pictures that revealed their state or dynamics, like this:

Source: USGS

This suggests that there’s some low-hanging fruit to be harvested through provision of simple feedback, as in the Climate Scoreboard.

The human body has a well understood mechanism for monitoring blood pressure changes, consisting of sensors embedded in the major arterial walls that monitor changes in pressure and then trigger other changes in the body to increase or reduce the pressure as necessary, such as the regulation of the volume of fluid in the blood vessels. This is known as the baroreceptor reflex.

So an interesting question is why this system does not respond appropriately as the body ages. Why, for example, does this system not reduce the volume of fluid in the blood to decrease the pressure when it senses a high systolic pressure in an elderly person?

The theory that Pettersen and co have tested is that the sensors in the arterial walls do not directly measure pressure but instead measure strain, that is the deformation of the arterial walls.

As these walls stiffen due to the natural ageing process, the sensors become less able to monitors changes in pressure and therefore less able to compensate.

Flickr/Eric Fischer

As Schelling showed, it doesn’t take draconian measures for this behavior to emerge. It’s a robust (but suboptimal) outcome of weak coupling by positive feedback.

Catastrophic Collapse Can Occur without Early Warning: Examples of Silent Catastrophes in Structured Ecological Models (PLOS ONE – open access)

Catastrophic and sudden collapses of ecosystems are sometimes preceded by early warning signals that potentially could be used to predict and prevent a forthcoming catastrophe. Universality of these early warning signals has been proposed, but no formal proof has been provided. Here, we show that in relatively simple ecological models the most commonly used early warning signals for a catastrophic collapse can be silent. We underpin the mathematical reason for this phenomenon, which involves the direction of the eigenvectors of the system. Our results demonstrate that claims on the universality of early warning signals are not correct, and that catastrophic collapses can occur without prior warning. In order to correctly predict a collapse and determine whether early warning signals precede the collapse, detailed knowledge of the mathematical structure of the approaching bifurcation is necessary. Unfortunately, such knowledge is often only obtained after the collapse has already occurred.

To get the insight, it helps to back up a bit. (If you haven’t read my posts on bifurcations and 1D vector fields, they’re good background for this.)

Consider a first order system, with a flow that is a sinusoid, plus noise:

Flow=a*SIN(Stock*2*pi) + disturbance

For different values of a, and disturbance = 0, this looks like:

For a = 1, the system has a stable point at stock=0.5. The gain of the negative feedback that maintains the stable point at 0.5, given by the slope of the stock-flow phase plot, is strong, so the stock will quickly return to 0.5 if disturbed.

For a = -1, the system is unstable at 0.5, which has become a tipping point. It’s stable at the extremes where the stock is 0 or 1. If the stock starts at 0.5, the slightest disturbance triggers feedback to carry it to 0 or 1.

For a = 0.04, the system is approaching the transition (i.e. bifurcation) between stable and unstable behavior around 0.5. The gain of the negative feedback that maintains the stable point at 0.5, given by the slope of the stock-flow phase plot, is weak. If something disturbs the system away from 0.5, it will be slow to recover. The effective time constant of the system around 0.5, which is inversely proportional to a, becomes long for small a. This is termed critical slowing down.

For a=0 exactly, not shown, there is no feedback and the system is a pure random walk that integrates the disturbance.

The neat thing about critical slowing down, or more generally the approach of a bifurcation, is that it leaves fingerprints. Here’s a run of the system above, with a=1 (stable) initially, and ramping to a=-.33 (tipping) finally. It crosses a=0 (the bifurcation) at T=75. The disturbance is mild pink noise.

Notice that, as a approaches zero, particularly between T=50 and T=75, the variance of the stock increases considerably.

This means that you can potentially detect approaching bifurcations in time series without modeling the detailed interactions in the system, by observing the variance or similar, better other signs. Such analyses indicate that there has been a qualitative change in Arctic sea ice behavior, for example.

Now, back to the original paper.

It turns out that there’s a catch. Not all systems are neatly one dimensional (though they operate on low-dimensional manifolds surprisingly often).

In a multidimensional phase space, the symptoms of critical slowing down don’t necessarily reveal themselves in all variables. They have a preferred orientation in the phase space, associated with the eigenvectors of the eigenvalue that’s changing at the bifurcation.

The authors explore a third-order ecological model with juvenile and adult prey and a predator:

Predators undergo a collapse when their mortality rate exceeds a critical value (.553). Here, I vary the mortality rate gradually from .55 to .56, with the collapse occurring around time 450:

Note that the critical value of the mortality rate is actually passed around time 300, so it takes a while for the transient collapse to occur. Also notice that the variance of the adult population changes a lot post-collapse. This is another symptom of qualitative change in the dynamics.

The authors show that, in this system, approaching criticality of the predator mortality rate only reveals itself in increased variance or autocorrelation if noise impacts the juvenile population, and even then you have to be able to see the juvenile population.

We have shown three examples where catastrophic collapse can occur without prior early warning signals in autocorrelation or variance. Although critical slowing down is a universal property of fold bifurcations, this does not mean that the increased sensitivity will necessarily manifest itself in the system variables. Instead, whether the population numbers will display early warning will depend on the direction of the dominant eigenvector of the system, that is, the direction in which the system is destabilizing. This theoretical point also applies to other proposed early warning signals, such as skewness [18], spatial correlation [19], and conditional heteroscedasticity [20]. In our main example, early warning signal only occurs in the juvenile population, which in fact could easily be overlooked in ecological systems (e.g. exploited, marine fish stocks), as often only densities of older, more mature individuals are monitored. Furthermore, the early warning signals can in some cases be completely absent, depending on the direction of the perturbations to the system.

They then detail some additional reasons for lack of warning in similar systems.

In conclusion, we propose to reject the currently popular hypothesis that catastrophic shifts are preceded by universal early warning signals. We have provided counterexamples of silent catastrophes, and we have pointed out the underlying mathematical reason for the absence of early warning signals. In order to assess whether specific early warning signals will occur in a particular system, detailed knowledge of the underlying mathematical structure is necessary.

In other words, critical slowing down is a convenient, generic sign of impending change in a time series, but its absence is not a reliable indicator that all is well. Without some knowledge of the system in question, surprise can easily occur.

I think one could further strengthen the argument against early warning by looking at transients. In my simulation above, I’d argue that it takes at least 100 time units to detect a change in the variance of the juvenile population with any confidence, after it passes the critical point around T=300 (longer, if someone’s job depends on not seeing the change). The period of oscillations of the adult population in response to a disturbance is about 20 time units. So it seems likely that early warning, even where it exists, can only be established on time scales that are long with respect to the natural time scale of the system and environmental changes that affect it. Therefore, while signs of critical slowing down might exist in principle, they’re not particularly useful in this setting.

The models are in my library.

… at a Ceilidh:

… in the MSU Cello Festival concert:

… during intermission at Bozeman’s Pecha Kucha night #7:

The Helmholtz motion of bowed strings is an interesting dynamic system, but that’s a topic for another day.

Support pika research on RocketHub.

COL: Let’s say tomorrow, evidence is found that proves to everyone that global warming as a result of human released emissions of CO2 and methane, is real. What would you suggest we do?

SPENCER: I would say we need to grow the economy as fast as possible, in order to afford the extra R&D necessary to develop new energy technologies. Current solar and wind technologies are too expensive, unreliable, and can only replace a small fraction of our energy needs. Since the economy runs on inexpensive energy, in order to grow the economy we will need to use fossil fuels to create that extra wealth. In other words, we will need to burn even more fossil fuels in order to find replacements for fossil fuels.

via Planet 3.0

On the face of it, this is absurd. Reverse a positive feedback loop by making it stronger? But it could work, if given the right structure – a relative quit smoking by going in a closet to smoke until he couldn’t stand it anymore. Here’s what I can make of the mental model:

Spencer’s arguing that we need to run reinforcing loops R1 and R2 as hard as possible, because loop R3 is too weak to sustain the economy, because renewables (or more generally non-emitting sources) are too expensive. R1 and R2 provide the wealth to drive R&D, in a virtuous cycle R4 that activates R3 and shuts down the fossil sector via B2. There are a number of problems with this thinking.

• Rapid growth around R1 rapidly grows environmental damage (B1) – not only climate, but also local air quality, etc. It also contributes to depletion (not shown), and with depletion comes increasing cost (weakening R1) and greater marginal damage from extraction technologies (not shown). It makes no sense to manage the economy as if R1 exists and B1 does not. R3 looks much more favorable today in light of this.
• Spencer’s view discounts delays. But there are long delays in R&D and investment turnover, which will permit more environmental damage to accumulate while we wait for R&D.
• In addition to the delay, R4 is weak. For example, if economic growth is 3%/year, and all technical progress in renewables is from R&D with a 70% learning rate, it’ll take 44 years to halve renewable costs.
• A 70% learning curve for R&D is highly optimistic. Moreover, a fair amount of renewable cost reductions are due to learning-by-doing and scale economies (not shown), which require R3 to be active, not R4. No current deployment, no progress.
• Spencer’s argument ignores efficiency (not shown), which works regardless of the source of energy. Spurring investment in the fossil loop R1 sends the wrong signal for efficiency, by depressing current prices.

In truth, these feedbacks are already present in many energy models. Most of those are standard economic stuff – equilibrium, rational expectations, etc. – assumptions which favor growth. Yet among the subset that includes endogenous technology, I’m not aware of a single instance that finds a growth+R&D led policy to be optimal or even effective.

It’s time for the techno-optimists like Spencer and Breakthrough to put up or shut up. Either articulate the argument in a formal model that can be shared and tested, or admit that it’s a nice twinkle in the eye that regrettably lacks evidence.

Two things about this surprise me.

Confronted with obvious and irrefutable errors, the authors double down and admit nothing. They also downplay the significance of the results, ‘… we are very careful in all our papers to speak of “association” and not “causality” …’

But of course the (amplified) message, Debt/GDP>90%=doom, was taken causally in the policy world; see the multiple clips in the intro to the Colbert video. Politicians are nuts to accord one paper in a sea of macroeconomic thought so much weight, but I guess this was the one they liked.