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Insights from the Lotka-Volterra Model

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All models are wrong;
the practical question is how wrong do they have to be to not be useful.

George Box

In science, there’s an inherent trade off between comprehensibility and realism. Realistic models tend to be intricate … even convoluted. But to be comprehensible, a model must be simple.

For a good example of this trade off, look to high-school physics. In the real world, we know that projectiles are affected by aerodynamics. (That’s why frisbees fly differently than baseballs.) But since aerodynamics are complicated, high school teachers ignore them. Instead, they teach students that earthbound projectiles behave as they would on the moon — blissfully unaffected by air drag. This simplification is a lie, of course. But it’s useful for teaching students about the essence of Newton’s equations.

Science is filled with this sort of simplification. We learn about the world by developing toy models — models which simplify reality, yet retain (we hope) an element of truth.

In economics, there’s no shortage of toy models. But most of these playthings belong in the landfill; they’re models that assume away the most pertinent features of the real world. (For example, neoclassical economic models capitalism by assuming ‘perfect competition’, whereas the real world is marked by pernicious oligarchy.)

In short, if we want simple models that capture key elements of human behavior, it’s best to leave mainstream economics behind. Instead, a good place to start is with population biology — specifically the Lotka-Volterra model of predator-prey dynamics. Like projectile motion that neglects aerodynamics, the Lotka-Volterra equations are a toy model of how predator and prey populations respond to each other. In a sense, it’s the simplest ‘systems model’ that still provides useful insights about the real world.

In what follows, we’ll take a tour of the Lotka-Volterra model, and see how it gives insights into human behavior.

The Lotka-Volterra model

Developed in the early 20th century by the mathematicians Alfred J. Lotka and Vito Volterra, the Lotka-Volterra equations are an early example of what we would today call a ‘systems model’ — a model that simulates feedback between two or more entities.

In the Lotka-Volterra model, we imagine feedback between a population of predators and a population of prey. Looking ahead, an important feature of the Lotka-Volterra equations is that they can’t be solved with algebra. Instead, they must be solved numerically by throwing numbers in and seeing what comes out.

Why is this algebraic intractability important? Well, because much of mainstream economics operates like a subdiscipline of pure mathematics, where the goal is to postulate models with neat (but worthless) analytic solutions. And since systems models defy this sort of rigid thinking, they’ve been neglected by economists.

Back to the Lotka-Volterra model. The model begins by imagining two populations, one of prey and one of predators. Now the presumption is that predators eat prey. However this predation isn’t captured literally by the model. (There’s no equation that tells us when or how a wolf eats a sheep.) Instead, the model simulates predation in terms of population dynamics. For example, if a wolf population expands, it will cut into the growth rate of the neighboring sheep population (on which it preys).

The Lotka-Volterra model assumes that if left in isolation, our predator and prey populations will have opposite dynamics. If left alone, our prey population will grow exponentially.1 (For example, if we put a group of sheep into an empty field, they will reproduce and their population will expand.) In contrast, if our predator population is left alone it will decline exponentially. (For example, if we deprive a wolf pack of food, its population will gradually starve to death.)

So those are the starting assumptions, which both yield straightforward predictions if they’re left to play out. Fortunately, the Lotka-Volterra model doesn’t stop there. Instead, it imagines what happens when we mix predators with prey. It’s here that we encounter the magic of feedback. If wolves kill sheep, then a larger wolf population will reduce the growth of the sheep population. But if the sheep population declines, that causes the wolf population to die off from starvation.

Now an economist might look at this model and imagine that it arrives at some sort of equilibrium with an ‘optimal’ number of sheep and wolves. But that’s not what happens. When we run the Lotka-Volterra model, we find that it’s defined by non-equilibrium dynamics of boom and bust.

Figure 1 illustrates the dance between predators and prey. Here, the blue curve shows the population of sheep, while the red curve shows the population of wolves. Initially, the population of wolves is small enough that the sheep population expands happily. But this sheep boom then causes the wolf population to grow. As the wolf population balloons, predation causes a collapse of the sheep population, leading to starvation amongst the wolves. Finally, once enough wolves have died, the sheep population starts to expand, and the cycle begins again.

Figure 1: Boom-bust dynamics — the characteristic behavior of the Lotka-Volterra model. A key feature of the Lotka-Volterra model is that it gives rise to cycles of booms and busts, here visualized by a feedback relation between sheep and wolf populations. A rising sheep population causes the wolf population to boom. Over-predation then causes the sheep population to collapse, leading to an eventual decline in the number of wolves. Once enough wolves have died off, the cycle begins again.

Now, the cyclical dance of the Lotka-Volterra model is well known to population biologists. In fact, it illustrates a fundamental feature of natural systems: they’re marked not by static equilibrium, but by what the physicist Ilya Prigogine called ‘order through fluctuation’. In short, if natural systems are stable, it’s because they fluctuate. And if they don’t fluctuate (if they veer in a single direction), that’s a sign that something abnormal is happening.

Harvesting non-renewable resources

Speaking of ‘abnormal’, let’s think about the nature of industrial society. It’s built on a one-time bonanza of fossil-fuel extraction, which means that there can be no long-term cycles. When it comes to fossil fuels, order through fluctuation gives way to a single extraction pulse — a wild party followed by a bad hangover.

The Lotka-Volterra model, it turns out, is a valuable tool for thinking about this fossil-fuel extraction pulse. That’s because, with a little tweak, we can transform the equations into a model of non-renewable resource extraction.

Rather sensibly, the Lotka-Volterra model assumes that the ‘prey’ population is self renewing — that sheep can replenish their numbers if some get eaten. But this assumption is just a model parameter, and parameters can be changed. If we set the prey replenishment rate to zero, then we transform the ‘prey’ population into something quite different: it becomes a stock of a non-renewable resource.

In this scenario, the Lotka-Volterra model produces a different set of dynamics. Figure 2 illustrates the new pattern. Here, the blue curve represents a non-renewable resource stock. (Let’s think of it as a stock of food in a laboratory Petri dish.) And the red curve represents a ‘predator’ population. (Petri-dish bacteria.) Initially, the bacteria population is small, and so they eat and reproduce merrily. Their numbers swell, and their fixed stock of food declines. When the foodstock reaches a point where further bacteria growth is impossible, the expansion switches to decline. A die off begins, and the bacteria head towards extinction.

Figure 2: When ‘prey’ is non-renewable, the Lotka-Volterra model creates a single pulse of predation. Suppose we put a few bacteria in a Petri dish filled with a fixed supply of food. Here’s what the Lotka-Volterra model says happens. Initially, the bacteria population (red curve) grows because food is plentiful. But because the foodstock is fixed (blue curve), it gradually declines, eventually reaching a point where further bacterial growth is impossible. At that point, growth switches to decline, and the bacteria population starves to death.

The extraction pulse

In Figure 2, the blue curve shows the remaining stock of the non-renewable resource. Now, the model presumes that we know, in advance, the total size of this stock. And in the case of a Petri dish filled with food, we certainly do know the size of the recoverable foodstock. However, in more realistic scenarios, the total recoverable resource stock remains uncertain. For example, if the Petri dish is large and the food is distributed unevenly, it may be that the bacteria never reach certain patches. So although this unreachable food ‘exists’, it doesn’t count towards the total stock of recoverable food.

Turning to humans, if the world is a Petri dish, fossil fuels are a ‘food’ that is unevenly distributed. Sure, we can guess at the total stock of fuel. However, much of this energy will never be extracted, because the cost is too prohibitive. Hence, the total recoverable stock of fossil fuels remains uncertain. For that reason, it’s more helpful to look at the dynamics of the Lotka-Volterra model in a slightly different light. Instead of measuring the remaining stock of a non-renewable resource, we can look at its flow — the rate that it’s extracted.

Figure 3 shows this flow-based view. In the case of our bacteria, the flow represents the rate of food consumption. It rises as the bacteria population expands, and then falls as the food gets exhausted and the bacteria population collapses. Now, the point is that in the Lotka-Volterra model, this bell-shaped pulse of extraction is a generic feature of non-renewable resource consumption. It applies to humans as much as it applies to bacteria. And unlike the stock-based view (Figure 2), this flow-based view is observable to human participants. We can watch the extraction rate of fossil fuels rise and fall. Indeed, in many places, we’ve already seen both sides of this extraction pulse.

Figure 3: An extraction pulse. Instead of plotting the stock of a remaining resource, this figure plots the flow of a non-renewable resource, as predicted by the Lotka-Volterra model. The result is a bell-curved pulse of consumption.

Predatory machines

Continuing the theme of non-renewable resource extraction, if fossil fuels are the ‘prey’, then who is the ‘predator’?

Well, in some sense, the fossil-fuel ‘eaters’ are literally humans. After all, we grow most of our food with fossil-fuel-based fertilizers, which means that in a way, we ‘eat’ fossil fuels. That said, the main ‘predator’ of fossil fuels is not people; it’s our fossil-fuel eating machines.

Think about it this way. We use our machinery to wrench fossil fuels from the Earth. Then we feed the harvested energy back to our machines, many of which help us extract more fossil fuels. This loop, it turns out, is exactly the sort of feedback envisioned by the Lotka-Volterra model. To use the Lotka-Volterra equations to simulate the extraction of fossil fuels, we let the ‘prey’ be fossil fuels; and we let the ‘predator’ be our extraction technology.

Now, many researchers have realized that fossil fuel extraction can be understood with simple systems models. However, it was Ugo Bardi and Alessandro Lavacchi who first proposed a direct link between the rate of resource extraction and the stock of extractive technology.

Figure 4 illustrates the connection envisioned by the Lotka-Volterra model. Here, the blue curve plots the extraction rate of a non-renewable resource. And the red curve plots the population of ‘predators’ — the stock of extraction technology. Notice two things about this simulation. First, both the resource harvest rate and size of the technological stock have a pulse-like behavior — a rise, peak, and fall. Second, the peak of the technological stock follows the peak of extraction.

Why this order? According to the Lotka-Volterra model, it’s because the extraction technology feeds on the resource being harvested. So when this resource input peaks and declines, the technological ‘predators’ begin to die off a short while later. (Which is to say that the machines are abandoned and left to rust.)

Figure 4: Feeding a technological predator. Here I’ve plotted the version of the Lotka-Volterra model envisioned by Bardi and Lavacchi. In this simulation, we imagine feedback between the extraction of a non-renewable resource and a stock of extraction technology. In essence, the technology ‘feeds’ on the resource in question, which means that its fate is linked to the resource itself. The key result is the stock of extraction technology peaks after the peak of extraction.

Predators in the Alberta oilpatch

At this point, I’m going to turn to a real-world example of Lotka-Volterra-like behavior. But before doing so, it’s worth reminding ourselves that the Lotka-Volterra model is a toy. It’s purposefully designed to be an over-simplification of reality. So it’s somewhat surprising the model has anything to say about the messy arena of human affairs. And yet when it comes to our exploitation of fossil fuels, it seems that humans behave unwittingly like the Lotka-Volterra model predicts.

For a good example of this unintended connection, let’s turn to the history of oil-and-gas production in the Canadian province of Alberta. Today, the province is (in)famous for its extraction of unconventional oil from the Athabasca tar sands. However, much of the 20th century was spent drilling for conventional oil and gas. Figure 5 shows the history of this geological bonanza.

Figure 5: The rise and fall of Alberta convention oil-and-gas production. The blue curve shows the history of conventional oil-and-gas production in Alberta, Canada. The red curve shows the rise and fall of active wells. Like the Lotka-Volterra model predicts, the peak of the technological extractive stock proceeds the peak of production. The inset map shows the over 650,000 wells drilled so far. [Sources and methods]

Following a few false starts in the early 20th century, the Alberta oilpatch got rolling after World War II, driven largely by an unquenchable American thirst for energy. Conventional oil-and-gas production expanded for the next fifty years, but peaked in 1998. Today, Alberta’s conventional oilpatch is in steep decline, and the big players have largely moved north to the unconventional oil sands.

Now, the oilpatch is driven by a simple principle, which is that you extract oil by drilling holes in the ground. The more holes you drill, the more oil you get. Or at least that’s how it works at first. Over time, the big reserves get depleted, and more and more wells become duds. Eventually, there are enough duds that oil production begins to decline even though the number of wells continues to increase. When that happens, the economics of the oilpatch shift. Drilling slows, unproductive wells are left to rust, and the number of active wells begins to decline.

The red curve in Figure 5 shows this pattern of active-well peak and decline. It is eerily similar to the Lotka-Volterra model in Figure 4. The message here is that the players in the Alberta oilpatch seem to be unwitting puppets of a toy model. As predicted by the Lotka-Volterra model, the stock of Alberta’s active oil-and-gas wells peaked shortly after the peak of oil-and-gas extraction.

Now, the effect of a good chart is always to collapse complicated behavior into a graphical pattern that’s simple enough to comprehend. So when we stare at a chart like Figure 5, it’s easy to lose sight of the antics beneath it. For that reason, I’ve included a map of the Alberta oilpatch, where each oil-and-gas well is an imperceptibly small dot. Today, there are over 650 000 wells in total, each with its own story of ambition, glory, and failure. Importantly, there was no grand plan to the Alberta oilpatch, other than to make money selling the riches of the Earth. But ironically, it’s this lack of plan that gives rise to the overarching pattern of rise and fall.

The Lotka-Volterra model assumes a basic instinct to eat when the pickings are good, and starve when the food runs dry. But it could be that humans, in all our wisdom, are able to suppress this urge. For example, we can imagine a scenario in which the government sets quotas on oil-and-gas drilling — quotas designed to keep production constant. In the face of such planning, the Lotka-Volterra model would have nothing to say about oil extraction.

Although humans are surely smart enough to enact such policies, rarely do we actually do so. Instead, when faced with a stock of exploitable resources, we’re gripped by an animalistic urge to consume them as fast as possible. The Lotka-Volterra model captures this urge, which is why it seems to predict the large-scale pattern of how we extract resources, without knowing anything about our small-scale antics.

Shocking the system

When we ‘play’ with a model, it’s important to be open about its limitations. On that front, the Lotka-Volterra model is an obvious over-simplification of the real-world, which means we expect to find many situations where it breaks down.

For example, we can imagine a population of sheep and wolves in which a farmer drastically culls the wolf pack. Or we can imagine a bacteria-filled Petri dish in which a scientist suddenly dumps in more food. Neither situation can be anticipated by the Lotka-Volterra model, which pretends that its modeled populations exist in splendid isolation. In the arcane language of economics, these system shocks are said to be ‘exogenous’; they are not part of the Lotka-Volterra model, which means they can’t be predicted.

That said, these shocks can be put in ‘by hand’. To add a system shock to the Lotka-Volterra model, we can arbitrarily change the predator/prey population midway through the model run. Then we see how the model responds.

To get started with system shocks, let’s return to our example of Petri-dish bacteria which are busy eating a finite stock of food. Left alone, the bacteria’s food consumption will follow the familiar resource ‘pulse’, plotted in Figure 3. Food consumption will rise as the bacteria population expands, and then collapse as the bacteria starve. Now suppose that partway through this consumption pulse, a benevolent scientist dumps more food into the dish. What happens?

Well, it turns out that the impact depends on the timing of the food dump. If the food dump happens early in the experiment, the shape of the consumption pulse remains essentially unchanged. Figure 6 illustrates. Here, the foodstock quadruples early on, before the bacteria population has had much time to grow. The result is a minor uptick in food consumption, followed by the expected pulse of growth and decline.

In contrast, if the food dump happens late in the experiment, the effect is drastically different. Figure 7 illustrates. Here, our scientist waits until the original foodstock has begun to wane before dumping in a new bonanza. The result is a massive increase in resource consumption, which creates a second extraction pulse.

Figure 6: An early-game resource shock. Here we imagine bacteria in a Petri dish with a finite stock of food. Early in the consumption pulse, a scientist quadruples the foodstock. According to the Lotka-Volterra model, not much happens. That’s because at the time of this early dump, the bacteria population is small, so it can’t do much with the extra food. So the consumption pulse plays out as though the larger foodstock was there all along.

Figure 7: A late-game resource shock. Unlike an early-game resource shock, a late-game shock changes the shape of the consumption pulse by adding a second peak. Here, we imagine a population of Petri-dish bacteria left alone to eat a finite stock of food. After food consumption has peaked, a benevolent scientist quadruples the remaining foodstock. This bonanza creates a second peak of consumption — one which burns more brightly and more briefly than the first peak.

So why does it matter when we dump in the food? Well, because the bacteria’s ability to harvest food depends on their population. If we add food when the population is small, the bacteria can’t do much with it — their numbers are too few. If, however, we dump food into the dish late in the game, there is a large population of starving bacteria ready to gobble up the resource.

Returning to humans, the same scale principle holds. For example, suppose that in 1870, a benevolent god somehow quadrupled the global stock of oil. Would anyone have noticed? Probably not. At the time, oil extraction was in its infancy; our seismic technology was non-existent, our drilling technology was juvenile, and our refining technology was rickety. In short, when faced with an early-game quadrupling of our oil reserves, pretty much nothing would have happened (at the time).

Now imagine that the remaining stock of oil quadrupled today. Would anyone notice? It’s a silly question that’s not rhetorical. As it happens, the United States is in the midst of an oil-and-gas bonanza — one created by the exploitation of tight oil and shale gas. Of course, these reserves haven’t just popped into existence — they’ve been there all along. What changed is our technology.

For most of the 20th century, oil and gas was extracted by drilling a vertical well, and then sucking out the reserve. This technique works well if the formation is porous enough for the oil to flow. But if the formation is imporous, the well will come up dry. Now suppose that instead of drilling vertically, we bent the borehole and extended it horizontally though the reserve. And then suppose that we pumped in a high-pressure liquid which fractured the formation. Well then, this previously inaccessible resource would flow like melted butter. That, in a nutshell, is how the fracking revolution has worked.

To see this revolution, let’s turn to Figure 8, which plots the history of US oil production. For decades, the United States was the poster child for peak oil. In 1956, the geologist M. King Hubbert predicted that US oil production would peak in the early 1970s. And that is exactly what happened. From the next four decades, production declined. But in the mid 2000s, the fracking revolution opened up new reserves, sending US oil production to new heights.

Figure 8: A second bonanza — oil production in the United States. After more than three decades of decline, US oil production exploded in the 2010s. The turnaround owes to the new technique of fracking — using high-pressure liquid to fracture oil formations that were previously too impervious to flow. [Sources and methods]

Now the pertinent question is — how long will this second oil-and-gas bonanza last? For their part, peak oil theorists have become less strident than they were in the mid 2000s, in large part because the fracking revolution has shown the importance of technological change. That is, the amount of recoverable oil is affected not just by the Earth’s geology, but also by the tools we use to harvest fossil fuels.

So while I won’t give a definite prediction for the second peak of US oil production (I’ve already made one that’s proved wrong), the Lotka-Volterra model does give us reason to be bearish about the timing of this peak. Looking at Figure 7, the Lotka-Volterra model predicts that a late-game resource shock creates a second consumption peak that burns more brightly and briefly than the first peak. Again, this is because when resources are added late in the game, there’s a huge stock of ‘predators’ ready to exploit them.

Likewise, in the United States, the fracking revolution is taking advantage of an immense technological stack that is hungry for oil (and that had been slowly starving for decades). Because of this latent capacity, the US will likely burn through its unconventional oil reserves more quickly than it did with the conventional stuff.

Killing off predators

Continuing the theme of system shocks, so far we’ve explored what happens if we shock the Lotka-Volterra model by adding new ‘prey’. In the same vein, we can also shock the model by killing off ‘predators’.

As before, what interests me is how this shock plays out in the context of a non-renewable resource ‘pulse’. Since it’s the ‘predators’ that do the consuming, killing them off has the predictable effect of temporarily dampening resource consumption.

Figure 9 illustrates the effects of a predator die off. Returning to our example of Petri-dish bacteria, suppose that midway through their growth pulse, a vindictive scientist dumps cyanide into the dish. The bacteria population crashes, as does their rate of food consumption. However, the population soon recovers, and the consumption pulse begins again.

Figure 9: Killing off predators. Here we suppose that early in the consumption pulse of a non-renewable resource, most of the predators die off. The resource harvest rate plunges, but not for long. As the predator population recovers, the pulse of resource consumption resumes its course.

Similarly, anything that kills off humans — be it famine, pandemic, or war — will disrupt our consumption of resources. And the disruption will be doubly severe if it also destroys our technology, as is the case during warfare. In this regard, fossil fuels have been both a blessing and a curse. On the one hand, they’ve enabled an unprecedented rise in our standard of living. But on the other hand, they’ve magnified our destructive power, making war far more terrifying.

Perhaps more than any country, Japan offers the best example of how a consumption pulse can go wrong when it is interrupted by war. For much of the early 20th century, Japan was busy doing what every other great power had done before, which was to conquer new territory. Japan’s main sin was that it was late to the imperial game, which meant its expansion tread on the toes of established colonial powers.

As part of this imperial game, the Japanese military decided, in late 1941, to prod the US empire by bombing Pearl Harbor. It was a foolish decision. At the time, the United States was by far the world’s most powerful country, consuming about a third of the world’s energy. So egging it into war was destined to end badly.

And for Japan, end badly the war did. Not only was much of the country flattened by conventional bombs, Japan remains the only population to have been devastated by a nuclear bomb. Figure 10 plots the scale of this wartime destruction, measured in terms of Japan’s share of world energy use. After prompting the US into World War II, Japan’s share of world energy use plummeted. It didn’t recover to pre-war levels until 1966.

Figure 10: Japan’s share of world energy use and the devastation of WWII. It’s easy to spot the moment when Japan provoked the US into declaring war (in late 1941). The ensuing US bombardment decimated Japan’s infrastructure (and of course, its population), sending Japan’s share of world energy use back to 1890 levels. The post-war recovery took decades. [Sources and methods]

Now, I realize that it feels crass to reduce the violence of war to an abstract systems model. But really, it’s no more crass than converting the violence of predation into a mathematical equation — which is exactly what the Lotka-Volterra model does. In this case, the destruction of imperial Japan offers a case study in what happens when machines and infrastructure (our technological ‘predators’) get destroyed.

Social collapse

Since human-built machines don’t (yet) have a life of their own, they needn’t be destroyed to be rendered useless. Anything that obstructs their human caretakers will have the same effect, freeing technology to sit idle. For this reason, periods of social collapse can (like war) be treated as a ‘predator shock’ — moments when our active technological stock suddenly decreases.

For example, during the Great Depression, much of the world’s machinery lay unused, simply because people couldn’t afford to use it. More recently, the collapse of the Soviet Union provided a similar experiment with idle technology driven by social chaos.

When the Soviet regime died in 1990, Western economists confidently declared that markets would emerge and pick up the slack left by the absence of state planning. Unsurprisingly, the market ‘miracle’ worked rather differently. In the aftermath of the Soviet collapse, former member states suffered a severe and prolonged depression.2

The history of Russian oil production offers a good window into this collapse. During the later years of Soviet control, Russian oil production had exploded, reaching a pinnacle in 1987. But in the aftermath of the Soviet collapse, Russian oil production was cut nearly in half. It didn’t return to the Soviet-era high until 2019. Figure 11 illustrates this free market ‘miracle’.

Figure 11: A market ‘miracle’ — Russian oil production implodes following the Soviet collapse. When the Soviet Union dissolved, Soviet-block countries experienced a severe and prolonged depression, visible clearly in the production of oil. For example, Russia’s oil production was cut nearly in half, and didn’t return to the Soviet-era peak until 2018. [Sources and methods]

Again, I think the Lotka-Volterra model provides some useful insight into the post-Soviet depression. Sure, it says nothing about how or why the Soviet Union collapsed. But when the ensuing social chaos rendered a large portion of Soviet machinery inoperable, we can treat the catastrophe that followed like a sort of ‘predator’ die off. Only recently, have Russia’s ‘oil predators’ recovered.

A humble toy

This concludes my tour of the Lotka-Volterra model, which I’ll remind you, is best considered a humble mathematical toy. The Lotka-Volterra model doesn’t make grand predictions about the future. It’s not compelling enough to attract acolytes. It’s not seductive enough to be enshrined in official dogma. And it’s not enthralling enough to be the subject of political debate. No, the Lotka-Volterra model is a simple thought experiment about the effects of population feedbacks.

And yet, I hope to have convinced you that the Lotka-Volterra model is useful. Life on Earth is dominated by feedback effects, and it behooves us to understand how they work. The Lotka-Volterra model offers a comprehensible entry point into the world of systems modeling, a world in which simple principles generate complex effects. In an academic landscape dominated by neoclassical economic fantasies, surely we could use more of this type of thinking.


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Minsky

It would be foolish to write a post about feedback modeling without mentioning the hard work being done by heterodox economist Steve Keen.

Backing up a bit, systems models consist of nothing but sets of differential equations. In the early days of modeling, scientists coded these equations by hand. But that gets tedious quickly. And so researchers developed graphical tools for creating systems models. Today, there are many such tools, but virtually all of them are proprietary and closed source. The exception is the systems modelling program Minsky developed by Steve Keen and coded by Russell Standish. Minsky is free and open source, and designed specifically with economics in mind. I encourage you to try it.

The Lotka-Volterra equations

The Lotka-Volterra model consists of a set of coupled differential equations, typically written as:

\displaystyle \begin{aligned} \frac{dx}{dt} &= \alpha x - \beta x y \\ \\ \frac{dy}{dt} &= -\gamma y + \delta xy \end{aligned}

Here, x is the prey population and y is the predator population, while dx/dt is the rate of change of the prey population and dy/dt is the rate of change of the predator population. The remaining terms are model parameters which, to me, make more sense if we rewrite the model in terms of population growth rates.

To reframe the Lotka-Volterra equations in terms of growth rates, note that a growth rate is simply a rate of change expressed as a portion of the thing changing. So the growth rate of x is:

\displaystyle \widehat{x} = \frac{dx/dt}{x}

Likewise, the growth rate of y is:

\displaystyle \widehat{y} = \frac{dy/dt}{y}

When we reframe the Lotka-Volterra equations in terms of growth rates, they simplify as follows:

\displaystyle \begin{aligned} \widehat{x} &= \alpha - \beta y \\ \\ \widehat{y} &= -\gamma + \delta x \end{aligned}

In English, these equations state:

  1. Without predators, the prey population will grow exponentially at rate \widehat{x} = \alpha .
  2. Without prey, the predator population will decline exponentially at rate \widehat{y} = -\gamma .
  3. The presence of predators y decreases the growth rate of prey by - \beta y .
  4. The presence of prey x increases the growth rate of predators by \delta y .

And that’s it! From these simple equations comes a host of dynamics that are not predictable from algebra alone.

Sources and methods

Alberta oilpatch (Figure 5)

This data is from my post ‘A Case Study of Fossil-Fuel Depletion’. Detailed methods are available here.

US oil production (Figure 8)

Oil production data is from the following sources:

  • 1949 to present: Energy Information Agency, Table 1.2, Primary energy production by source
  • 1860 to 1948: Historical Statistics of the United States, Table DB157

Japan’s share of world energy use (Figure 10)

World energy use is from Our World in Data, Energy Production and Consumption.

Japan’s energy use is from the following sources:

All Japanese series are indexed backwards from the Statistical Review data. Also note that the carbon-based estimates are indexed twice. First, I index the carbon data to Japanese energy use in 1900. The resulting series assumes that Japan’s energy use prior to 1900 directly tracks its carbon emissions. The problem with this estimate is that it ignores non-fossil fuel sources of energy, which become more important as we head back in time. To correct this problem, I then add to the time series the constant value of 26,000 KCal of energy per person per day. Finally, I re-index this updated energy estimate to the statistical data from 1900.

(For those who are interested, I’ve used the same carbon-based method to estimate the historical energy use of the Soviet Union.)

Russian oil production (Figure 11)

Russian oil production data is from the following sources:

Notes

  1. In the real world, the sheep population will eventually plateau as it reaches the field’s carrying capacity. But the Lotka-Volterra model assumes that predation keeps the sheep population well below this upper limit.↩
  2. Dmitry Orlov’s book Reinventing Collapse gives a fascinating account of the suffocating environment in post-Soviet Russia.↩

Further reading

Bardi, U., & Lavacchi, A. (2009). A simple interpretation of Hubbert’s model of resource exploitation. Energies, 2(3), 646–661.

Orlov, D. (2008). Reinventing collapse: The Soviet example and American prospects. New Society Pub.

The post Insights from the Lotka-Volterra Model appeared first on Economics from the Top Down.

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cjheinz
1 hour ago
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Worth a read.
Lexington, KY; Naples, FL
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Saving Money on Groceries by Understanding Food Product Packaging Dates

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When shopping for groceries, it’s easy to get confused by the different food product dates stamped on packaging. “Sell by,” “Best if used by,” “Use by,” and “Expiration” dates don’t all mean the same thing, and misunderstanding them can lead to throwing out perfectly good food—or worse, wasting money. By learning what these dates really mean, you can stretch your grocery budget and reduce food waste.

Importantly, food package dating is not federally regulated except in infant formula. 

Types of Packaging Dates

  • Sell By Date: This is meant for the store, not the customer. It tells retailers how long to display the product for sale. Foods are usually still safe to eat for days (sometimes weeks) after this date if stored properly.

  • Best If Used By/Before Date: This refers to quality, not safety. It’s the manufacturer’s estimate of when the product tastes its best. Many packaged foods—like cereal, pasta, and canned goods—are safe long past this date.

  • Use By Date: This is the last date the manufacturer recommends for peak quality. It’s not necessarily a safety cutoff (except on infant formula, where it is federally regulated).

  • Expiration Date: This is the closest thing to a real safety deadline. If you see “expires on,” it’s best not to consume the product after that point.

Tips for Saving Money and Reducing Waste

  1. Shop Smart Around Dates: Grocery stores often discount items nearing their “sell by” or “best by” dates. Buying these and using or freezing them quickly can save you money.

  2. Trust Your Senses: Look, smell, and taste (safely) before tossing something. Many foods are fine well beyond the printed date.

  3. Use Your Freezer: Freezing meat, bread, and even dairy products before their date can extend shelf life for months.

  4. Practice FIFO (First In, First Out): Rotate items in your pantry and fridge so older items are used first.

  5. Know the Shelf Life: Canned goods, dried pasta, and rice can last for years if stored properly. Don’t rush to throw them away just because of a “best by” label.

Why It Matters

According to the USDA, Americans waste about 30–40% of the food supply each year, much of it due to confusion over date labels. That’s money out of your pocket and food out of the supply chain. By understanding packaging dates, you can save money, reduce waste, and make your groceries stretch further.

To learn more, check out this USDA resource https://www.fsis.usda.gov/food-safety/safe-food-handling-and-preparation/food-safety-basics/food-product-dating 

Find all of our UF-IFAS Blogs here https://blogs.ifas.ufl.edu/about/

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cjheinz
10 hours ago
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My family has a running discussion of this.
Lexington, KY; Naples, FL
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Podcast: AI Slop Is Drowning Out Human YouTubers

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This week, we talk about how 'Boring History' AI slop is taking over YouTube and making it harder to discover content that humans spend months researching, filming, and editing. Then we talk about how Meta has totally given up on content moderation. In the bonus segment, we discuss the 'AI Darwin Awards,' which is, uhh, celebrating the dumbest uses of AI.

Listen to the weekly podcast on Apple Podcasts, Spotify, or YouTube. Become a paid subscriber for access to this episode's bonus content and to power our journalism. If you become a paid subscriber, check your inbox for an email from our podcast host Transistor for a link to the subscribers-only version! You can also add that subscribers feed to your podcast app of choice and never miss an episode that way. The email should also contain the subscribers-only unlisted YouTube link for the extended video version too. It will also be in the show notes in your podcast player.



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cjheinz
23 hours ago
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I am still jealous of Gary Marcus for inventing the term "Slopacolypse Now", IMO better than my "Bullshit Apocalypse" - although I think my term is more accurate & inclusive.
Lexington, KY; Naples, FL
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Often give in

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In an oft-quoted speech, Winston Churchill said:

Never give in–never, never, never, never, in nothing great or small, large or petty, never give in except to convictions of honour and good sense. Never yield to force; never yield to the apparently overwhelming might of the enemy.

The problem with this advice is that it means we spend an enormous amount of time in senseless battles with senseless folks who are also following this advice.

In a community, perhaps it makes more sense to only have battles about honour and good sense. In everything else, sure, give in. It’ll help you focus on what really matters.

      
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cjheinz
1 day ago
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Lexington, KY; Naples, FL
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Congressional Oversight of Bill Pulte

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I'm hoping that Senate Banking and House Financial Services, or at least some members thereof, will be sending Bill Pulte an oversight letter seeking answers to some questions: 

(1) How did FHFA learn of supposed issues with the mortgage loans of Letitia James, Adam Schiff, and Lisa Cook?

(2) For each of James, Schiff, and Cook, if there was a tip or a whistleblower, how and when did that person make contact and what information was presented? Did FHFA follow up and make contact with any whistleblower or tipster?

(3) Once FHFA had information about supposed issues, how did obtain the underlying mortgage files?  If they were obtained from Fannie/Freddie, to whom at the GSEs did it send the request for the underlying mortgage files and the date of that request? 

(4) Has FHFA requested the mortgage files of any other individuals in federal or state executive, legislative, or judicial branch positions? 

I'm sure there are some other questions that might be asked, but this set all strikes me as well within the bounds of legitimate oversight. Pulte should answer if he's not hiding something. 

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cjheinz
4 days ago
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Seriously, fuck this guy.
We bought a DiVasta home in Naples, FL, in 2009. Poured concrete walls, roof rebarred to the walls like you wouldn’t believe. A hurricane-proof pillbox.
DiVasta got acquired by Pulte, quality went down, down, down. Sad.
Lexington, KY; Naples, FL
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When MAGA Prophecy Fails

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Charlatans, Pseudo-Experts, Snake Oil Salesmen, Phonies, Frauds, Wanna-Bes,  Hobbyists, Hang-Arounds, Opportunists…. and Psychopaths.

Like everyone else, I’ve been following the latest Trump-Epstein revelations. Or maybe we should call them confirmations: Unless you were deep in the cult, you already had a pretty good idea of who Trump was and were aware that he and Epstein went way back.

But many Trump loyalists are cultists, sufficiently so that they believed that Donald Trump — Donald Trump! — was heroically defending the world against pedophiles. And we know what cultists do when confronted with facts that refute their beliefs: They engage in denial.

Now, Epstein and all that is — thank God! — not my department. But MAGA’s cultish nature is relevant to matters that are in my usual domain.

For Trump made many prophecies about the economic miracles he would achieve as president. “Starting on day one, we will end inflation,” he promised. “We will be slashing energy and electricity prices by half within 12 months, at a maximum 18 months.” He promised to get gasoline below $2 a gallon. And of course he insisted that he would deliver a jobs boom, especially in manufacturing.

Obviously none of that is happening. Tomorrow’s report on consumer prices will probably show inflation running at close to 3 percent, with most economists expecting it to rise in the months ahead. Electricity prices are rising rapidly, while gas is solidly above $3 a gallon. And job growth appears to be stalling.

Furthermore, much of the bad news is Trump’s own fault. His tariffs and deportations are both adding to inflation and, by creating uncertainty, slowing the economy.

But cultists never admit that their prophecies were wrong. Rather than admit that the promised economic miracle isn’t happening, Trump and his minions have gone after the people reporting the bad news, specifically the Bureau of Labor Statistics, which produces both jobs and inflation data.

Trump has already fired the head of the BLS for reporting job numbers he didn’t like, claiming falsely that the bad numbers were rigged to hurt him politically. We can expect further claims of partisan bias as the inflation numbers get worse, and eventually, probably quite soon, an attempt to purge and politicize the agency.

The push to politicize the BLS has been reinforced by yesterday’s report from the agency, which revised downward its estimates of past job growth. The White House claimed that it shows that “the BLS is broken.”

It showed no such thing. As a helpful post from the Economic Policy Institute says,

These BLS data revisions are not corrections of mistakes. Revisions are part of the regular, transparent process to update employment counts with the most comprehensive data possible.

As the EPI explains, monthly job numbers don’t literally track every job in America. They’re estimates based on a partial survey of employers. We only get comprehensive data from unemployment insurance tax records, which become available once a year. Revising the estimates based on that data is normal and in no sense a sign that the BLS is doing anything wrong.

But the administration will try to use the revision to discredit the agency, and in particular its recent reports showing a worsening labor market.

So what you need to know is that the BLS is doing its job the way it should, and that there is plenty of additional evidence confirming that the labor market has gotten worse under Trump.

For example, the widely respected Conference Board survey of consumers shows that between last December and August there was a sharp decline in the number of people saying jobs were “plentiful” and a sharp rise in those saying they were “hard to get.”

The New York Fed reports that the percentage of respondents who believe that they could quickly find a new job if they lost their current one has dropped sharply.

And the Federal Reserve’s Beige Book, a regular informal survey that often serves as a useful check on formal data, gave a clear picture of stalling employment:

Eleven Districts described little or no net change in overall employment levels, while one District described a modest decline. Seven Districts noted that firms were hesitant to hire workers because of weaker demand or uncertainty. Moreover, contacts in two Districts reported an increase in layoffs, while contacts in multiple Districts reported reducing headcounts through attrition …

This is not a booming economy.

It's not really surprising that Trump is failing to deliver on any of his promises, which never made sense in the first place. Nor is it surprising that he and those around him, rather than making a course correction, are trying to shoot the messengers. But it’s a tragedy that the attempt to suppress bad news may well destroy the Bureau of Labor Statistics, a highly competent and professional agency whose services we need more than ever.

MUSICAL CODA

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cjheinz
5 days ago
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I agree, the Orange Turd is a prophet & prophecies. I identified and placed prophecy in my Taxonomy of Bullshit.
Lexington, KY; Naples, FL
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