Logistic equation model for collapse, or?

jimk

I've been playing with some simple differential equations... I am starting to think that maybe the Logistic equation used by Hubbert may not be such a good model for global collapse.

My current thinking is that the Logistic equation is great for when the world at large is roughly constant, but we keep adding more powerful equipment and expertise to coax more fuel out of some pocket. As the resource starts to give out, we can just coax even harder. My thought is, this is because the world at large is still providing a steady stream of resources to invest in extraction capability.

But when we look at the whole world, then as our pocket of fuel is depleted, we have no place else to go to give us the capability to coax even harder. So I think a different system of equations may be called for.

Here is my proposed fix:

and here is a comparison of two simulations:

I tweaked the parameters to get the peaks to line up. The coupled system has more parameters, so actually I can generate a family of simulation with the same peak, but different combinations of investment and depreciation rates.

Anyway, I'd love any feedback!

jimk

One first observation: this R/P system will not burn through all the resource. Eventually there just isn't enough resource to keep investment above depreciation and the capability to extract just disintegrates before all the resource has been extracted.

Doly · Expert Joined: Dec 03, 2004 Posts: 4035

The reason Hubbert used the logistic equation is because it was easier to calculate, nothing else. In fact, in the known cases of peak, a Gaussian seems to approximate better.

If you want to calculate depletion based on realistic parameters, you have to take into account economic factors, and even a simplistic model of economic factors is a tad more complex than that. I have been working on one, but I'm not sure the depletion curves look right yet.

pup55 · Expert Joined: May 26, 2004 Posts: 3694

I like it.

Your equations give an asymmetrical function: a lower and slightly earlier peak, and also a few more resources toward the end, at roughly year 166.

This is similar to what we were getting when we used the verhulst function in one of the earlier threads a couple of years ago.

What we were doing at the time was trying to fit one or more of these models to the historical oil production data. The theory on this was if we could get as good a fit as possible for the left hand side of the curve, fitting the existing data to some values on your equation, we could predict the peak and/or the right hand side of the curve.

It has been awhile since we updated some of this, and there are a couple more data points to work with.

But it is kind of an interesting exercise, and at the very least, using your equation, there is some suggestion that the downward slide might not be quite so steep.

jimk

Here's a fun tweak. I'll call this next system of differential equations the "open R/P system":

The c term is a steady stream of input from some external source, like the sun. With this, the system undergoes a spikey kind of oscillation:

I imagine many similar systems have been explored by ecologists. So much to learn, so little time!

Is there a way in excel to plot two different series on the same graph, but where the ranges of the two series are quite different, so one would need something like axes both on the right and the left?

jimk

jimk

So here is a plot of P and PR on the same graph. Of course this is just one simulation with one set of parameters.

bobcousins

[quote="jimk"]...
edit: I'm confused by your caption, shouldn't it read "the more that has been extracted, the harder it is to extract more?".
_________________
It's all downhill from here

jimk

bobcousins

Ok, thanks. I always wondered how investment was incorporated into the logistic equation. Now I just need to get my brain around why the rate peaks and decreases.

jimk

Here's another comparison of the R/P system against the Logistic equation. This is a plot of (dR/dt)/R, i.e. of the power of extraction, what fraction of the remaining resource gets extracted per year (or whatever time interval).

This is for a different set of parameters than the other simulations - I keep tweaking the simulations just to see what sorts of behaviors are possible, and I confess I am not very good about keeping a good lab notebook!

One thing I would love to try, is to fit these equations to the global history, with P being world population, and PR being world energy consumption from all sources, wood, coal, petroleum, whatever.

jimk

ohanian · Light Sweet Crude Joined: Oct 17, 2004 Posts: 1089

I'm really confused by your equation

What is R

Is R the amount of Resources remaining in the ground?

Is R the amount of Resources extracted from the ground?

Also, why is dR/dt a positive number? It can only be a positive number if R is the amount of Resources extracted from the ground.

What is P?

Is P the production of Resources?

Is P the number of human beings?

Could you defined your equations exactly and give us a sample run of your equation with some sample values of R and P at t=0 , t=1 , and t=10 , so that I can have some feel for your equations.

jimk

Sorry, the equations are a bit abstract!

R is indeed the amount of resource in the ground. dR/dt is mostly negative: dR/dt = -PR. In the open system I add in a positive constant, so dR/dt = c-PR. Over the millions of years when we are not burning fossil fuel, sunlight gets turned into stored fuel which slowly builds up. Then our rate of burning quickly burns through what has built up. So in the open system dR/dt oscillates between positive and negative.

P is the fractional rate of resource extraction, i.e. like what percentage of the resource gets extracted in unit time.

Somehow this fractional rate of extraction changes over time. So that's what I am trying to look at and change a bit from the logistic equation. The fractional rate has to do with whatever facilities we build up to enable extraction. These grow through investment and decay through depreciation. The facilities are some mix of human beings and machinery and whatever else.

Of course one can easily build up a more complicated model with many more dynamic variables and parameters. The advantage of a simpler model is not just that the simulations run faster. It can actually improve predictive power. If the number of parameters exceeds the number of data points, then one can fit the data exactly with a variety of different parameter value combinations, each predicting a different future. If one can find a pretty good fit to many data points with very few parameter values, then the prediction of the future evolution is not so arbitrary.

Whether the R/P system I have proposed here will give a better fit, I don't know. I am not claiming that it will! I am just offering these equations as a way of exploring the family of equations that are quite similar to the logistic equation. Sketching out a range of plausible models is one step, then narrowing down to the best fit is another step. My proposal is just one contribution to that first step.

Here is another plot of a simulation run, giving R and P. The horizontal time axis should run from 0 to about 2200 - I don't know how to fix up the excel axis labelling, sorry.

This is the open R/P system, with parameter values c=1; a=0.00004
; b=0.05; the initial values are R=1000; P=1E-07.

WebHubbleTelescope · Intermediate Crude Joined: Jul 08, 2004 Posts: 909

Why are you writing out predator-prey equations? Is it because you think that the population of oil-depletors is the predator and the oil is the prey?

Lotka-Volterra equations usually model cyclic oscillations of "renewable" resources such as foxes and rabbits.

So R is the prey and P is the predator. R starts at some value and decays nonlinearly and proportionally to how much prey there is and how many predators there are to eat it.

The predator P starts at some value and grows proportionally to how many predators there are and how much prey exist (i.e. exponential birth rate sped up by food supply). The predator population also dies in proportion to its current population (i.e. classical death rate).

This only differs from the conventional L-V in that the prey does not have its own birth rate. So it generates a "non-renewable" variant of the L-V solution.

Interesting model but as in the logistic model, the parameters have no physical meaning with respect to oil depletion. Plus, this set of equations can only be solved numerically since it has nonlinear dependencies among the variables. Which makes it hard to come up with a URR.

So given the choice of two heuristics, people will likely choose the simpler, which remains the logistic. Knock yourself out trying to pursue this. As for myself, I prefer to derive a model from more firmly grounded first principles. Which of course excludes the logistic.