Oh, wait, did I forget the word “eventually” there?

Doomsday predictions from science go a long way back. The reverend Thomas Malthus has probably one of the most famous ones. His prediction didn’t become reality, but just because he and others have falsely cried wolf doesn’t mean there aren’t any. The Dagupta review also shows that we are likely to run into hard ecological limits, if we don’t rapidly change our behaviour.

Hans Stegeman, sustainable investment banker, drew my attention in one of his posts to another such model that was published in scientific reports (a journal from the Nature/Springer group) in 2020. The authors of the study draw a grim picture of our probability of survival as a species. From the abstract:

Based on the current resource consumption rates and best estimate of technological rate growth our study shows that we have very low probability, less than 10% in most optimistic estimate, to survive without facing a catastrophic collapse.

Bologna & Aquino (2020) Deforestation and world population sustainability: a quantitative analysis. Scientific reports 10, 7631. https://doi.org/10.1038/s41598-020-63657-6

The paper combines two models: a relatively simple renewable resource model (although they don’t call it that) and a stochastic model of technological progress. What they then do is compare the time it takes for society to collapse because we meet ecological limits if we maintain current harvest rates, based on the resource model, with the expected time it will take to develop technology that will allow us to colonize space, based on the technological growth model. Their conclusion is as above: we have 10% probability of developing sufficient technology before collapse is a fact.

Although I am aware that we may run into hard ecological limits, I was wondering how realistic this prediction was, so I read the article in more detail, with a special focus on the renewable resource model, as that is my expertise.

The renewable resource model used in the paper was originally used by the same authors to model the collapse of Easter Island. Easter Island has been studied by a number of ecologists and economists (see e.g. the work of James Brander and Scott Taylor and of Scott Taylor and references therein) and although the original paper of Bologna & Aquino, does reference the first of those papers, it doesn’t do much with it. That is a pity, as it is in fact the combination of those papers that would be a more realistic portrayal of what happened. As it stands, the model of Scott Taylor is in my view more realistic and telling than this current one.

For those interested in the mathematics I explain the paper below, but my conclusion was that the statement about 10% risk of survival is overly pessimistic because they do not allow for policy feedback or technological progress. In addition, their definition of collapse is debatable, but let’s leave that for another round. When policy or technological progress is introduced in some scenarios society still collapses, but in the more optimistic ones, even with very mild policy or technological progress, there is no collapse at all.

So are we doomed? Well, eventually very likely, yes, if your time horizon is long enough. And admittedly the IPCC projections are pretty worrying (see for example conclusion B.3 and and B.4 in the summary for policy makers of the latest report). But overly pessimistic conclusions don’t help either, they only paralyze.

The model

The renewable resource model is based on two equations: one equation describing the growth and harvest of forests and the other describing population growth. Forest amount available in hectares in year t is denoted as R(t), and similarly the number of people in the world at time t is denoted as N(t). The forest evolves as follows:

\frac{dR(t)} {dt} = r_f R(t) ( 1 - \frac{R(t)}{R_c}) - a_0*N(t)*R(t)

The first part (before the minus sign is a standard logistic growth curve. r_f is called the internal growth rate of the forest: the percentage growth if we had almost no forest. R_c is the carrying capacity. If we reach that point the forest no longer expands and if it goes beyond it will shrink rather then expand.

The harvest function (or destruction of forest) is the second part. This also has the standard form that we know from e.g. fisheries models. Harvest is determined by some standard parameter (here a_0), a measure of effort (here population size N(t)) and the stock size R(t). The important difference is that in resource models the level of effort is a choice variable, usually driven by costs and benefits, whereas here it is the population which is given, not chosen.

The evolvement of the population over time is a similar but not completely equal process. For one there is no harvest, and the carrying capacity is different.

\frac{dN(t)} {dt} = r_p N(t) ( 1 - \frac{N(t)}{\beta R(t)})

Again the r_p is the internal growth rate of the population. The carrying capacity depends on the forest stock. The reasoning is that forests provide vital ecosystem services, such as oxygen, so reducing the forest stock reduces the total number of people the earth can sustain. This also generates a feedback loop: if our harvest exceeds the natural growth and the forest stock reduces, this reduces the carrying capacity, which lowers population growth.

The authors define a crash as the point where population exceeds carrying capacity. Their assumption is that if the population exceeds this threshold, we won’t have a gradual decline of the population, as predicted by the model, but a violent chaotic crash. There is something to say for that, but I’d like to emphasize that that’s what it is: an assumption. Also note that given the model structure the ONLY way the population can ever decrease is if it exceeds carrying capacity.

Next, the authors come up with estimates for the parameters r_f,r_p \textnormal { and} \beta of the model, and run simulations to see when the population hits carrying capacity. They have two values of \beta they consider: a low one which means that we reach carrying capacity quite soon and a high one, where we don’t reach carrying capacity for another 100 years or so (I reproduce their figures below).

How can we introduce policy or technological progress into this model? The a_0 is a good candidate. Given that a_0*N(t)*R(t) is the harvest, then a_0 is the share of the forest that is harvested per person. If we have technological progress we can be more efficient with our resources and we can bring down this share per person. Similarly, taking measures to reduce destruction of forests will bring this share down. I will therefore assume that a_0 is slowly driven down by a small % each year, that is I replace a_0 with:


where x is the reduction per year.

All simulations I did where done using R, and package GGplot 2.

The original figures of the paper are reproduced below. Note that in the paper the time axis on the forest stock is different from the one on the population. In my figure they use the same time scale, which makes the forest figures look slightly different.

Figure 1: Evolvement of population in original model with β=170 in low and β=700 high
Figure 2: Evolvement of forest in original model with β=170 in low and β=700 high

In the next figures (3&4) I show what the introduction of technological progress does. I include the original level of technological progress (0%) for reference. In the optimistic (\beta is high scenario introduction of minimal technological progress of 0.5% per year lengthens the time until carrying capacity by about 50 years (Figure 3). Using slightly larger technological progress (1% per year) and we never exceed carrying capacity. As we can see in the forest amounts the forest doesn’t get destroyed either (Figure 4). Things look less promising in the pessimistic scenario. In that case we need at least 5% technological progress per year not to exceed the carrying capacity.

Figure 3: Evolvement of population in model with technological progress with β=170 in low and β=700 high
Figure 4: Evolvement of forest in model with technological progress with β=170 in low and β=700 high

Note that due to the model structure the population will always reach carrying capacity eventually, but overshooting no longer happens in this model in some scenarios.

In a way these conclusions are very similar to similar contributions on this blog. There are some hard ecological limits we need to deeply worry about, but overly pessimistic conclusions don’t help either, they only paralyze.

Cover photo credit: Easter Island, Ahu Tongariki. © Arian Zwegers.