Population dynamics

This article goes to the very core of the overpopulation objection: If we defeat ageing, will we actually cause a huge spike in population? And if so, how long will it take before we reach an unsustainable population?

Are we overpopulated?

That is a good question, and while answers range from ‘You f*cking blind? Of course we are!’ to ‘lol, no’, there doesn’t seem to be scientific consensus on the matter. As pointed out in this paper, estimates for the Earth’s carrying capacity vary wildly, from a few hundred millions to hundreds of billions, and that’s perfectly understandable. The carrying capacity is probably a difficult variable to estimate as it is, let alone in the case of humans: Our lifestyle is unlike that of any other species on the planet, and together with our technology it influences the ability of the planet to support us. Generally speaking, technology has increased the carrying capacity of the planet, but side-effects like GHGs emissions and overconsumption of resources caused by a consumerist lifestyle may easily decrease number of people the Earth can support.

I don’t have the expertise to give a conclusive answer to this question, but I can briefly tell you my opinion and let you come to your own conclusions. As I said here, overpopulation is not a black-or-white matter at all; we may be overpopulated with respect to a certain environmental factor, but not another. I’ll list here a few of these factors, and explain why I think we are (or aren’t) too many with respect to them.

First things first: Just how many of us are there?

updated-world-population-growth-1750-2100-750x525
Image credit: Our World in Data

From the picture above, we see we’re somewhere between 7.4 and 7.5 billion people, with a growth rate (more about it later) of a bit less than 1.2%. (As a side note, if we forget rejuvenation and all of that for a moment, as things stand world population isn’t projected to double even at the turn of the century, let alone ‘by 2060 at current growth rates’, as it is said on the first website I linked above. Perhaps they shouldn’t have left out the minor detail that the ‘current growth rates’ won’t stay current for long. The graph is quite clear that the growth rate has been plummeting since the 60’s and is showing no sign of stopping.)

So, as promised, here come my two cents. Are we overpopulated with respect to…

  • space? Not even close. As I was discussing here, in terms of room alone we could afford a population several times larger than the current one. Infrastructure is a different matter altogether, though, and we probably could do with better infrastructure already now. Hopefully, 3D-printing will prove helpful in this department. 3D-printed buildings are already a thing, afterall.
  • resources? It depends on the resources. When it comes to food, for example, I’m more keen on thinking that we don’t have a shortage problem, but rather a management problem. When you consider that in the US obesity is a national epidemic and that we’re constantly decreasing inequality of food supply and undernourishment all over the world despite the population is still growing, it’s easy to conclude the problem is resource distribution, not really lack thereof. So, the answer is no in terms of food; I am not sure about the rest.
  • environmental impact? Yes. As I said before, we’re already emitting too many GHGs, and not even all people in the world contribute in the same way. However, the answer is not going on a legalised killing spree or shutting down all hospitals to trim the population down a bit, but rather strive to improve our means of energy production and resort to technologies that cause lower emissions. We’re doing that, little by little.
  • jobs? Probably yes, since unemployment is already a real problem. As I discussed here, though, the solution to this problem lies in the changing nature of work, not in population purges.

As said, the above are only my guesses, and they aren’t very important. The central question to be answered is, would rejuvenation cause a significant population increase? In other words, if people stopped dying of old age and therefore lived much, much longer than they do today, would that be sufficient to lead to an unsustainable population? And if yes, what would be the expected time frame? In order to answer these questions, we need to deal with a little bit of maths. If maths ain’t your cup of tea, don’t worry—I’ll keep it simple.

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Population growth

For the benefit of those readers who have no idea what an exponential is, I’m going to spend a few words on the concept; if you’re already familiar with it, you may want to skip right to the next part.

Exponential growth for dummies

Even if you’re unsure what it means, you probably hear the expression a lot. People often say that something grows ‘exponentially’ when it grows really fast. Not all things that grow fast grow exponentially, but rest assured that all things that grow exponentially grow pretty damn fast. So, what does ‘exponential growth’ actually mean?

Let’s start from the so-called power notation. Imagine you need to multiply the same number with itself many times. For example,

2 ✕ 2 ✕ 2✕ 2 = 16.

That’s already quite lengthy as it is, and imagine if you had to multiply 2 by itself 20 times rather than 4. We can do this more compactly by just keeping track of how many times the multiplication occurs, and agreeing on a symbol meaning ‘the multiplication of this number by itself occurs this many times.’ In our case, ‘this number’ is 2, and ‘this many times’ is 4. So, we agree to write

24

to mean

2 ✕ 2 ✕ 2✕ 2.

Long story short,

24 = 2 ✕ 2 ✕ 2✕ 2 = 16.

To make our notation more general, we can write nm, where n and m can be any two numbers. If we say n=2 and m=4, we fall back to our example of 24. The number nm is called a power of n; n is said to be the base of the power, while m is called its exponent. The expression nm is read ‘n to the power of m’, so for example 24 is two to the power of four.

Imagine we kept the exponent fixed at some arbitrary value, but let the base change. For example, consider n2, where n can be any natural number, i.e. n can be 0, 1, 2, 3, 4, and so on. (We consider only natural numbers for simplicity. As said, we can use any number at all.) As the number n grows, how fast does the expression n2 grow? Let’s see.

n n2=nn
0 0
1 1
2 4
3 9
4 16
5 25
6 36

Table 1. The values of n2 for the first 6 natural numbers.

To have a better idea of how fast it grows, have a look at the picture below. It shows the graph for n2, n3, n4, and n5, not just for natural numbers but for positive numbers in general.

powers

Figure 1. The black curve is n2, n3 is red, n4 is blue, and n5 is green.

Powers aren’t too slow, but not especially fast either. I mean, 32=9 isn’t too far from 3 itself; however, you certainly have noticed how higher exponents imply faster growth—32=9, while 35=243 doesn’t even fit in the graph. What would happen if we kept the base fixed and let the exponent vary? For example, how fast does the expression 2m grow?

m 2m=2 ✕ 2 ✕ … ✕ 2 m times
0 0
1 2
2 4
3 8
4 16
5 32
6 64

Table 2. The values of 2n for the first 6 natural numbers.

Tables 1 and 2 show clearly that 2n catches up fast with n2, and quickly gets to the point of no return—2n will grow faster and faster, and n2 will never catch up with it again. Expressions like 2n are called exponentials, for the very reason that the variable is the exponent, not the base. The picture below shows the growth of four different exponentials.

exps

Figure 2. The black curve is 2n, 3n is red, 4n is blue, and 5n is green.

Of course, as the base gets larger, exponentials also, grow faster. As a matter of fact, nn grows crazy fast. However, what matters here is that exponentials grow faster than powers, and after a certain point, any exponential will be larger than any power. In the case of 2n and n2, the situation is like this:

p-exp

Figure 3. The black curve is n2, while 2n is light blue.

As a matter of fact, after a certain point, any exponential will be forever larger than any linear or polynomial expression. (Though there are growth types faster than exponential growth, such as hyperbolic growth, which unlike exponentials can reach infinity in a finite time1.) To give further demonstration of just how fast exponentials grow, consider this: If n = 100, then n2 = 10.000, while 2n = 1.267.650.600.000.000.000.000.000.000.000.

The fact exponentials will eventually outrun any powers and linear and polynomial expressions can be proved rigorously, but I won’t go there. Rather, I want to point out the intuitive reason why exponentials grow so fast. The number 24 is twice 23 which is twice 22 which is twice 2. Every time the exponent grows by 1, our total doubles. In the general case am, each time m increases by 1, our total becomes a times its size—size which increases more and more each time.

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Births vs. deaths

Populations grow exponentially, right? That’s what they say. Well, it depends. Populations can grow exponentially, but they don’t have to. For example, cells reproduce by division: You start with one cell, which splits into two, each of which splits into two, and so on. If there are enough resources around for all the cells to thrive, you get something like this:

cells

Figure 4. Exponential growth of reproducing cells.

At the beginning (‘generation 0’), we have 1 cell. The first generation has two, the second has four, and so on. Sounds familiar? It’s an exponential in base 2: 20=1, 21=2, 22=4, 23=8, 24=16, etc2. If a population reproduces like this, then it’s growing exponentially for sure, because with each generation the population doubles its size. Notice that, in this case, one individual can reproduce on its own and it ultimately gives rise to two individuals total. Even if your sex life isn’t very enthusing, you’ve probably figured out by now that this isn’t quite like how humans reproduce: Traditionally, it takes two humans to create a single new one (with the exception of twins). This fact makes for an interesting example.

Suppose we have a seriously weird bunch of cells that reproduce kind of like humans. It takes two cells to make a single new one, and cells of newer generations don’t reproduce with cells of older generations, which age and eventually die. How would this population change? To answer this question, it’s enough to turn the previous picture upside down:

cells-rev

Figure 5. Hypothetical cell population dies off because of insufficient offspring.

The population dies off. The reason is simply that each generation has fewer individuals than the previous one, eventually reaching a generation with a single individual who has no one to mate with. What happens if, all other conditions being equal, each cell pair has exactly two offspring? We have two offspring for each couple, therefore each generation has exactly the same size as the previous one. (We don’t really care if and how the newborn cells shuffle before mating again.)

cells-2

Figure 6. Stable cell population.

It would appear as if the population remains the same when each generation has the same number of individuals as the previous generation, and grows exponentially or shrinks to death when there are more (or fewer) individuals in each new generation than there were in the previous one. However, this isn’t quite true. For example, if we had a population such that each generation had only one individual more than the previous generation, this growth would be very slow and nowhere near being exponential. (It’d be a linear growth, n + 1.) What’s really important here is the number of offspring per reproductive unit—by ‘reproductive unit’ I mean the set of individuals necessary to make new individuals: In our first example, a single cell was a reproductive unit, whereas in the second one, a reproductive unit was a pair of cells. If the number of offspring of each reproductive unit is larger than the number of individuals within the reproductive unit, we get exponential growth: In our first example, the ratio offspring:reproductive unit size was 2:1, so we had exponential growth; in the second, the ratio was 1:2, and the population died off. In the third, it was 2:2, and the population stayed perfectly stable. Of course, in real life not all reproductive units have the exact same number of offspring, and some may have none; however, this reasoning still works for the average number of offspring per reproductive unit. In the example of Figure 6, if not all cell pairs had exactly 2 offspring but the average number of offspring per cell pair was 2, we would still get the same number of individuals in each generation and the same type of growth (zero growth, in this case). It follows that, in order for the human population to grow exponentially, the average number of children per reproductive unit (or, equivalently, per couple or per woman) must be larger than 2.

If this average number (also known as fertility rate) was exactly 2, then assuming no female mortality before the end of reproductive age, the population would stay at replacement level and not grow at all. However, the mortality rate during and prior childbearing age varies between industrialised and developing countries, so that in the latter replacement level is attained with an average number of children per woman ranging between 2.5 to 3.3; anything above this threshold would trigger exponential growth. This difference takes into account that developing countries have a higher mortality rate, unfortunately, and thus some of those 2.5-3.3 children have good chances of dying before they can have children themselves. Therefore, in order to reach replacement level, each woman in developing countries needs to have a higher number of children than in developed countries. At the moment, the overall fertility rate in the world is around 2.4, so the growth is presently exponential, but not a very fast exponential. Also note how the linked graph shows the corkscrew-diving of the fertility rate in the past 50+ years.

Now that we have a clearer picture of how births contribute to population growth, we can finally answer one of the main questions subject of this article: Can deaths offset births and prevent overpopulation from happening? If the fertility rate is above the replacement level, growth becomes exponential, and the answer is no, not a chance. This is because within a given generation, you can’t have more people dying than there are alive, and since exponential growth produces generations that are always larger in size than the previous ones, the amount of people who die is always smaller than those who are born. If we’re not too far above the replacement level, the growth won’t be lightning fast, but it will continue nonetheless; things will still go apeshit, only later. If we’re much above the replacement level (i.e. so that each generation is at least twice the size of the previous one), each new generation will be larger than all previous ones put together, and our numbers will skyrocket no matter what. If the fertility rate is below the replacement level, then our population will start decreasing. Bottom line: Are you worried about overpopulation? Fine, there’s only one way to avoid it: Keep the fertility rate no higher than replacement level—i.e., make fewer children. As a rule of thumb, 2 is more than enough. Deaths aren’t going to prevent overpopulation3.

Of course the fertility rate isn’t always the same, so population growth isn’t all the time exponential, stable, or plummeting. However, a sufficiently high (albeit oscillating) fertility level can make population grow crazy fast even if everyone eventually dies, while a sufficiently low fertility rate can dramatically slow down population growth even if nobody dies of ageing. This is what we will see in the next section.

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Population projections

It’s time to finally check how fast rejuvenation would cause overpopulation, if ever. We will do a bit more than just that, and in order to do it, we’ll make use of the exponential growth equation. Given a growth rate, it will tell us the population size at any point from now. The logistic growth equation would make for a more accurate model, because it takes into account the carrying capacity of the planet. However, I will stick to the exponential growth equation for two reasons: First, as said the carrying capacity isn’t very easily determined, and can change over time; second, the logistic equation returns values closer and closer to the carrying capacity itself as time approaches infinity, and we aren’t interested in that. We want to see how large the population would get without age-related deaths in the worst-case scenario of unbounded growth. If things don’t go too badly in a worst-case scenario, we can rest assured they’ll go better in a better-case scenario. The exponential growth equation is the following:

P(t) = P(0)ert.

In a nutshell, we start off with P(0) as our initial population. After a certain number t of years, the initial population will have grown by a factor of ert, thus becoming P(0)ert. The number e = 2.718… is the so-called Euler’s number, the base of natural logarithms. The reasons why it appears in the formula are not important here, so don’t worry about it. The number r is the yearly population growth rate, expressed in percentage. Basically, it’s the red line in the population chart at the top of this page; that red line was drawn basing on UN data available here. (It’s an XLS file called Population Growth Rate.)

Note that, in the formula above, the growth rate is constant; in other words, it doesn’t change with time. However, the growth rate does change with time, so if we computed

P(t) = P(0)er1 + r2 + … + rt,

where r1, r2, …, rt are the growth rates for years 1, 2, …, t, our estimates would be more accurate4. However, we’re soon going to assume that no age-related death occurs, and therefore we won’t be able to use the UN growth rates any more, because the lack of ARDs will change them. Making assumptions on how each of these rates will vary would introduce far too much uncertainty in the computation, therefore we’ll stick with a constant rate. This will make our worst-case scenario even ‘more worst’, if you pass the expression, because the plummeting birth rates of the past 50 years despite population growth suggest even no-ARD growth rates would go down with time. We’ll choose our constant rate to be the average for 2015, i.e. r = 0.0108, or 1.08%. We will also choose 2015 as our starting year, and therefore our initial population will be P(0)= 7,349,472,0995. (That’s the population for 2015 measured by the UN. You can find this data on the same UN page, this time in an XLS file called Total Population – Both Sexes.)

If the growth rate stayed fixed at r = 0.0108, this is what the population numbers would look like up to 2100 (i.e., 85 years after 2015).

t (year) Population after t years
10 (2025) 8,187,662,822
25 (2040) 9,627,547,181
45 (2060) 11,948,771,710
85 (2100) 18,405,112,649

Table 3. Population projection at a fixed growth rate r = 0.0108.

The situation is definitely worse than as it was on the chart above, and this is to be expected; in our case the growth rate isn’t decreasing, unlike in real life. However, note that even with this constant growth rate the population does not double by 2060, as claimed by the first website I linked atop the page. Also note that this growth rate still takes into account all kinds of mortality.

What we want now is a population projection with no age-related deaths (ARDs for short). We assume that, magically, in 2015 we already has full rejuvenation and starting that year nobody died of ageing any more. This means we need a new growth rate that accounts for this change. How do we get one?

The UN calculated their growth rates using the following formula:

rate

(this information is available on the usual UN page, in the description of the Population Growth Rate file.) The symbol ln indicates the aforementioned natural logarithm, but you don’t really need to understand what it is or how it works. You calculator can help you double-check my maths, if you wish to do so.

The rate r = 0.0108 we used was also calculated according to the formula above, as the average rate for the timespan between 2015 and 2020. Thus, t= 5 (the length of that timespan), P(0) is again the world population in 2015, and P(t) is the world population in 2020 according to their projections. We can use this same formula to calculate a rate rnARD such that nobody ever dies of old age. All we need is an estimate for the world population in 2020, assuming no one died of ageing in the previous 5 years. We can obtain this estimate easily as

P(5) = P(0) + 5·BY – 5·ODY,

where BY is the average number of yearly births, and ODY is the average yearly number of deaths due to causes other than ageing. If we wanted a population estimate for 2020 that accounted for age-related deaths, we would have to subtract five times the average number of age-related yearly deaths, i.e. the term -5·ADY would appear at the bottom of our formula. Now, how do we get BY and ODY?

The average numbers of daily births and daily total deaths can be found here, while average daily ARDs can be found on this PDF. They’re estimated as roughly BD=360,000 daily births, DY= 150,000 daily total deaths, and ADD = 100,000 daily age-related deaths. The average number of daily deaths not related to ageing is thus ODD = DDADD = 150,000 – 100,000 = 50,000. Between 2015 and 2020 there are two leap years and three normal years, for an average of 365.4 days each year. Thus, we have an average of BY=365.4·360,000 = 131,544,000 births each year, and an average of ODY=365.4·50,000 =18,270,000 deaths due to other causes than ageing each year. If we plug in these numbers in our expression for P(5), we get

P(5) = P(0) + 5·131,544,000 – 5·18,270,000 = 7,349,472,099 + 657,720,000 – 91,350,000 = 7,915,842,099.

We can plug this estimate in the rate formula above to get our ARD-free rate with value rnARD = 0.0148, expectedly larger than r. It is therefore no surprise that, if we had cured ageing in 2015, our population numbers would look like this:

t (year) Population after t years, no ARDs
10 (2025) 8,521,807,680
25 (2040) 10,640,085,157
45 (2060) 14,305,276,960
85 (2100) 25,858,218,670

Table 4. Population projection at a fixed growth rate rnARD = 0.0148. In this scenario, nobody dies of ageing.

The difference between Table 3 and Table 4 starts off small, but just as you’d expect when dealing with exponentials, it becomes more and more important as time goes by. However, you need to take into account that 360,000 births a day occur now that the fertility rate is above replacement level; in other words, 360,000 births a day is above replacement level. We have already established that as long as fertility is above this threshold, population is going to grow no matter what, so we would anyway reach the numbers in Table 4 eventually, even if we didn’t cure ageing, only later. We have also already noted that the only real way to prevent population from growing too large is to stay near (and possibly below) fertility replacement level. Bottom line: 360,000 births a day are too many. What would happen if we reduced them down to 70% of that number?

Seventy percent of 360.000 amounts to 252,000. With this figure, the yearly average number of births between 2015 and 2020 would be BY=365.4·252,000 = 92,080,800, and the estimated population for 2020 (again, assuming no one dies of ageing) would be P(5) = 7,718,526,099. The new growth rate is thus rnARD = 0.0098, which is only about 91% of our original rate r=0.0108. This means that, if we cut yearly births by 30%, we would not only compensate for the population increase caused by the defeat of ageing, but population growth would also be slower than it is in a world where people still die of ageing.

t (year) Population after t years, no ARDs Population after t years, ARDs
10 (2025) 8,106,194,215 8,187,662,822
25 (2040) 9,389,842,194 9,627,547,181
45 (2060) 11,422,995,666 11,948,771,710
85 (2100) 16,905,322,063 18,405,112,649

Table 5. The left-hand column shows the population projection at a fixed growth rate rnARD = 0.0098. In this scenario, nobody dies of ageing, and births have been cut by 30%. For comparison, the right-hand column shows the same data as Table 3, population projection at a fixed rate r = 0.0108, in a scenario where births haven’t been cut down and everybody still dies of ageing.

Surprised? You shouldn’t be. This is due to a very simple fact. I could perhaps have spared you this lengthy explanation and just pointed out the obvious: 360,000 daily births are much more than 100,000 ARDs. Their amount is 3.6 times larger than that of ARDs. It’s obvious that a relatively small percentage of the births will therefore amount to all ARDs; daily ARDs are in fact 27.7% of daily births, which is why cutting births by 30% results in the lack of ARDs being more than compensated for. As a side note, the total daily death toll is 41.6% of the daily births. In principle, if we cut births by 42%, we could even eliminate all causes of death and still grow more slowly than we currently do.

Naturally, this does not mean we can keep packing more and more people on the same planet indefinitely. If nobody ever died but new people were still born, we would still keep growing, and no matter how slow that growth was, we would eventually get to a point when we couldn’t have more people on the planet. However, if the growth was very slow, it would give us plenty of time to adapt and increase the carrying capacity of the planet through technology, and possibly become a space-faring species well before we hit much too large numbers. We’re talking centuries, of course, but with a sufficiently slow growth that’s not a problem.

Of course, we can’t expect births to suddenly drop down to 70%, so our ride will be a bit more bumpy than in my projections, but we can still expect births to go down over time. How? Population control? Nope.

First, let’s be realistic. Even if we defeat ageing and eliminate more and more causes of death, you can expect some people will still die, because we probably won’t always know how to save everyone. That being said, consider that births have been going down on their own for quite a while now. Why is that?

  • Fertility is inversely correlated with wealth, and falls with development. This means that, as people become more wealthy, have access to better services, higher education, and generally have a better quality of life, they have fewer children. It is worth pointing out, though, that if standards of living go very high, then fertility rises again. I’ll talk about this in a second.
  • Women are no longer just baby-churners. While women’s role used to be pretty much that of a biological incubator, this is finally no longer the case. Women have other priorities than just popping out a baby as soon as possible, and are consequently postponing their first pregnancy longer and longer (though until rejuvenation kicks in, there’s a limit to how long they can keep postponing without consequences.)
  • Contraception is more widespread, and people are more aware of it. Especially in developing countries, lack of effective contraception is the third parent of numerous unwanted children. As the use of contraceptive increases (even in the developing world, as it slowly becomes developed), fewer unwanted children are born.

The above are at least a few reasons why fertility is going down. Are they going to be enough to cut down our growth rate as much as we need to afford rejuvenation? Hard to say. However, there are other factors that are likely to kick in, as a consequence of rejuvenation itself. These factors may drive population growth down even if—as we all hope—our living standards were to grow sky high.

  • Rejuvenation is bound to change people’s attitude toward life. Presently, our life is tightly dictated by our biological clocks. There’s a time for school, one for work, one for having a family, and one to be sick and die. Even though we still are subject to this imposed rhythm, we’re starting to part ways with if we can—for example, think of people who go back to university in their 40’s. Rejuvenation would remove this imposition and make the boundaries between life stages meaningless, giving people virtually unlimited time to dedicate to their own personal growth and interests. In the circumstances, for many people reproduction might simply not be much of a priority any more. There would no longer be children born only out of their parents’ fear of their own limited fertile days.
  • Rejuvenation will allow for longer generations. In a post-rejuvenation world, you could have a family at any adult age, and would no longer be pressured to have one while you’re sufficiently young to keep up with it. (This is especially true of women, whose ability to give birth eventually disappears entirely.) This pressure lacking, people may well decide to explore life more in depth, become wiser and more experienced before rushing into parenthood. Other goals than parenthood may take priority, since in a post-rejuvenation world, parenthood can be postponed without consequences for as long as you like. This means that generations (the timespan between your birth and your children’s) could become much longer than they are now, thus slowing down the growth rate.
  • People who have already had children won’t necessarily have more. In a post-rejuvenation world, people who are, say, 150 years old or more and have already had children may simply want to be out of the baby-making business for a long while, during which they would not contribute to population growth. Some of these people might even never want to be a parent again. They can try parenthood, and if turns out it’s not for them, after they’ve borne with it for 20 years they’re not an old person with most of their life gone; they’re biologically young and healthy, with all the time in the world, and free to take up whatever new challenge they wish. On the other hand, if they did like parenthood, they have all eternity to do it again. This is one of the beauties of rejuvenation: Wrong choices don’t become life sentences; right choices can be made again and again.

(I might add, how many times would women be willing to undergo pregnancy? But anyway.)

Our long journey into the overpopulation objection ends with these considerations. As always, it is up to you to decide whether this objection is as valid as people think it is or not. Keep in mind that my projections and calculations are inevitably not as accurate as the UN’s, but they’re sufficiently accurate to illustrate the point. If you wish to investigate the matter further, I recommend the following sources:

Our World in Data
Worldometers
Gapminder
UN Population Division
Demographic Consequences of Defeating Aging (A SENS-funded study)

Whether I have convinced you or not, I hope you enjoyed the read, and that it may be of use for you.

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^ 1. In my old overpopulation article, I claimed, erroneously, that there was no faster growth than exponential growth, thus neglecting hyperbolic growth entirely. Shame on me, I guess… Go back up
^ 2. It may seem strange that 20=1, but there are valid mathematical reasons into which I won’t go because it is unnecessary to understand this article. Go back up
^ 3. This is true only if exclude things that kill people too fast, i.e. catastrophes: wars, pandemics, asteroid impacts, etc. These things kill people much faster than ageing, accidents, or non-pandemic diseases, and would not only prevent overpopulation, but could also decimate us. A bit of an overkill, if you pardon the pun. Go back up
^ 4. The formula

P(t) = P(0)er1 + r2 + … + rt

is easily explained. Say our starting year is 2015, with population P(0). One year after 2015, the population size is given by P(1)=P(0)er1·1 = P(0)er1. How do we compute the population 2 years after 2015? Notice that this is the same as the population 1 year after 2016: We can therefore take 2016 as the new starting year and P(1) as the new starting population, thus obtaining population P(2)=P(1 + 1)=P(1)er2. We can then replace our old expression for P(1) and obtain P(2)=P(0)er1er2=P(0)er1+r2, because of the laws of exponents. If we repeat the trick until year t, eventually we’ll get to the formula I wrote above. Go back up
^ 5. If you’re confused about why our starting population is P(0) and not P(2015), it’s simple. Our formula tells us the population after t years since a starting year have passed, and what we call that starting year doesn’t matter. We could very well rewrite our formula as P(t)=P(2015)er(t – 2015), with the restriction that t must be larger than or equal to 2015, and it would work just the same. However, in this formula t isn’t the number of years passed since the starting year 2015, but rather the actual year number of a year after 2015. Go back up

The overpopulation objection
Moral implications Space, environment, resources, jobs Population dynamics
leafOverpopulation answered on LEAF (See also Lack of resources on LEAF)
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