# Predictions of Environmental Doom

Discussion in 'Earth Science' started by madanthonywayne, Apr 21, 2011.

1. ### iceauraValued Senior Member

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26,872
It is an error, regardless of what you mean by "current trends continue" (you have defended a couple of mutually exclusive possibilities - that the population is plateauing, that the second derivative is on average constant, etc. I'll let you choose).

For example, if you mean that the second derivative remains at or near its current negative value - as a constant, or average constant, or variation around the current mean, or whatever - your conclusion of a plateau is false. Your model would produce a crash - the doomsayer's vision.

A horizontal line, a plateau in a population graph, has a second derivative of zero - approached from above or below.

Or consider the example I posted above - the square root function. That could model the effect of a braking force of some kind, right? Graph it, and look at the second derivative. Compare.

Takehome: the possibility of a bust of the current boom cannot be dismissed, or even cast into doubt, by pointing to a negative second derivative in the growth function. The question of overshoot and permanent loss of carrying capacity - like rabbits on an island, or the fate of the Easter Island settlers, both of which undoubtedly exhibited negative second derivatives in their population growth functions as they approached the mass dieoff - is not answered by pointing to negative second derivatives.

Last edited: May 16, 2011

3. ### TrippyALEA IACTA ESTStaff Member

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Nonsense.
Again, you make the assertion that it is an error, without demonstrating such.

"The second derivative can remain negative while the population continues to increase, indefinitely."
The analogy that I raised demonstrates that this statement is blatently wrong. Fraggle Rocker pointed this out:
"No, not indefinitely. The closest you can come to that is to start with:
A positive first derivative
A second derivative that is slightly negative but rising toward zero asymptotically.
With the right initial values you could have a population that is increasing but at a slowing rate. It will rise indefinitely but only toward a maximum value it will never quite reach--the asymptote."
"eventually it has to reach a point where, because of the natural variation in the total human population, and the growth from year to year, the population growth becomes indistinguishable from zero growth. If it's not zero growth at that point, what, precisely, are you supposed to call it?"
Fraggle Rocker went on to make the following Assertion:
"His assertion, that population can continue to increase indefinitely when its second derivative is negative, is, therefore, falsified."
Even this scenario:
"It is also possible for it to avoid even that, and increase significantly forever with no asymptote - think the curve of the square root function: negative second derivative, no maximum value or asymptotic line. This is common in situations involving self-damping exponential growth, btw - exactly the situation we are considering."
Is addressed - because at some point the population growth must become so low that it becomes indistinguishable from a situation of zero population growth - which is precisely what I suggested in my comment on Fraggle Rockers post.

Oh, and incidentally neither of these things:
"you have defended a couple of mutually exclusive possibilities - that the population is plateauing, that the second derivative is on average constant, etc. I'll let you choose."
Are things that I have actually claimed.
I have not claimed that the global population is plateuing - only that it has in some countries.
I have not claimed that the second derivative is on average constant, only that if it were to be so then we must eventually reach a situation of zero growth.

As I have already explained to it was an analogy (not a model) and the problems you (and it is only you that is running into them) are running into, are only there because you insist on extending it past the bounds of the analogy.

"A negative second derivative, in itself, does not mean the population will plateau, approach a plateau, or even slow down it's growth very much."
Which is precisely what I demonstrated with my analogy - that a population that has a negative derivative WRT population growth must neccessarily slow it's growth. The only thing that mystifies me about this discussion is that you seem incapable of infering from the context that I may have been considering a braking force - a braking force produces a negative acceleration, however you will not get your car to go in reverse simply by keeping your foot pressed firmly against the brake.

I'm confused about nothing - unless you can provide an example of a form of y=mx+c where the second derivative has a none zero value.

Where, precisely, did I suggest otherwise? Oh right - I didn't.
The only thing I have claimed is that if the second derivative is even slightly zero then the population growth - the first derivative - must slow, and eventually reach zero. Consideration of what happens after it reaches zero is beyond the scope of that statement, and attempts to assert that it is in error on the basis of what happens outside the scope of the statement are fallicous at best. It's beyond the scope of the statement that I was addressing which was this:
"A negative second derivative, in itself, does not mean the population will ... slow down it's growth very much."
Because in order to discuss whether or not a negative second derivative is a sufficient condition for ZPG, it must first be established whether or not a negative second derivative is insufficient to even slow down the growth of the population. You have asserted that this condition alone is sufficient, where I have asserted that it must eventually arrest momentum and reduce growth.

A scenario which I discussed 17 days ago, on the second page of the thread, but which you seem to have missed, or skipped.

The question of whether or not it can happen in time to prevent an ecological catastrophe is a seperate question to whether or not it's actually happening, or going to happen ever, and not a question I have actually made any effort to address, either implicitly, or explicitly, save for the suggestion that with a paradigm shift, and some research into what at this point appear to be novel technologies, then the levels of global population which current models suggest will be plateud at may be sustainable.

Last edited: May 16, 2011

5. ### Fraggle RockerStaff Member

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* * * * NOTE FROM ANOTHER MODERATOR TO ICEAURA * * * *

Your bad science, flawed mathematics and unsound reasoning have crossed the line into trolling. Your assertions have been challenged, in accordance with the process of peer review which is a key component of the scientific method. You have failed to successfully rebut the challenges, so your assertions have been falsified. To continue to repeat them (in this thread or any other, on this subforum or any other) is, therefore, a violation of the scientific method, which in a place of science and scholarship constitutes intellectual dishonesty, the worst form of trolling.

To occasionally host a "debate" such as this is instructive to our younger members who need practical experience with the scientific method. However, we don't have the bandwidth to humor you and keep it going forever.

Please desist immediately or you will be given a vacation.

7. ### iceauraValued Senior Member

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26,872

This?
The statement is completely correct, not even subtly wrong. The second derivative can in fact remain negative while the population continues to increase forever.

Look, you guys are making a fairly simple mathematical error. Very common, ordinary functions exist that increase without bound, despite a continuously negative second derivative. If your chosen and fitted population growth curve is described by such a function (and they are the right kinds of functions, often, to fit small stretches of a population growth curve) it too will not plateau or even approach a plateau - ever - merely by following that function.

To plateau in such a case, something else must happen - current trends must change, and factors not currently dominant must become so. The second derivative must at least approach zero closely, and not remain other than infinitesimally negative.

The relevance is that many human population growth curves, with current negative second derivatives and otherwise well fitted to current data on human population, quite possibly yield increase far past the currently established (present technological) carrying capacity of the planet. They may in fact already have done so. The negative second derivative is therefore almost irrelevant in a discussion of predictions of environmental doom. If you want reassurance about environmental doom, you need a lot more than a negative second derivative in the population growth function.

Trying again: Trippy and you are wrong - mathematically in error - to claim that a negative second derivative, that criterion alone, promises a plateau in the population. More restrictions must be placed on the second derivative to allow that conclusion - and Trippy came up with one, after my comment: that the second derivative be constant, or as he elaborated after my further comment, constant on average, as well as negative. That works to produce a maximum, not a plateau, but we can overlook that if no conclusions depend on plateau - if the possibility of disastrous crash is not ruled out, say. If all you want is a maximum, and don't care about the height reached or the subsequent crash, then a constant negative second derivative will do.

A continuously negative second derivative does not necessarily produce a plateau, an approach to a horizontal line, in a function. This is simple, first year, single variable calculus.

Examples relevant to population growth curves: Logarithm (to any base). Fractional exponent or root functions (square root, cube root, etc).

I am sorry if you find this kind of technical quibbling to be bad science, but I assure you that it is perfectly sound, ordinary, simple mathematics. I find your handwaving produced "paradigm changes" that are assumed to preserve whatever features of the current setup you want to have ("wealth" reducing fertility by enough to offset lifespan) while discarding the ones you find inconvenient (the future effects of the new setup on the relations between "wealth" and survival/fertility are not known for sure), to be bad science fiction - along with the notion that everyone is going to become wealthy processing each other's data, or whatever you mean by automated food production and "information work".

And I deny that anything I have said here has been challenged in "accordance with a process of peer review". You are no peer, for starters, and simple error is not challenge.

Last edited: May 17, 2011
8. ### Me-Ki-GalBannedBanned

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4,634
Go read what the U,N, has to say about population growth . They are working on the problem now . It ain't what your Daddy Hippie told you either . The world will once again look up to the woman . Fraggle ! I see the future . Developers make it there job to do so. We gots to plan way out into the future . Risky business bro I must admit

9. ### iceauraValued Senior Member

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26,872
Been there, done that. What specifically did you have in mind?

Since I am apparently to be banned for giving calculus lessons to the careless, amy as well tuck in the corners:
The claim was that it has plateaued from the effects of prosperity, in some countries. I pointed out that it hasn't, and invited you to name a country or two. You then posted data on current fertility rates in some prosperous countries, which were below theoretical (current lifespan assumed, etc) replacement values. You did not name any prosperous countries with plateaued populations. There aren't any.
Not "reach", which implies a destination - pass through, in an instant. Followed immediately by increasing negative growth - a crash.

Fraggle was talking about a plateau, an approached horizontal asymptote, and you were agreeing with him. He was talking about it as a reality, claiming that it really existed as a property of current population dynamics, and you were agreeing with his statements. Your later amendment, the addition of the word "constant", was after you had agreed with Fraggles assertions; your elaboration of "average" constant was in response to my objecting to the plateau claim; your original claim about negative second derivatives was in agreement with his basic error.

Last edited: May 17, 2011
10. ### TrippyALEA IACTA ESTStaff Member

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Yes, I did - there were several (more than two) examples given in the passage that I quoted of countries that had declining populations - I suggest you go back and re-read it.

No it doesn't. In order to pass through a point, I must first reach it.

The only basic error is yours.
"The double derivative of population growth has already turned negative (the rate at which population growth is increasing) and has been that way for around 20 years. All but the worst case scenarios predict world population plateuing or decreasing by 2050."

From there I went on to say that:
"What I said was that the second derivative was negative. The first sign of a declining population."
And then explain, specifically, what that meant, and then make the point that in order for a plateu to occur "The only thing that would be required is for current trends to continue."
All of which predated Fraggle Rockers involvement in this thread.

I then, subsequently agreed with Fraggle Rockers assertion, that because population was a discretized variable, even in a situation where the trend line approaches some asymptote, pragmatically speaking, it must eventually reach the point, due to interannual variability that it is indistinquishable from a situation of zero growth, that is to say, there is no conceivable test or measurement that could be made that could determine whether growth was continuing, or had plateued - even if we treat population as a continuous variable. This is a point which you have repeatedly failed to address.

Now, because I have a few moments of free time on my hands, I have prepared the following graphs to illustrate that I am, in fact, correct.

You will no doubt remember this graph, which I posted earlier in this thread, which displays the world population (blue) the first derivative (brown) and the second derivative (green).

Subsequently, I have prepared this graph:

This graph reflects what will happen if "Current trends continue" - the claim I originally made. I derived a linear least squares fit for the second derivative (R[sup]2[/sup]=0.85) and projected that trend into the future. It predicts that if current trends continue, then by 2120 the world population will have plateued at 20.9 billion people, and begun to decline.

Once again, allow me to reiterate - The ONLY assumption I have made in prodcuing this graph is that the current trends continue into the future.

I have also prepared this graph:

This graph reflects what will happen if the "Second derivative remains constant" - I'm not sure, precisely, where this claim has come from, except I think on one occasion where I said "even if..." as a contrast to the statement "If current trends continue". This scenario predicts that the world population will plateu at 19.2 billion people in 2330.

Either way, my assertions are supported.

Incidentally - it occurs to me to point out the fact that, originally, stating that it was a sufficient condition was a strawman hypothesis invented by you. That isn't the claim I originally made.

11. ### TrippyALEA IACTA ESTStaff Member

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Now that I'm at home, and have some more spare time - here's a third scenario.
This scenario models a situation where the second derivate has already acheived its global minimum, and behaves in a way that oscillates about an average value that asymptotically approaches zero (from below).

It still plateus and begins to decline, except this time it plateus in the year 3020 with a global population of 37.9 Billion people.

12. ### iceauraValued Senior Member

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26,872
There was one candidate - Japan - and it has some problems, including an idiosyncratic treatment of non-Japanese residents, and the possibly temporary nature of the hiatus in growth (Japan's population increased in 2009, and shows signs of imitating Germany's rebound of the 70s).

Adoucette named more - three - and they make interesting examples: looking at Germany, for example, we see two bumps in a curve of overall increase. http://upload.wikimedia.org/wikipedia/commons/2/22/Germany_demography.png If you want to assume the second hilltop there is a plateau, and the population has plateaued this time instead of merely pausing in its overall rise, no one can say no at the moment - but on what grounds?

We also note that the correlation between prosperity measured as income and population growth is occasionally positive, in Germany: during recessions and the like, pop growth apparently slows.

And the replies went like this:

and this:
Now these quotes keep talking about constant - elaborated elsewhere as "average constant", for some reason - negative second derivative, and referring to a plateau that it promises.

And this, chewing over a point with Fraggle:
You are clearly not talking about passing through a zero, but plateauing - one person per century. And you are discussing Fraggle's scenario - the one in which a negative second derivative necessarily plateaus.

So this is fine, but still without mechanism ("prosperity" is not a mechanism, especially if wealth is to be redefined shortly):
Which brings us around to an actual issue: Since the predictions of environmental doom have had a reasonable record of success - compared with most speculation re future, anyway - can the population of the globe, or any part of it, avoid disaster without significant reduction in its future population growth, or possibly even current population? If so how?

13. ### TrippyALEA IACTA ESTStaff Member

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There were three candidates in the section of text I quoted, Japan was one of them, but there were two others.

What Adoucette did or didn't do is irrelevant to anything I've said. I didn't reference the same countries as he did, nor did I reference his post.

Is this the best you can muster? Outright dishonesty?

There are, occasionally, exceptions to the rule (for example, Israel) however, for the most part, those exceptions have a reason for being exceptions.

And I seem to recall throwing around two other points which can act against, and in some cases counter sub replacement fertility, now, what were they again - oh, I remember, population momentum and immigration.

How many times have I pointed out the baby boomers now?

And the replies went like this:

Oh good lord. You're wrong, get over it already - I've already proven that my point holds in all three cases - the trend in the second derivative continues, the second derivative remains at some average value, and the second derivative acheives a global minimum and asymptoticaly approaches zero from below.

That would be because, I'm discussing your scenario.

:Sigh:
The link between prosperity and birthrate is an emperical observation, I linked to a wikipedia page earlier in the thread which discusses several mechanisms that explain the relationship.

14. ### iceauraValued Senior Member

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26,872
Japan was the only prosperous one, and the only even possibly plateaued one.
That would be my point, yes. Actual examples of possible prosperity plateaus have little to do with your posts here.
That would be numerous, not occasional, exceptions. And the existence of reasons for the rule not being reliable in many cases are assumed by us all.
You need to make up your mind which of those three mutually exclusive possibilities you want to bet on, in a discussion of the contributions of population increase to environmental doom.

And whatever your point was is long lost - if those three possibilities all illustrate it, it isn't anything you've graced us with yet. Your claim that the population has plateaued in some countries due to prosperity, for example, remains unsupported by any of those graphs - they all just look like pieces of Germany's population graph, linked by Adoucette and me above. The matter of such a plateau answering the concerns of the doomsayers - which would be the sole relevance here, and my central issue - is not even addressed by you.

Last edited: May 17, 2011
15. ### TrippyALEA IACTA ESTStaff Member

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10,890
:Yawn:
:LOL:
So you're saying you bought up an irrelevant point to try and counter something I said?

I'm not so sure about numerous, but certainly well characterized - for example, ecxept for a very small minority, all of the exceptions to the rule have populations less than 5 million, and of those that aren't, two of them are theocracies, and one of them I would expect that it's a small mnority holding all of the wealth 'artificially' raising the GDP - oh, and the US shows up as the outlier, a country that has a strong religous sect, a strong growth from immigration, and then there's the baby boomers.

LOL

It is, it's just been buried in page after page of your irrelvancies, nitpicking and ever finer pedantism - your 'argument of the gaps'.

Two assertions, both bullshit.

16. ### iceauraValued Senior Member

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Uh, no, Adoucette's contribution was relevant - to the thread, my issues and posts, and the argument. It was a direct and cogent response to one of my posts, which he apparently comprehended.
And China. And Germany. But then you were including the Ukraine, so all your assessments are suspect.

But we still don't have a firm commitment from you about what your rule is. At least two thread-relevant and not immediately (mathematically) invalid possibilities exist, from your posts agreeing with Fraggles calculus error: prosperity creates a negative second derivative in the population growth curve. Prosperity will, from its effects alone, cause the population of the prosperous area to plateau.

These are not the same rule, restated - that's one of my observations. The existence of a theoretical plateau some day (if some unspecified "current trends" continue) does not address the concerns of the environmental doomsayers - that's another. You need mechanism, and cannot simply invoke an extrapolation of current Western industrial wealth creation - a third.

Last edited: May 17, 2011
17. ### TrippyALEA IACTA ESTStaff Member

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No, China and Germany fall broadly within the curve and as far as I'm aware don't register as outliers.

1. You have demonstrated no calculus error on Fraggles part.
2. I have demonstrated that in all three scenarios the population will will plateu, and then begin to decline. In all three of the thread relevant scenarios, the population will eventually plateu, and begion declining. The only way that outcome can be avoided is if the second derivative becomes zero, or changes its sign.

LOL
The trends are not unspecified - and the plateu exists in all three cases, irrespectiuve of whether or not current trends continue. The only way to avoid a plateu followed by decline is if the second derivative undergoes a further sign change.

18. ### TrippyALEA IACTA ESTStaff Member

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Oh and just an FYI, it's 'Ukraine', not 'THE Ukraine'. To the best of my... Recollection 'THE Ukraine' is an American invention, and in my experience, the quickest way to upset a Ukrainian is to talk about 'THE Ukraine' instead of 'Ukraine' when referring to their country.

My mistake, I think I just remembered that it was a (ethnic) Russian invention to marginalize Ukraine and Ukrainians - the reason why they might wish to do so escaped me until I discovered researching my family history that the Russ empire was originally found by Vikings from Kiev.

Last edited: May 17, 2011
19. ### TrippyALEA IACTA ESTStaff Member

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Here's a point you have utterly failed to consider, and it amazes me that you have failed to consider it - to the point where I find it difficult to believe that you might have inadvertantly failed to consider it.

Keep in mind one of the points that I have made, repeatedly, is that imigration can be a key factor 'bouying up' population growth in spite of sub replacement fertitilty.

1985, the year the growth resumes (approximately) saw the signing of the original Schengen agreement, between Germany, France, Belgium, Luxembourg, and the Netherlands which was fully implemented in these countries by 1995. 1985 also saw the beginning of Gorbachev introducing Perestroika, 1987 saw Regan give his famous "Bring this wall down Mr Gorbachev" speech, and finally 1989 saw the destruction of the berlin wall. All of which co-incides with the renewed growth, and all of which can be expected to cause an increase in immigration to West and/or East Germany.

To prove my point, and additionally support Adoucette's point, among other things, this graph:

Is a representation of the demographics of the german population since 1900.

The purple line is the natural change of the population. It represents live births - deaths. Note the effects the WWI and WWII had on the line. Also note that in 1960, the year the contraceptive pill was introduced,, and during the '60s - the sexual revolution, there is a minor peak in growth, which seems to be more attributable to momentum than any other single thing, there is a sharp decline, followed by a period since 1972 - 39 years, 2.5 traditional generations (16 years), where, although some variation has occured, the natural change in german population has remained negative.

The pinkish coloured line that has no title is what happens if we start with the German population in 1900, and do nothing more than add each years net change to the population. of the previous year. It excludes population changes due to migration, and population changes due to changes in territory.

And if you're still struggling to follow what I'm saying, according to the Federal Statistics Office of Germany, the immigration curve looks like this:

Which proves that the observed population growth in Germany between 1987 and today is as a result of Immigration, rather than natural population growth.

And given that Immigrants have effectively been counted twice, all of this completely supports Adoucette's assertion that the population of Germany has platued, or is in decline as a result of prosperity.

20. ### iceauraValued Senior Member

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26,872
So by "current trends continue" part of what you mean involves including all the effects of immigration on fertility, prosperity, etc, but not including the immigration itself.

And ignoring the emigration entirely - not including the source population in the graph at all - or the effects of emigration on its prosperity. It too is to be isolated, and arbitrary exclusions made?
Which brings up the key issue of mechanism, and what exactly the current trends are that you are counting on to magically plateau the population of interest (global or local) short of the unknown disaster level.

Because if you simply look at the population graph of Germany, and wave your hands at the second derivative and say "current trends continue" the way you are handling this, it will make a big difference to your projections exactly where - what year - on that graph you do that.
I provided counterexamples of functions that have continuously negative second derivatives, are relevant to population growth models, and do not plateau at all. They contradict not only Fraggle's assertions, but several of your own.
Plateau and decline are mutually exclusive. Accelerating decline - crash - is the doomsayer's prediction. You need mechanism, to differentiate your projections of the future of the current negative second derivatives from the doomsayer's projections, which fit the current data and curves just as well.
Germany's population has not plateaued, and is trending upward with prosperity - no plateau is visible, unless particular mechanisms are postulated. The population of China has several times this century been reduced in its rate of growth without increases in prosperity, and the present rate reduction predated the very recent prosperity increase - an effect even more marked if geographic areas of China are considered separately.

Btw: are we including female infanticide and selective abortion among the effects of prosperity - projecting their increasing influence and continuance until 2050?

? They look like pieces of the Germany curve - and Germany clearly has no established plateau. The three cases each involve their own particular assumed "trends" assumed to "continue" (quite restrictive ones, actually - unrealistic) if they are to be taken as modeling population growth.

21. ### adoucetteCaca OccursValued Senior Member

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Last edited: May 18, 2011
22. ### TrippyALEA IACTA ESTStaff Member

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Both of these points have already been addressed by a couple of points that have already been made in this thread, including the portion of the wikipedia article.

The first point is that it has already been explicitly aknowledged in this thread that immigration can force a country into growth that would otherwise be in decline.

The second point is that you're ignoring the factors which seem to have driven the growth of immigration in the first place.
1. Perestroika - Perestroika was an attempt to stimulate the economies of the USSR and the eastern bloc countries. East Germany, already an attractive destination from the perspective of someone living in (for example) the Balkan states, would have become a more attractive destination.
2. The Schengen Agreement - The Schengen Agreement was an arrangement between (initially) 5 countries (already named) that relaxed border controls virtually to the point of non-existence.
3. The Collapse of the Berlin Wall.

You've failed to consider naturalisation.

But there's a fourth point that I haven't mentioned, and you've failed to consider - the German Constitution, up until 1993 allowed pretty much anybody to seek political asylum, in 1993 the criteria for political asylum were restricted. 1993, coincidentally, is the year that we see the german population graph, and the foreign population graph plateu for a second time.

No hand waving required on my part.

You have given two, and suggested that a third might exist.
One of these I have addressed by making the point that at some point the population curve reaches a state where it's indistinguishable from ZPG because of the noise of interannual variability.
The second of these was so vague and waffly as to be virtually unusuable.

But, once again, we come back to one of my subsequent points - I have proven, in this thread, that in any of the scenarios that have been discussed - The second derivative continues on it's current trend, and becomes increasingly negative at a constant rate. The second derivative stays negative, and maintains a constant level. The second derivative undergoes an inflexion and begins increasing aysmptoticaly towards zero. That in these three cases, the behaviour is the same - the population eventually plateus, and then begins to decline.

Sure, but only in that they can't happen at the same time, just like growth and plateu are mutually exclusive. All three, however, can occur on the same graph.

So you keep saying, but I don't see anyone here actually arguing that a crash is imminent.

No I don't, the first thing that I need is for you to realize that they're illustrative analogs being used to illustrate a specific point, rather than predictive models being used to neccessarily fortell the future of humankind.

I have demonstrated otherwise - that the natural change in population of Germany is one of decline, and that the population rise that you keep referring to is being driven by immigration, however, the foreign population of Germany also seems to be leveling of (according to the Federal Department of Statistics), and this appears to be reflected in the average population graph.

Wrong curve. Again, go back and look at the link I provided you.

Of course the look like pieces of the Germany curve - however, the first two plateu at around 19 Billion, the second one plateus around 40 billion, but again, I forget how long for. It might have been more obvious if I had included some form of model for interannual variability, but eh, I thought that would only muddy things. All three graphs have a point of inflection. All three graphs plateu around the point of inflexion. How wide the plateu is, is a point that might be argued, however, you'd be hard pressed to call it less than twenty years.

23. ### TrippyALEA IACTA ESTStaff Member

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First I'm going to demonstrate your point.
Then I'm going to demonstrate why - for at least one class of solutions it's invalid.

But first, I'm going to address another, salient point.

I originally stated that it was the first sign, not that it was the only sign.

In responde to your objection, Fraggle Rocker posited the following:

Which mentioned two criteria.

You then replied with this:
Which ignores one of the criteria posited by Fraggle Rocker.

The function $y=x-\frac{1}{x}$
Has the first derivative of $1+\frac{1}{x^{2}}$ which is positive,
And has the second derivative of $\frac{-2}{x^{3}}$ which is negative for x>0.
Which, at first glance, appears to meet all criteria - certainly it meets both of the criteria described by Fraggle Rocker.

However, there are two features of the data which it fails to model.

The first derivative can not undergo a sign change. The first derivative of this function is positive for all values of X, and so it fails to meet the requirement of allowing a sign change - as has been observed in the recorded data for individual countries.

The second feature is that while the second derivative is capable of undergoing a sign change, as has been observed in the real world data, it is discontinuous around that sign change - both the first derivative, and the second derivative are discontinuous about X=0, as is the original function. And sure, whilst it might be possible to arrange the equations just so by changing the constants in the original equation, I'm going to leave it up to you to prove that, because when push comes to shove - you've also said this:

So then it seems that we've come to the point where something that is "just a technical quibble, of little bearing on the reality" is "relevant to population growth models" because they "do not plateau at all" even though "no one presumes an indefinitely increasing population."

Last edited: May 18, 2011