picture two exponential growth curves, one steeper than the other, over time the a trend moves from the steeper curve to the less steep curve. The "population" whatever it is, is still growing exponentially at the less steep curve.
I never said that it would continue to decline indefinitely. I said it was exponential before, and it's still exponential, even though the rate has changed. In spite of the massive negative response here, that statement is perfectly true.
> I never said that it would continue to decline indefinitely.
It's been continuously declining for half a century after peaking, the assumption that it's suddenly reached a bottom at the most recent measurement and returned to an exponential steady state is unwarranted.
No, but we are talking about a formula of the type where a growth rate G is a factor of a base population P, and P in Period n+1 = P(n) + G*P(n)
In this case G is the difference between the birth rate and the death rate for time period t. As long as a G is positive, you get exponential growth. Every time. This is how compound interest works. The fact that your balance next year is based on your starting balance + the balance multiplied by a positive growth rate ALWAYS leads to exponential growth in the balance.
> In this case G is the difference between the birth rate and the death rate for time period t
OK then. Let the birth rate be 1 person per second and death rate 0 (nobody dies). Do you get exponential growth? No. This is still linear growth.
Your another mistake is that ax^b is exponential growth, and ax^c is also exponential growth (if b and c are constants), but these are different processes. If you get a single process with changing exponent, this is no longer exponential growth, it can be arbitrarily defined.
Thank you. You finally made it clear exactly what you don't understand.
One person per second is linear.
.00001 person per year * current population is exponential, assuming only that the (fractional) person is added to the population factor so that the base is bigger next year.
If you don't understand that, you shouldn't be commenting on growth rates.
Indeed. However you also made it clear what you don't understand. Consider 1 person per year being born.
If you assume the growth rate is exponential, then you will see that the first year, the growth rate is 100%, the second year the growth rate is 50%, the third year the growth rate is 33% and so on. At any point, you can fit an exponential to the graph tangential to the current point. It won't fit much, but you can still do it.
A decreasing exponential may be exponential, but it also very well may not be. But we'll need to look a few years into the future to be sure. That said, expert opinion seems to be that growth rate is at this point linear (or iow an exponential that decreases each year so as to appear linear) [0]
No, a "decreasing exponential" is not exponential. It might return to an exponential after the decreasing period, but during the period of decline, the growth is not exponential.
The population growth rate isn't dependant on the base population. You can pretend it is by expressing it as a percentage, but that doesn't make it so.
Look at [0]. 2016 had a higher population than 2015, yet the growth was smaller. Not just a smaller percentage, but actually smaller by about half a million people. If growth was exponential, then the higher population would have a larger increase, but it doesn't, and so the growth isn't exponential.
Sigh.
The growth _rate_ is not dependent on the population, but the actual numeric growth _amount_ is. It is the _population_ that is growing exponentially.
You are describing the shift from a steep exponential growth curve to a lower exponential growth curve that I mentioned earlier.
If you get 10% compounded interest on your money it grows exponentially at 10%.
If you get 9% compounded interest it grows exponentially at 9%.
And if you _were_ getting 10% and the bank changes the rate to 9% your money is now growing exponentially at 9% interest, assuming only that the interest is reinvested.
Assume my bank gives me a linear interest of $10 per year.
Then going from $100 to $110 that would be 10%.
Going from $110 to $120 would be ~9%.
Going from $120 to $130 would be ~8%.
In any year you could conclude my savings grow exponentially. 10%! 9%! whoohoo! But if you look over a longer time frame, it clearly isn't.
And as nevdka mentioned, the population growth is less than linear. If my bank gave me $10 of interest last year and $9 this year, and $8 the next, there is definitely no exponential growth.
