How to calculate rate of future population decline with loss and gain parameters?

PistolSlap

I need help figuring out what I think is an algebra problem, but I don't know how to set it up.
I have a population starting at 1.6 billion. The annual growth rate is 2%. The annual mortality rate is 0.6%.

The problem is this: In addition to the above parameters, there is a constant loss of 52,560,000 people per year.

How many years would it take for the population to drop to 620,000,000?

(also could you please show how you did it?)

Thanks!

skeeter

population (in billions) as a function of time in years ...

$P(t) = 1.6(1.014)^t - 0.05256t$

$0.62 = 1.6(1.014)^t - 0.05256t$

$t \approx 45.5$ years (solved with a calculator)

note that after about 62 years, the population will begin increasing again ...

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DenisB

Hmmm...I get not quite 27 years.

160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
n = 26.968614....

Population at zero at end of 40th year.

skeeter

Hmmm...I get not quite 27 years.

160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
n = 26.968614....

Population at zero at end of 40th year.
there is a constant loss of 52,560,000 people per year.
?

DenisB

That's where we differ.
I'm assuming this "loss" reduces the effect of the 1.4% growth.
Like, I'm looking at it as "taking it to the Bank"!
A savings account earning 1.4% annually,
from which an annual withdrawal is made:
Code:
YEAR   WITHDRAWAL   INTEREST    BALANCE
0                             160,000
1     -5,256       2,240      156,984
.....
26     -5,256         986       66,193
27     -5,256         927       61,863
That's how I see it...probably wrong !
But I'll stick to my water pistols PistolSlap

Hey guys, I appreciate you tackling this problem! Let me give you a bit of background on where this is coming from, so you understand what I'm trying to accomplish.

I'm an author. I'm creating a science fiction universe and I have to have internal consistency while determining the rate of decline of the inhabitants of my extremely overpopulated nation-state as they go into the future with some very unethical measures of drastically culling the population. They exterminate a set number of individuals per year, with an aim to get down to a particular population number. I want to know how long it would take them, but I can't just take the current population, subtract the desired population, then divide that number by the constant number eliminated per year, because that doesn't take into account population gain and loss parameters of birth and mortality rates. Since these gains and losses are based on percentages, as the population changes each year by the constant extermination number, the resulting numbers of gains and losses must change dynamically as they reflect that updated total each and every year. Therefore, unless I can determine all three of these factors together, I can't get an accurate estimate on how long it will take my population to be culled to its desired size.

And of course the reason I am asking if you can show me how you did it is because I might have to do it again in the future, if aspects of the plot shift which affect the population of the state. (for example, to begin with, the nation occupied the whole continent of Australia = 8.5 million km^2, then I decided to devote 40% of that continent to wastelands, so the area dropped to 6.4 million km^2. To keep the population density proportionate, the entire starting population also has to drop by 40%. Therefore, I have to do all of these prospective population calculations again. And I'd like to know an equation I can just plug the numbers into, in the likely case that I have to shift things again.

I hope all that casts a little more light on the convoluted situation? Hahaha
Thanks! I really do appreciate the help. As is the case with many authors, math is not my strong suit. ;-)

skeeter

Hmmm...I get not quite 27 years.

160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
n = 26.968614....

Population at zero at end of 40th year.
did it again using the DE, $\dfrac{dP}{dt} = 0.014P - .05256$

my solution agrees (real close) to Denis ... t about 26.78 years

DenisB

1.6*exp(0.014*t) - 0.52*t
True nuff, Mr. Votan.
BUT how precise do we need to be with a population approaching 2 billion
and applying growth/decline factors that are guessed at to start with?

Geezzz...you'd need to end up with people not quite dead, plus a few
housewives pregnant for 4.123456789.... months As far as the "spitit of the problem goes", I see no problem
in rounding out (even to the closest 1000) ALL calculations.

PistolSlap

did it again using the DE, $\dfrac{dP}{dt} = 0.014P - .05256$

my solution agrees (real close) to Denis ... t about 26.78 years

Not sure the meaning of the dP/dt, equation, but it is agreed that this works?

160000(1.014^n) - 5256(1.014^n - 1) / .014 = 62000
n = 26.968614

So the answer being approximately 27 years. Thanks!
(Does the "n-1" in this case have anything to do with degrees of freedom, or is that another, unrelated n-1? )