I need some help in creating a population growth function for a story.

Basically, we start with a population of around 35 million, with a birth rate of (on average) 0.04 children born a year to each couple. Which makes about 480k children born a year, we shall say.

But then (don't ask why) people stop dying. Or least, almost all people stop dying.

So, we have a birth rate of 600k/year (population growth accelerates instantly because no infant mortality) and no death rate. This will occur for about 300 years. After these 300 years, birth rates have dropped dramatically. To about 120k/year. Carry on for another 700 years at 120k/year.

Death rates are about 0.5% of the population/year.

My problem is that the people being born will then have children too, so an exponential sequence will be needed (I think) rather than a simple one.

Anyway, does anyone know how to create a sequence (or rather 2 sequences, one for the first 300 years and another for the next 700) to give me my current population, using the above information?

Edit: Some numbers adjusted.

 

Taure, you've already got "kids having kids" in your exponential growth model.

Written in terms of differentials, your model is

dP = P(t) r(t) dt

dP is the change in population over a small interval of time
dt is that time interval
P(t) is the current population
r(t) is the rate of population growth (which, in principle, is a function of time).

initially, r(t) = r_b - r_d, where both r_b (birth rate) and r_d (extinction rate) are constant.

r_b = 0.04 / 2 = 0.02 (2 percent growth from having kids). You divide by two because it takes two to procreate.

r_d = 0.005 (0.5% death rate)

Employing some elementary calculus, you can integrate the differential equation to get the population as a function of time. For instance, over the initial period, the population grows at a rate of 1.5%:

P(t) = P(t0) exp( (r_b - r_d) (t - t0) )

where t is in years and P(t0) is 35 million. If you want to turn off death, just set r_d = 0 in the above equation and fix the normalization constant P(t0) to be what the current population is at the time you start the new epoch of growth. For instance, you can let the growth run for 300 years, then adust the birth rate again, reset the constant out front, and run it forward some more.

The general case of a time-varying rate requires a quadrature.

[Edit: Had some time to plug in a number or two.]

First 300 years, starting with a population of 35 mil, no deaths, only births at 2%/year:

P(300) = 35e6 exp( 0.02 * 300 ) = 35e6 exp(6) = 14.1e9. 14 billion people at the end of 3 centuries.

I am assuming that by "120k/year", you mean a rate "120k/year per 35 million", i.e., a rate given in terms of the initial population. (If not, then births are negligible at 120k/year compared with 14 billion). This works out to 0.34% growth/year. If you turn on deaths at 0.5% of the population/year, then you have a net decrease in population of 0.16%/year:

P(t>=300 years) = 14.1e9 exp( -0.0016 * (t - 300) )

At the end of 900 years, the population is 14.1e9 * exp( -0.96 ) = 5.4e9, about 5.4 billion people.

 

Ah one of the reasons I love DLP. Batshit insane folks, gun nuts, geniuses, kids, gays, liberals, all different sorts.

 

5.4 billion... wow, glad I did this, because that's far too much for my purposes.

Hmm...gonna have to change something to get around that. Maybe some kind of one-child policy.

 

Okay, I decided that I wanted a population of 500 million and I'd work out the birth rate for the 1090 years that have passed based off of that.

Is my maths right?

P(1090) = 35e6 e^( x * 1090 ) = 500mil

e^(1090x) = 500mil/35mil = 100/7

ln [e^(1090x)] = ln [100/7]

1090x = ln [100/7]

x = (ln [100/7])/1090 = 0.00244 (3sf)

So birth rate of 0.002 babies a year per couple. That's one baby every 500 years per couple. I think what I'll say is that you need a permit to have children, and they're hard to get hold of.

 

Looks right to me.

 

How about the inevitable wars over food/space/energy?

There aren't enough resources on Earth to support anything like that population.

 

Hmm. The problem with that is that a part of the back history is that there's only been one war and that after it no one wants to war again.

I think I'll go with the "child permit" idea, along with upping the death rate a load (which means upping the price of immortality treatments).

The society I'm positing is pretty much a magical SF one, i.e. they effectively have SF kind of technology but its basis is magic, not science. But since the rate of progress was so quick, and most people don't die (thus enhancing the social conservatism movement) society hasn't really caught up with the level of technology they have access to yet. You'll have people flying about in one area and then a couple hundred miles away there'll be someone farming with a horse.