Formula for population growth

[sanabas]

What are you smoking? Fibonacci's model doesn't include mortality. Starting with a young (infertile) pair, we have:

One day I will learn to read, honest.

I don't know what fibonacci's model actually says, but I've done a model to demonstrate the fibonacci sequence to someone, and mortality was included. Start with one infertile pair, then we have:

1 young, has 1 kid
1 old, 1 young, they have 2 kids
1 dies, 1old, 2 young, 3 kids
1 dies, 2 old, 3 young, 5 kids
2 die, 3 old, 5 young, 8 kids
3 die, 5 old, 8 young, 13 kids
And they continue to breed like rabbitrs, and supply both rabbit stew and fibonacci numbers. And the growth rate is the golden ratio, which makes the rabbit stew taste even better.

 

[Erik Mesoy]

I haven't done matices math in 10 years, and so I must admit that I'll need some time to relearn them.

*blink blink*
"-in 10 years"? I'm supposed to start learning them in two years! I had no idea you were so old.

One day I will learn to read, honest.

Reading is always good for you. I read at 2000WPM and it lets me pick up large mountains of information in a very short time. Useful skill.

I don't know what fibonacci's model actually says, but I've done a model to demonstrate the fibonacci sequence to someone, and mortality was included. Start with one infertile pair, then we have:

1 young, has 1 kid
1 old, 1 young, they have 2 kids
1 dies, 1old, 2 young, 3 kids
1 dies, 2 old, 3 young, 5 kids
2 die, 3 old, 5 young, 8 kids
3 die, 5 old, 8 young, 13 kids
And they continue to breed like rabbitrs, and supply both rabbit stew and fibonacci numbers. And the growth rate is the golden ratio, which makes the rabbit stew taste even better.

Interesting. The model says:
F(0) = 0
F(1) = 1
F(n) = F(n-1) + F(n-2) for n>1

ie. the number of rabbits one month is equal to the number of rabbits last month (for obvious reasons) plus the number of rabbits you had two months ago (because all those were fertile and gave birth last month).

Another interesting sequence displaying the same property is (simplified) bee ancestry. Since unfertilized eggs grow into males and fertilized eggs grow into females, a male has one parent and a female has two parents.
N generations back, a male has F(N) parents.

 

[Masquerouge]

Fibonacci made up a somewhat similar model back in the 12th century.
He supposed a pair of rabbits would give birth to a new pair every month, which would start reproducing one month later. This results in the Fibonacci numbers 1,2,3,5,8,13,... respectively the Fibonacci series defined by x(n)=x(n-1) + x(n-2) [where n is the month and x(n) is the population in month n]. It lacks mortality though, so it is not really what you are searching for. But maybe a wiki on Fibonacci can help you digest that mustard seed.

Yup. I came too late, but Fibonacci's sequence is closely linked to pop growth.

 

[Masquerouge]

What are you smoking? Fibonacci's model doesn't include mortality. Starting with a young (infertile) pair, we have:

1 young pair = 1
1 old pair = 1
1 old pair, 1 young pair = 2
2 old pairs, 1 young pair = 3
3 old pairs, 2 young pairs = 5
5 old pairs, 3 young pairs = 8
8 old pairs, 5 young pairs = 13

1, 1, 2, 3, 5, 8, 13... That's the Fibonacci's sequence.

 

[Sidhe]

I haven't done matices math in 10 years, and so I must admit that I'll need some time to relearn them.

Thanks for the help, though. I figured there were enough experts here.

Just a piece of advice, when you do produce a model, make sure it actually matches something close to reality or how you would expect a human populaiton of Pn to grow in reality, if it doesn't it's useless obviously. There's no harm in knowing the answer before you get it, after all that's what statisticians do in order to produce future models. As you can see the current form of the equation ala the contributions so far need alot of work as the figures produced are nothing close to what would happen in 200 years if based on a human model. There are some false premises here anyway which would and do blur the results. I'm not sure exactly where the figures in the matrix comes from as their not explained, but assuming there based on something I can't see in the text, there's no population cut off so the figures just increase dramatically and unrealistically. This is called an exponential model and was discarded in the victorian era, it only tends to work for small population levels, at large ones as you can see it looks nuts

for example In 1900 the human population of planet Earth was 2.52 billion. In 2000 it was a little over 6 billion. The Equilibrium value, which you can work out from real world data but it's a tad complicated and involves simultaneous equations and a little more equation manipulation, and anyway I wont bore you is about 12 billion, make sure your model however it eventually looks feature all these criteria and you'll have a model that agrees with current models. If it doesn't then it's innacurate and a poor model. Good luck. I'm not sure how many models there are out their which start with a population of 200 so you may have to wing it a bit, small populations in both humans and animals grow differently than large ones anyway, which you can account for by alowing two different growth ratios(increasing the number of kids in your case or decreasing it for an age category) In your equations or more even.

 

[sanabas]

*blink blink*
"-in 10 years"? I'm supposed to start learning them in two years! I had no idea you were so old.

You're two years away from matrices? Either you're younger than I thought or you learn about matrices later than I thought. We first had them in yr 11, at age 15/16.

Reading is always good for you. I read at 2000WPM and it lets me pick up large mountains of information in a very short time. Useful skill.

So I've been told. I say I'm reading, but really I just look at the pictures.

Interesting. The model says:
F(0) = 0
F(1) = 1
F(n) = F(n-1) + F(n-2) for n>1

That bit I knew.

ie. the number of rabbits one month is equal to the number of rabbits last month (for obvious reasons) plus the number of rabbits you had two months ago (because all those were fertile and gave birth last month).

Whereas I explained it as the number of births is equal to the number of births last month, who are now fertile, plus the number of births two months ago, who are still fertile, with those more than two months old dead. Because a model based on immortal rabbits doesn't make much sense.

 

[Masquerouge]

for example In 1900 the human population of planet Earth was 2.52 billion. In 2000 it was a little over 6 billion. The Equilibrium value, which you can work out from real world data but it's a tad complicated and involves simultaneous equations and a little more equation manipulation, and anyway I wont bore you is about 12 billion, make sure your model however it eventually looks feature all these criteria and you'll have a model that agrees with current models. If it doesn't then it's innacurate and a poor model. Good luck. I'm not sure how many models there are out their which start with a population of 200 so you may have to wing it a bit, small populations in both humans and animals grow differently than large ones anyway, which you can account for by alowing two different growth ratios(increasing the number of kids in your case or decreasing it for an age category) In your equations or more even.

I think the Fibonacci model does not account for interbreeding, also.

 

[Narz]

I like graphs.

 

[Perfection]

I like graphs.

We know.

 

[El_Machinae]

Because a model based on immortal rabbits doesn't make much sense.

How about an immortal mouse?

Sidhe; don't worry about the modeling. This problem occured to me years ago when developing a theoretical scenario; but I've never been able to solve it.

 

[sanabas]

How about an immortal mouse?

I hate to be the one to break it to you, but Mickey's not real.

Sidhe; don't worry about the modeling. This problem occured to me years ago when developing a theoretical scenario; but I've never been able to solve it.

give us the scenario, with the complicated numbers, and I'm sure a spreadsheet or formula will appear within a day or so for you.

 

[El_Machinae]

Naw, I really don't need much more than Erik matrices filled into a formula. The scenario is pretty well what I described, I needed a formula to determine a population when breeding starts at 20% of the life expectancy and lasts for 45% of the expectancy.

I had no idea you were so old.

Ouch. My maturity and reasoning skills clearly leave something to be desired. OTOH, I remember Knight Rider and Air Wolf.

 

[cairo140]

Well, although your situation isn't the FIBONACCI sequence, it's very similar. For you Fibonacci afficionados (sp?), this is an excellent site which details Fibonacci and Phi:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Your desired "formula" is a sequence, and like the Fibonacci sequence, it is recursive (that is to say a partcular term of the sequence relies on at least one previous term in order to calculate its value).

Using some rough work, this is what I came up with.

Let n be year for which we would like to find the population.

Let t_n be the population (existing population+births-deaths)for that year.

Let t_1 be 4 (two females and two males - as far as I can see 4 is the minimum initial population we can use that doesn't end up in a needlessly complicated formula)

ASSUME that for every cohort of births, exactly half of it is male and half of it is female (again, this is an extremely necessary, although adventurous assumption)

Let m, a subset of n, be the range of integers (-infinity,0], and let t_m be 0. This is to say at n=1, 4 people spontaneously spawned (if someone wants to explore a way around this condition, it would be greatly appreciated).

Thus, for all n...

t_n = t_(n-1) + (sum{i=0 to 43} t_(n-i-20))/2 - t_(n-100)

I'm not on the computer with Maple (my math publishing software) right now, but I will be in the afternoon, and at that time, I'll rewrite it in normal math notation and upload it.

In the meantime, if someone could please check/revise my recursive series and conditions, it would be much appreciated.

By the way, what is this for? I doubt anyone would give a homework assignment as vague as this one, so is it just out of curiousity, or is there a hidden purpose?

In any case, time to eat lunch. Toodles!