[Guest]

A friend of mine posed this problem for which I am certain a mathematical function can be derived, but I have been unable to establish one.

On day 0, scientists discover a pair of aliens in stasis (nicknamed 'Adam' and 'Eve', since it is assumed that their purpose is to colonize the planet), one male and one female. After 100 days, the aliens emerge from their stasis at the beginning of their sexual maturity, which later study shows usually begins at 18 days. Not unlike cockroaches, once impregnated, female aliens will continue to produce offspring consistently for the duration of their ability to reproduce, which is determined to be 40 days. They produce one offspring at a time, either male or female, with an equal probability for each, and at a rate of one every two days. If necessary, a male can impregnate more than one female. Once an alien's age has reached 58 days, they are no longer able to impregnate females or bear offspring, and simply continue to live out the remainder of their lifespan, which is 2 years.

So, in short, on day 101, alien 'Eve' produces offspring 1.1, which may be a male or a female. On day 103, she will produce offspring 1.2, which has an equal chance of being male or female. On day 119, the first offspring reaches sexual maturity, and may begin contributing to the production of offspring (of course, if the offspring were male, he would need to wait until another female reached maturity).

The offspring appear extremely resilient, so there should be no need to factor in infant mortality. Assume that every offspring survives. If necessary, assume that the first offspring is male, and the gender alternates for each consecutive offspring.

How many aliens exist on earth for any given day D?

I appreciate your help, and hope that this is clear.

[Guest]

That's funny, I made an excel worksheet to find out how long it would take for Adam and Eve to reach 6.5 Billion offspring to determin if it were mathmaticaly possible in 6,000 years or so. Turns out you only need about 200 years given that no one ever dies and the women produce 1 child every year after reaching 16.

If I get a chance while at work I will try to modify it and give my best answer.

[Guest]

on Earth? if i see the history of human kind, there will be zero alien, maybe some scientist will keep few.

But its easy to see that the number of aliens is increasing extremly fast, so its a threat.

so the number of aliens at a random D day is zero with 99,999%. its sad...

Edited March 9, 2010 by det

[Guest]

That's funny, I made an excel worksheet to find out how long it would take for Adam and Eve to reach 6.5 Billion offspring to determin if it were mathmaticaly possible in 6,000 years or so. Turns out you only need about 200 years given that no one ever dies and the women produce 1 child every year after reaching 16.

If I get a chance while at work I will try to modify it and give my best answer.

I suspect that is along the same lines as my friend is thinking (which is why I added names the aliens 'Adam' and 'Eve', and because I thought it made the description more clear).

I still think it would be a fun mathematical exercise. But I'm just not that good.

[Guest]

Using a rule to favor a female birth in the event of an odd number of new babies (9 babies will be 5F and 4M) this is what I got. On day 245 I have the total alien population at 152,693- 76,360 are female and 76,333 are male. There should be 25,771 viable females able to reproduce.

I think I may have missed a birth or 2 along the way but this should be the answer +/- 50. Got to get back to work.

[Guest]

I cannot express the algorithm in mathematical language. I can't do it even with pseudocode. This is indeed complicated. Especially knowing, that four years contain 1461 days (Not 1460). I spent 2 hours solving this. I give up finding an algorithm. But, we can calculate the number of aliens in each day manually. I know this is long and "buggy", that's why I'll write a C# program for calculating the number of aliens. By knowing how many aliens we have in a certain day, we should be able to calculate dependencies between the day and number of aliens. I'll also publish the source code.

Just wait a bit (The program will require .NET Framework 2.0)

[Guest]

It looks like this takes a recursive formula, so although it isn't simple algebra, you should be able to calculate the change in a given day and then using some math (which I have long since chosen to forget) to find the final growth rate. You could also stick a few of the formulas into Excel and calculate that way. Some starter formulas (I think they are right) are in the spoiler. Happy to send a sheet if anyone desires.

B = Birthrate

Bf = Birthrate of females

Ff = Fertile Females

D = Day (excluding stasis)

DR = Deathrate

D’ = Day (including stasis) = D+82

Birthrate (expected)

B(D) = Ff(D)/2

Birthrate of Females (expected)

Bf(D) = (Ff(D)/2)/2

Bf(D) = B(D)/2

Virile Females (expected)

Ff + = Bf(D-20)

Ff - =Bf(D-40)

ΔFf(D) =Bf(D-20) – Bf(D-40)

ΔFf(D) =(Ff(D-20))/4 – (Ff(D-40))/2

Deathrate

DR(D) = B(D-720)

Population Change

ΔP(D) = B(D) – DR(D)

= 2*Bf(D) – B(D-720)

= 2*Bf(D) – 2*Bf(D-720)

= 1/2*Ff(D) – 1/2*Ff(D-720)

Integrate ΔFf(D) and ΔP(D) and you should find a mathematical answer, Otherwise, use Excel.

[Guest]

B = Birthrate

Bf = Birthrate of females

Ff = Fertile Females

D = Day (excluding stasis)

DR = Deathrate

D’ = Day (including stasis) = D+82

Birthrate (expected)

B(D) = Ff(D)/2

Birthrate of Females (expected)

Bf(D) = (Ff(D)/2)/2

Bf(D) = B(D)/2

Virile Females (expected)

Ff + = Bf(D-20)

Ff - =Bf(D-40)

ΔFf(D) =Bf(D-20) – Bf(D-40)

ΔFf(D) =(Ff(D-20))/4 – (Ff(D-40))/2

Deathrate

DR(D) = B(D-720)

Population Change

ΔP(D) = B(D) – DR(D)

= 2*Bf(D) – B(D-720)

= 2*Bf(D) – 2*Bf(D-720)

= 1/2*Ff(D) – 1/2*Ff(D-720)

Integrate ΔFf(D) and ΔP(D) and you should find a mathematical answer, Otherwise, use Excel.

After looking at all of that, I don't feel so bad about not being able to come up with a formula. I, too, considered writing a program that would take the number of days as input, then just cycle through the generations. I was hoping for something I could write out, and we could both understand, but it looks like this goes beyond that. Maybe I will do one in Flash so we can see the screen fill up with little people...

Is the problem greatly simplified with the assumption that during the 40 day period of fertility, each female produces 10 male and 10 female offspring? The matter would be far simpler still if we were calculating based on the generation instead of the day.