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How do!

Never mind the why, as the why isn’t suited to Dakka. But I need help with the formula, and understanding the formula, for the following poser.

A very simple organism lives for only 10 seconds. But, it completes a cycle of cellular mytosis every two seconds.

Starting with a single example of said organism, how do you calculate its population and growth rate after say 60 seconds?

It’s not not simply “1 to the power of 29”, because that doesn’t take into account the death rate, which as the population grows will remain stable (life expectancy staying the same throughout), but will apply to increasing numbers of organisms from second 10, 12, 14, 16 etc, which will impact the (eewww) breeding stock, as once dead they can’t further contribute to the growth.

The first 9 seconds are easy. 1, 2, 4, 8, as we’re just doubling. But from second 10 it would be 15 (8x2, -1, the -1 being the first death), then 28 (15x2, -2, the second and third deaths)

Is the death rate the same as the birth rate, just ten seconds later? Can you take end population and subtract the population from ten seconds prior?


Automatically Appended Next Post:

 BobtheInquisitor wrote:
Is the death rate the same as the birth rate, just ten seconds later? Can you take end population and subtract the population from ten seconds prior?

Nevermind. It’s more complicated than that, but I need to drive. Will come back to it later.

I am in no state to do maths right now, i changed to a 0600-1400 shift today.

Calculate the death rate with respect to generational interval and use that as a negative value added to the spawn rate with respect to the generational interval.


Edit: and then rearrange with respect to t

Erm….wot?

Simple terms people, please 🤣

I do have my maths GCSE, but that was from 1996.

And this is why I took all those law classes.

It definitely isn't 1^29, as that's 1.

So you need a few values established first

t=time
l=lifespan
g=generational interval
n=number of organisms

You use them as dimensionless numbers so you can essentially have a formula which equals t (whatever t happens to be) which you can then rearrange so one of the other values is on the far side of the equals sign instead of t.

Edit: what i suggested above was to make two formulas with g on the far side and then merge the two together and then rearrange the new formula with t on the far side.

Having thought about it, I'd just cheat and use Excel (or Google Sheets).

One column for the interval, one column for the population.

Each step of interval represents 2 seconds.

At interval 0, population is 1.

For intervals 1-3, population is previous population times 2.

For intervals 4 onwards, population is (previous population * 2) - (population 4 generations ago)

Using that approach, at interval 30 (aka 60 seconds), the population is 117,897,840.

+ + +

What ends up looking interesting is how quickly, and substantially, this progression deviates from the powers of 2.

Up to generation 3, it's a 1:1 ratio. By generation 7 your population is 3/4 the equivalent power of 2. By generation 12, you're below half. By generation 20? Roughly 1/4. By generation 31, your population is just under 11% of the equivalent power of 2.

 Dysartes wrote:
It definitely isn't 1^29, as that's 1.

Not unless you’re Terence Howard.

Which I’m not.

Should be 2^29 + 1?

I mean, not the problem I’ve set. But if there was no loss across the 30 Generations.

I'm sorry, but what's the Terrence Howard reference referencing?

 Lathe Biosas wrote:


I'm sorry, but what's the Terrence Howard reference referencing?

"Terryology"

Part of which involves his 'logic language' in which he discussed 1x1=2

"How can it equal one?" he said. "If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what's the square root of two? Should be one, but we're told it's two, and that cannot be."

He also has some other impressive maths ideas, revealed for all the world to bask in on the Joe Rogan Experience a few months ago.

Hang on, I've realised a couple of errors in my Excel-based approach, but it ties to errors in the original expression of the problem.

At interval 0 (0 seconds), we have 1 organism.
At interval 1 (2 seconds), we have 2 organisms (1 Gen 0, 1 Gen 1)
At interval 2 (4 seconds), we have 4 organisms (1 Gen 0, 1 Gen 1, 2 Gen 2)
At interval 3 (6 seconds), we have 8 organisms (1 Gen 0, 1 Gen 1, 2 Gen 2, 4 Gen 3)
At interval 4 (8 seconds), we have 16 organisms (1 Gen 0, 1 Gen 1, 2 Gen 2, 4 Gen 3, 8 Gen 4)

We on see our first death at interval 5 - the 10 second mark - so the reductions kick in a step later. Easy fix.

Making that fix sees the population at 60 seconds jump to 417,825,921.

But there's another issue.

To quote Doc:
"But from second 10 it would be 15 (8x2, -1, the -1 being the first death), then 28 (15x2, -2, the second and third deaths)"

On the second step, you're not getting deaths 2 & 3 - there was only one organism in that second generation. The overall population was 2, but one of those two was your initial organism, which died a step earlier.

As I said earlier in this post:
At interval 4 (8 seconds), we have 16 organisms (1 Gen 0, 1 Gen 1, 2 Gen 2, 4 Gen 3, 8 Gen 4)

When we move to interval 5, at the 10 second mark, our Gen 0 organism dies without reproducing, meaning our population becomes: (1 Gen 1, 2 Gen 2, 4 Gen 3, 8 Gen 4, 15 Gen 5)
At interval 6, our single Gen 1 organism dies without reproducing, giving us (2 Gen 2, 4 Gen 3, 8 Gen 4, 15 Gen 5, 29 Gen 6)

What we need to be removing at each step from step 6 onward isn't the population a few steps earlier, but the population growth from a few steps earlier, as that growth represents the next generation. That next generation will die off at the same interval - sorry, Jean-Luc - but will be a much smaller figure than the total population at the same stage.

Taking that into account, I now make the end figure to be 415,962,592

+ + +

If we have any actual mathematicians kicking around, feel free to tell me where I'm wrong here.

Your Terryology reminds me of a science class deep dive where a guest lecturer showed us wave math where 1+1 is observable as being any number between, and including, 0 and 2.

...And then I started taking Family Law classes.

As a very much reformed biologist, I'm not sure cell division works that way. You don't keep the original cell plus a child cell, the cell literally divides in half forming two new cells. I'm not sure how you track the cell age for cell death - but I don't think you can look at 16 cells after 4 divisions and say "that one is the original one, that one dies now".

That might be a problem with the original posited scenario, then - I'm just trying to work with what I've been given in Doc's OP

I guess we could reformulate the problem by stating that we have a self replicating machine that performes 9 identical replications before self-destructing.

It is definitly an interesting problem.

First of: how to implement it in Excel:
the first 9 generations would simply be: =2^(n-1)
from there onwards every next item would be the item above *2 - the difference between the items 9 and 10 places before it.
=> so in the 10th generation the initial machine (generation 0) would be destroyed
=> in the 11th generation the machine that was new in the 1st generation (Gen 1-Gen0)
=> in the 12th generation the two machines that were new in the 2nd generation (Gen2-Gen1) etc.
(see attached pic)


Now, if we try to put this into a formula for the n-th element it gets a bit more complex. Looking at the numbers we can see that the first 9 generations are simply generative:
= 2^(n-1)

from the 10th to 19th we get a relatively simple "dying term" of 2^(n-10)*(1+0.5*(n-10))
so the formula for the 10th to 19th element would be:
= 2^(n-1) - (2^(n-10)*(1+0.5*(n-10))

But from 20 onwards the pattern shifts again and then every further 10 items it shifts again and I can't figure out the correct formula, even if I do see some patterns there.

On Terryology (no, really. That’s what he called it) I’ll leave a link below to Professor Dave eviscerating the nonsense.

Mild swears, so link only.

https://www.youtube.com/watch?v=lWAyfr3gxMA&t=714s

But thank you all for your input. Been a long time since I studied Maths, but this has been genuinely interesting. And I’ve enjoyed understanding the corrections being given!

There's definitely a reason I wasn't even going to try to formulate it for n, just figure out a way to work it inside a spreadsheet.

What did you end up with after 30 intervals, Pyroalchi? Curious as to whether your population figure lined up with mine.

Crispy78 wrote:
As a very much reformed biologist, I'm not sure cell division works that way. You don't keep the original cell plus a child cell, the cell literally divides in half forming two new cells. I'm not sure how you track the cell age for cell death - but I don't think you can look at 16 cells after 4 divisions and say "that one is the original one, that one dies now".

If it helps, it’s in response to a silly anti-evolution question as to “how am have liff when there are deff”, to demonstrate population expansion, and how provided a life form can reproduce multiple times across its own life, population growth is high, whilst trying to take into account that the earliest “life as we know it” would be been incredibly simple.

 Mad Doc Grotsnik wrote:

Crispy78 wrote:
As a very much reformed biologist, I'm not sure cell division works that way. You don't keep the original cell plus a child cell, the cell literally divides in half forming two new cells. I'm not sure how you track the cell age for cell death - but I don't think you can look at 16 cells after 4 divisions and say "that one is the original one, that one dies now".

If it helps, it’s in response to a silly anti-evolution question as to “how am have liff when there are deff”, to demonstrate population expansion, and how provided a life form can reproduce multiple times across its own life, population growth is high, whilst trying to take into account that the earliest “life as we know it” would be been incredibly simple.

In that case this entire endeavour seems way overblown. The simple, and intuitively easy to understand, answer to population expansion basically comes down to each organism birthing more than one child. In the case of sexual reproduction, each couple needs to produce more than two children for the population to grow overall. I know evolution deniers are idiots, but this doesn't seem in any way controversial or difficult to understand.

Or maybe it's the fact that, as we're discovering, the maths needed to figure out the exact numbers is difficult, so you can't just give them a quick answer. Even if you could, I suspect they wouldn't understand it. Even then, it's pretty obvious the growth will be exponential, without needing to know the exact growth rate. From what others have said, it looks like the key to the formula is figuring out a general rule to apply to each group of 10 generations to account for the deaths at that point.

The answer I proffered was “effing loads of the little B’s”.

But intrigued by the underlying maths, this thread came to be

Crispy78 wrote:
As a very much reformed biologist, I'm not sure cell division works that way. You don't keep the original cell plus a child cell, the cell literally divides in half forming two new cells. I'm not sure how you track the cell age for cell death - but I don't think you can look at 16 cells after 4 divisions and say "that one is the original one, that one dies now".

I think this is broadly the case. Certainly in the case of microorganisms, I don't think they have anything like programmed senescence in the same way as animal cells, where there are a limited number of divisions available per cell.

Instead individual cells will die for random reasons, which means cell lifespan will be on a distribution curve that is probably right-skewed (some cells will get lucky and survive really long, but no cell can live shorter than 0 seconds). You could use an average lifespan for calculations (mean or median probably both work) but probably you would get a more accurate result using the probability of death over time. I reckon you could probably plot this on a very pretty 3D graph but I sure as hell don't have the skill to do it. It was bad enough building a Cox hazards model.

Also this assumes population size and survival probability are independent, but in practice they are not. If the population gets too large it will begin to run out of resources and the death rate will increase until a more-or-less stable equilibrium is found (again, these tend to wax and wane rather than stay flat).

 Dysartes wrote:
There's definitely a reason I wasn't even going to try to formulate it for n, just figure out a way to work it inside a spreadsheet.

What did you end up with after 30 intervals, Pyroalchi? Curious as to whether your population figure lined up with mine.

After 30 Generations I end up at 1.062.230.271 living individuals.

@Pyroalchi: they start dying at generation 5, not 10.

Uh, sorry, in that case the formula given above is also wrong. My bad.
Just using Excel I come to 687.534.062 living individuals in generation 30 (given that generation 1 is the starting generation with one cell/replicating machine)