Population growth formulas

[Mister Internet]

Hey all,

Calling on someone in academia (probably means CM or Rourke) to answer me this: is there a simple formula that describes geometric population growth? I tried finding some stuff online, but it was mostly exams and quiz material, nothing terribly instructional. Basically, I was trying to explain to someone how one of my female roaches gives birth to ~30 nymphs at a time... and those nymphs take roughly 60-90 days to mature. They were like "wow, so how many would one female be responsible for in a year?" I tried figuring it out in my head, and very quickly realized that it involved math I haven't used since my Senior year of high school in Calculus. So, is there a simple formula I can plug "1 Female", "30 offspring at a time", "maturity in 60 days" and "n generations" and get a quick answer?

I was trying to figure it out in my head today for kicks, and didn't make it past generation 2... there are a LOT of branches in that tree, let me tell you.

As soon as someone gives me a formula, I'll turn it into a computer program so if I never need it again, I won't have to dirty my hands with actual math.

 

[Spaceman_Spiff]

Anybody please correct me if this is wrong (It's early in the morning)!

the growth rate (per capita) r = 30 individuals/ 60 days
With 50% females and the population size N the growth of the population is
N/2 * r

It s exponential population growth.

The other possibility is that I'm tired and just made an idiot out of myself

 

[Spaceman_Spiff]

Had some sleep and and googled a bit!

http://en.wikipedia.org/wiki/Exponential_growth#Technical_details

I hope this helps!

greets
Bernhard

 

[Malkavian]

Mr. I:

The formula spiff links to-- X(t) = Xo * e^(Kt)
is correct---to use this you need to know two things: your initial population (Xo = 1 for your example) and the growth constant (k)

The growth pattern (if i'm visualizing this properly) should be someting like X(n) = 30^(n) where n is whichever generation you're on. Start counting your generations at 0 btw (So the "top" level, is X(0) = 30^0 = 1 which is correct since you're starting with a lone female. X(1) = 30, she has 30 children, and so forth)

X(0) = 1
X(1) = 30
X(2) = 30^2 = 900
X(3) = 30^3 = 27000
X(4) = 30^4 = 810 000

Forgive the edits, my math apparently wasn't correct, tryin to fix it

 

[james]

Man you guys are funny!!!

I can count to five :? . Oh wait another hand make it ten . Typically roaches take closer to 90-120 days to mature and then once they become pregnant another 3-4 weeks. I have about 250 female dubia and they produce between 3,000-5,000 babies a month. Give or take. Imagine if I actually held that many back. :razz:
James

 

[Malkavian]

Gestation period, knew i was forgetting something.

Also, is it safe to assume 50% females?

 

[Mister Internet]

spiff, malkavian,

Alright, I'm with you so far... now how would one modify this to include the fact that only 15 of the 30 offspring are producers? Malkavian, your growth rate chart seems to assume a 100% female offsping... so instead of:

X(2) = 30^2 = 900

We would have more like:

X(2) = (30^2)/2 = 450

Is there a good way to incorporate this into the base equation, or is it best to just keep it as part of the growth pattern calculation?

 

[Mister Internet]

thanks james... so now we have more values:

(for simplicity's sake)
offspring per female: 20 (50% females)
maturity: 90 days
gestation: 30 days

this creates a generational "period" of 120 days for a generation to fully reproduce itself. This is only making things more complicated... aAUGUHGH

 

[james]

Numbers

I think your numbers are a little high. Most of females I see give birth have closer to 20 babies. Females average 15-30 babies. So you need to take that into account.
James

 

[james]

One more!!!

Not to complicate the matter, but some number of mortality is also a factor. Roaches molt seven times on their way to being adults, and sometimes right after they molt they are attacked by their buddies. This is another reason proper food, water, and even space are important. Most roaches will try and molt in isolated areas in the open away from the pack. The dubia have much less mortality than other species like the E. prosticus which are known cannibles. Toss a dead mouse or rat to orange heads and you will be amazed.
James

 

[Malkavian]

You're correct Mr I, and I hadnt edited the charts yet, sitting here playing with Matlab and trying to figure out what i'm not taking into account.

I think I figured it out though, just need to re calculate the growth per generation for a few generations, as you said. (and you're right btw, originally I had assumed 100% females and didnt think about it till james chimed in)

so,

X(0) = 1
X(1) = 30 (15m/15f)
X(2) = 30 * 15 = 450 (225m / 225 f)
X(3) = 30 * 225 = 6750 ((3375 each)
So that's a more reasonable factor of 15 growth.

Now, what's the average lifetime of a roach? Reason i ask is, becase for example, X(2) lays X(3), X(1) and (0) will have gone through another reproductive cycle and had MORE babies right? Or do they not live that long
In other words, how many times could you expect a female to go trhoguh a birth cycle


James: Thought about mortality as well, but frankly couple years ago in calc i did HORRIBLY on problems that introduced mortality into the mix, and I'm not sure how we'd estimate that resonably anyway.

 

[james]

Life span

Females typically live 18-24 months after reaching adulthood. This does vary greatly though depending on the species.
James

 

[Malkavian]

So for the sake of a "how many in one year" we could assume that they dont die off of old age?

 

[james]

lol

I would figure 5% to cover all basis.

 

[Mister Internet]

So for the sake of a "how many in one year" we could assume that they dont die off of old age?

Yes, let's... this is complicated enough already!!

j/k... obviously, if we were building an honest to God population model, mortality/attrition would be vital, but for the one year projection, let's ignore it. Pretty please?

 

[james]

Sorry man!!!

Didn't mean to be a pain. :wall: Just having some fun with these numbers. ;P At least where not trying to figure out my peppered roaches that take nine months to mature, females hang on to the ootheca for six months.
James

 

[Malkavian]

THink im gonna have to email my old calc teacher for this one fellas. I thought I had it covered until I remembered the previous generations would still be having babies, that threw my sums all off (i.e. not a consistent growth rate anymore)

 

[Mister Internet]

THink im gonna have to email my old calc teacher for this one fellas. I thought I had it covered until I remembered the previous generations would still be having babies, that threw my sums all off (i.e. not a consistent growth rate anymore)

That's the exact point for me that took it from "simple differential/exponential" to "I'm so lost" I was cooking right along with a simple exponential, but I realize that that required each successive generation birthing exactly one generation, and so on... a breeding age female will likely give birth again 30ish days after giving birth to a batch of nymphs. They all live in such close proximity, no reason to think they wouldn't be immediately knocked up again. :clap:

 

[Mister Internet]

Didn't mean to be a pain. :wall: Just having some fun with these numbers. ;P At least where not trying to figure out my peppered roaches that take nine months to mature, females hang on to the ootheca for six months.
James

Not a pain man... we can put it in in version 2.0.

Those tesselata have SERIOUSLY long wait times... you'd have to have colony of 5000 just to feed a couple BDs...

 

[Malkavian]

I have the feeling the issue i'm having has something to do with how i'm defining my time constant. I have the popultaion laid out in terms of a "generation" which i'm defining as 120days, so roughly 3/year. I'm wondering if i shouldnt be just defining the growth in terms of X generations rather than X generations equalling a year....