Population growth formulas

[james]

Peppereds

Anything thats five bucks each is a bad choice for feeding. :8o There body mass is pretty incredable. Same goes for the pure Blaberus craniifer. They would make a good feeder, but at $4 each that would be a bad ideal. The discoids, orange spots, orange heads, and fusca are the feeder stock.
James

 

[Malkavian]

I fired off a plea for help to my old math instructor fro mthe comm college, keep you posted if he responds with a hint

 

[TroyMcClureOG82]

It's going to be different for different species. I am working on these in calculus right now. Basically you'll start out with a function that says the population of after a given amount of time is expressed by the function f(x)=(insert function here); (x) being your time inteval. To find the growth rate you take the derivative of that function that will be your inital growth at f(0) you can then insert whatever time you are looking for into the derivative of that function to find the percent the pop grows at that time. The exponential growth formula is primitive and won't work well in this case.

Short answer: your equation will be different for different species also depending on the age of individuals in the pop at the start

 

[Malkavian]

Here's what my teacher sent back. It's largely done in word's equation editor so you'll hafta have that installed to have it render properly

 

[Mister Internet]

That's very, very insightful... I see he actually accounted for the lone "Eve" female in the entire female population equation. Means a lot of adding "1". We could probably use the male formula*2 to get total population, and just disregard Eve. Hmmm... PHP is not a language I'm used to writing math in.

 

[Malkavian]

Once you have the actual math equation it's pretty simple... you just declare your variable $n = 0; and then its a matter of arranging parenthesis to get the equation to format up properly and use a loop to iterate through it however many times you want

 

[Mister Internet]

Anyway... what I get when disregarding Eve is that after 3 "generations", you get 40 total roaches. If we assume that a "generation" is 120 days (90 days to maturity and getting knocked up, 30 days gestation), and there are 3 times 120 days in a year, and each batch of nymphs is 20. however, let's assume she drops the babies on Jan 1, this would give us 4 "generations" or 840 total roaches. Generation 5 gives you 16,840, and generation 6 gives you 336,840. This *sounds* feasible, but obviously there are many other factors at work here.

Do these equations take into account that a female must take 90days to mature, but can give birth every 30 days after that?

 

[Malkavian]

Do these equations take into account that a female must take 90days to mature, but can give birth every 30 days after that?

Crap...no :8o I'm not sure i knew that was the case when i sent the email, so his equation assumes that all the elligable roaches are dropping at the same time the newest generation matures

 

[TroyMcClureOG82]

Another way to do it is if you can figure out the rate that the colony has been growing at then take the antiderivative of that you will have your growth formula

 

[siliconthoughts]

You guys are seriously math-impaired

Use a difference equation (best model for discrete generations)
Males don't matter in the equation (just multiply population of females by 2)
Run time in 30 day units for simplicity.

Population at time T+1 =
Population at time T-3 (90 days old) * female babies/ female +
Population at time T

Assuming we start with an adult virgin pair, and half the avg 20 offspring are female, the recurrence relation is
P(0) = 1, and P(-1) and P(-2) = 0
P(T+1) = P(T-3) * 10 + P(T)

Plugging in the numbers for the first year: (assuming no arithmetic errors)
P(0) = 1
P(1) = 10 + 1 = 11
P(2) = 10 + 11 = 21
P(3) = 10 + 21 = 31
P(4) = 110 + 31 = 141
P(5) = 210 + 141 = 351
P(6) = 310 + 351 = 661
P(7) = 1410 + 661 = 2071
P(8) = 3510 + 2071 = 5581
P(9) = 6610 + 5581 = 12191
P(10) = 20710 + 12191 = 32901
P(11) = 55810 + 32901 = 88711
P(12) = 121910 + 88711 = 210621

These are females. Total population is double this. Why these make good feeders should now be obvious.

You can model mortality by modifying the equation to be
P(T) = P(T-3) * 10 + P(T) * s
where s is the survival rate (i.e. 0.95 for 5% fatalities per month) In real populations both the birth and survival rates change with population size or density, which is why we aren't hip deep in roaches.

My quantitative ecology degree finally came in useful for something.

 

[MilkmanWes]

Interesting proof to this would be several colonies started with with single breeding pairs. Could find out that way if the formulas is accurately predicting the median population of all the colonies. But who wants to raise 2 million roaches?

Essentially this is a compounding intrest question. Population growth formulas and banking formulas are remarkably alike.

 

[JonDaAzn]

I'm sorry to ressurect this thread, but a quick and dirty way to estimate the roaches reproduction rate is just to weigh them, and then use weight as your variable rather than number of roaches, that way instead of reproduction rate, you just find the rate of the change of weight, you could convert this back into roach #'s by just finding the average weight of a roach (in all growth cycles), it wouldn't be totally accurate, but it could be close enough for some purposes

 

[Jakob]

a quick and dirty way to estimate the roaches reproduction rate is just to weigh them

Or use volume...haha, good luck with that one!

 

[reverendsterlin]

one thing to consider as well is the probability of 100% successive pregnancies every time. A reduction to about 50-75% high end reproduction or lower due to whatever reasons per female would seem more realistic to me.
Rev

 

[JonDaAzn]

discounting the added weight of the buildup of spoor, weighing the roaches wouldnt actually be that hard, in fact, as long as you removed the food/water, you could probably just sit the entire container on a scale, your population count wouldn't be correct, but you could find out the growth rate