Reproductive potential: One insect lays four eggs per day. These eggs take 30 days to

[beternal]

Ok so I am an entomologist (I work with insects) and am trying to work out reproductive potentials of different species. Is there a *one formula works for everything* that I can use? - or at the very least, can someone please help me with one example so I can see how it's done?

So here is the question:

One insect lays four eggs per day. These eggs take 30 days to grow from egg to adult, so after 30 days, we now have 5 adults (one original plus four babies), 31 days we have 9 adults etc. As these adults emerge, they too lay eggs (four each).
Assuming they all produce four eggs (no male needed), how many adults will there be each day from 0-90 days? - assuming all survive

I realise this is not a simple formula as after 30 days, a mass of new individuals arise, but how would I work it out?

Any help is appreciated!

Jon

 

[Ishuda]

Ok so I am an entomologist (I work with insects) and am trying to work out reproductive potentials of different species. Is there a *one formula works for everything* that I can use? - or at the very least, can someone please help me with one example so I can see how it's done?

So here is the question:

One insect lays four eggs per day. These eggs take 30 days to grow from egg to adult, so after 30 days, we now have 5 adults (one original plus four babies), 31 days we have 9 adults etc. As these adults emerge, they too lay eggs (four each).
Assuming they all produce four eggs (no male needed), how many adults will there be each day from 0-90 days? - assuming all survive

I realise this is not a simple formula as after 30 days, a mass of new individuals arise, but how would I work it out?

Any help is appreciated!

Jon

Generally, all of the 'population growth' problems fit an exponential formula of some sort. For example, if the population doubles every 30 days we would have
P = P₀ 2^d/30
where P₀ is the population at day zero and d is the number of days since day zero.

Your problem fits into this category but it is a facor of four instead and made a little more complicated by the delayed 'adulthood' and what happens in between. If the starting insect lays 4 eggs a day then at the end of the 30th day you have you have 121 insects, the starting one plus the 120 eggs laid by it. At the beginning of the 31th day you have 141 insects, 124 laid by the original plus 16 (4 each) laid by the first four the original laid plus the original. And so forth.
End of day 30: 30 * 4 + 1
End of day 60: 60 * 4 + 30 * 16 + 1
End of day 90: 90 * 4 + 60 * 16 + 30 * 64 + 1
...

Of course, if, as you say in your write up, the insects don't lay eggs on days 2-30 of their lives, the equation does become simpler.

Which is it?