# calculus

A population of a very rare insect in an area in a discrete rural region in ASEAN will grow at a rate that is proportional to their current population. In the absence of any outside factors, the population will triple in two weeks’ time. On any given day there is a net migration (into the area) of 15 insects and there are also an extinction cause by identified factor such as 16 are eaten by the local bird population and 7 die of natural causes. If there are initially 100 insects in the area, predict either the population of the rare insect will survive or not? If not, when do they will die out?

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1. Hmmm. If the population is p, then we have

dp/dt = kp
dp/p = k dt
ln(p) = kt + c
p = c e^kt

Now we know that p triples in 14 days, so we can forgo using e, and just use 3 as our base:

p(t) = Po * 3^(t/14)

so, starting with 100, and gaining 15 per day and losing 23 each day, making a net loss of 8 per day.

p(t) = 100*3^(t/14) - 8t

so, take a look at the graph and decide:

http://www.wolframalpha.com/input/?i=100*3^%28t%2F14%29+-+8t

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2. 34

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