find the logistic function that satisfies the given conditions?

Question

Intitial value 10
Limit to growth 40
Passing through 1,20

Initial value 12
limit to growth 60
passing through 1,24

any help would be great
Im stumpped

Answer 1

The logistic function is of the form:

P(t) = K P exp(r*t) / { K + P [ exp(r*t) - 1] }

Here, P is the initial value and K is the limit to growth, so we have P = 10 and K = 40.

You can verify this by:

P(0) = K P / { K + P * 0 } = P.

Rewrite P(t) as KP / { P + (K-P) exp(-rt) }. Now take the limit of P(t) as t goes to infinity, and it is clear that this is K P / P = K.

Simplifying, we now have P(t) = 40 exp(rt) / {4 + [exp(rt) - 1] }.

P(1) = 40 exp(r) / [4 + exp(r) - 1] = 40 exp(r) / [ 3 + exp(r) ] = 24.

40 exp(r) = 72 + 24 exp(r)
16 exp(r) = 72
exp(r) = 9/2
Taking natural logs, r = log(9/2) = 1.50408.

The other problem is solved similarly.

Answer 2

You don't have enough data to solve the general logistic function and your mention of "growth" suggests you are really looking at the Verhulst equation, which is a variation:

http://en.wikipedia.org/wiki/Logistic_function

That equation says:

P(t) = (K P0 e^(rt)) / (K + P0 ((e^(rt) - 1))

where P0 is the initial population, K is the carrying capacity, and r is the rate of growth parameter.

You know P0, K, and P(1), so you only have to solve for r.

The easy way to do that is to solve for e^(rt) first, and then take the natural log. (A value of t = 0) makes this particularly easy.)