I have a growth curve defined by a logistic term as: $$N_t = rN_0(1-\frac{N_0}{\kappa})$$ I would like to find when a specific value of N is reached. Let's say I have these parameters: $$\kappa = 10^{13}$$ $$r = 2$$ $$N_0 = 10^{4}$$ when would $10^6$ copies be reached? How many iteractions are exactly needed?
If I plug the values in I get: $$10^6 \not = 2 \cdot 10^4 \cdot (1-\frac{10^4}{10^{13}}) \not \sim 20,000 $$ What is the right way to solve it? If I run it with a for loop, I get:
t = 1, N = 20,000
t = 2, N = 40,000
t = 3, N = 80,000
t = 4, N = 160,000
t = 5, N = 320,000
t = 6, N = 640,000
t = 7, N = 1,280,000
Which means N = 1,000,000 is reached before interaction 7. But at what exact $t$?
Thank you