Difference between growth formulas

Question

What is the difference between $$N = N_0 \cdot e^{kt}$$ and

$$N= N_0(1+r)^n$$

I'm trying to find the best formula to calculate population growth and sources seem to vary between these two?

Answer 1

The first one is in a continuous time setup, the second one in a discrete time setup. Both with a constant growth rate.

Answer 2

They are equivalent.

Let $e^k = 1+r$, and the first becomes $N_0 (1+r)^t$ and the second becomes $N_0 e^{kn}$.

$n$ and $t$ just measure how far things have gone, with $n$ usually having integral values and $t$ usually having continuous values. If $t$ is measured at equally spaced intervals, then $t = a+bn$, so $N_0(1+r)^t =N_0(1+r)^{a+bn} =N_0 e^a((1+r)^b)^n =N_1 u^n $ where $N_1 = N_0 e^a$ and $u = (1+r)^b$.