I'm looking into the formula of growth, namely
$$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'.
What is the growth constant and how do I find it?
I'm looking at a bug that has on average 1,67 offspring each month and lives 5 years, the offspring becomes mature in 55 days and then has an average of 1,67 offspring each month itself.
How would I use this information with the growth formula to find out how many bugs we have in 1 year? Or is it out of its league?
On
Months, days, years: too many time units!
Let's say the bug has an average of $\alpha$ offspring per day from age $A$ days to age $B$ days. We can't describe the growth just by the number of bugs, we need the age distribution. Let's say $f(t)$ is the rate at which bugs are born at time $t$. The parents of these bugs were born between times $t-B$ and $t-A$. Thus $$ f(t) = \int_{t-B}^{t-A} \alpha f(s)\ ds $$
Now plug in $f(t) = C e^{kt}$, divide both sides by $C e^{kt}$, and simplify. You should get
$$ 1 = \alpha \dfrac{e^{-kA} - e^{-kB}}{ k} $$
This can't be solved in "closed form" for $k$, but you can use numerical methods.
Let $t = 0$ represent your starting point with $N(0)$. Well, if $N = N_0 e^{kt}$ is the right model for their growth then we would need $N(0) = N_0 e^{0}$, thus $N_0$ is your starting number of bugs. Now, choose a unit for $t$, for instance seconds. Then calculate the number of seconds in a month, $m$.
For any number $N_0$ of starting bugs we have $N(m) = (1 + 1.67)N_0$ So solve for $k$: $e^{km} = (1 + 1.67)$. I don't know if you can do this without considering more about the fact that $1.67$ is a monthly average.
$k$ controls how quickly the population grows as $\frac{d}{dt}N(t) = N_0 k e^{kt}$. Also at any given time $t$ we have that $\frac{N'(t)}{N(t)} = k$. So $k$ is a constant of the function. So define a constant of a countable collection of functions to be a real value $k$ such that if your functions are $f_1, f_2, \dots$, then there exists a rational function $R(t) \in \Bbb{R}[f_1(t), f_2(t), \dots]$, such that $R(t) = k$, and $R(t)$ viewed as a rational polynomial in the $f_i$ is not the constant polynomial $K(t) = k$. That might work.