A Poisson distribution models tells us the number of times an event occurs in an interval of time or space, but $\ \lambda\ $ is a constant, so this rate is fixed.
What if the rate changes with time or space, e.g. instead of $X \sim Po(10),\ $ what if we had, for example, $X \sim Po(\ln(t+1))),\ t>0,\ t \text{ is time in minutes since math.se website's launch?}\ $
This could represent many other real-world things, but for this example let $X$ be the random variable: "number of questions asked each hour on math.se (e.g. in the first few years of the website's launch)".
Is is known how to do calculations like finding $P(1300\leq X\leq1400) $ in the first week of the website's launch?
[Note the I am just making up the numbers of course, but I have tried to do it in a way so that the probability is not obviously close to $1$ or $0$.]
I think maybe we would need to do some sort of mean of a function or integral or double integral but I'm not sure what or how or if there is any literature on a $\ \lambda\ $ that varies.
If the varying parameter at time $t$ is $\lambda(t)$ then the number between times $t_1$ and $t_2$ will be distributed $\mathrm{Po}(\mu)$ where $\mu = \int\limits_{t_1}^{t_2} \lambda(t) \, dt$.
Then the probability that this number is from $a$ through to $b$ is then $\sum\limits_{n=a}^b e^{-\mu}\frac{\mu^n}{n!}$,
and if this calculation is too difficult then a normal approximation may be $\Phi\left(\dfrac{b+\frac12-\mu}{\sqrt{\mu}}\right)-\Phi\left(\dfrac{a-\frac12-\mu}{\sqrt{\mu}}\right)$ and you can drop the second term if $a=0$.
You will need to ensure consistency in the time units: your example mixes minutes and weeks.