Extinction probability in a population with competition Suppose we have a colony of bacteria. At the end of each day, each bacterium produces an exact copy of itself with probability $p$ and then dies with probability $q$. However, $q$ is not constant, but a function of $N$, the total number of bacteria:
$$q=p\bigg(1-\frac{1}{N}\bigg)$$
So in larger populations of bacteria, each bacterium is more likely to die (because of competition, say).
To clarify, $N$ counts the number of bacteria before new ones were born. For instance, if there are $2$ bacteria on one day and they both reproduce to form $4$ bacteria, both of them still have exactly $p/2$ chance of dying (not $3p/4$). And the babies that have just been born cannot die immediately.
Let $P_N$ be the probability that a bacteria colony consisting of $N$ bacteria initially eventually goes extinct. Can we find an asymptotic formula for $P_N$? I suspect that we will have
$$P_N\sim \alpha^N$$
for some $\alpha$, but I don’t know how to calculate this constant.
I did manage to figure out that if we keep $q$ constant, then the probability of eventual extinction starting with $N$ bacteria is exactly equal to
$$\bigg(1-\frac{p-q}{p(1-q)}\bigg)^N$$
for $p>q$, and equal to $1$ for $p\le q$. But that problem was much easier because “newborn” bacteria were independent from their parents, whereas in this problem the chance of each bacterium’s survival is dependent on the overall population size.
So, really my question is: what is the value of
$$\lim_{N\to\infty}P_N^{1/N}=\space ?$$

(query on the exact terms of the problem, too long for a comment)
One bacterium, at time $t$ , with an extant population $n(t)$, has
 - $p$ , constant, probability to generate an additional bacterium, so that it contributes $+1$ to $n(t+1)$;
 - $q=p(1-1/n(t))$, depending on $n(t)$,  probability to die and contributing $-1$ to $n(t+1)$;
 - and consequently $r=1-p-q$ probability of just surviving and contribute $0$ to $n(t+1)$.
This is a classical [birth-death process][1] ,  which fundamentally is continuous in time, referring to live organism as bacteria.
The standard approach is to assume that in a small interval $\Delta t$ the probability of having more than one birth/death is negligible (higher order infinitesimal
wrt $\Delta t$).
In the post, the example of two bacteria that replicate in one day hints that the adopted discretization in time is not obeying to the above hypothesis.
Although it would be possible to reduce the time unit from a day to an hour, or even less,  so as to achieve that the hypothesis could be realistic,
it seems that the OP is considering the one day lapse as a sort of "juvenile quisciency" of no fertility and no mortality.
Is this the correct interpretation of the scheme being adopted ?
