I need help at the following model. I dont really know if it makes any sense. Consider a system of $N$ individuals at $t=0$. I want logistic growth up to a carrying capacity $K = N + \sqrt{N}$, only birth of particles and no dead, born particles cant give birth, and not a deterministic differential equation. Further I assume that when a particle gives birth, it splits into two new particles. So what I did is the following:
I consider a stochastic process $N(t)$ with the following assumptions. First, the probability for a particle to split into two is of order $\nu dt$, where $\nu$ models the birth rate. Second, the probability that more than one particle splits in the time $dt$ shall be neglible. Third, descendants are not able to proliferate and fourth $\mathbb{P}(N(t)=N) = 1$ at $t=0$. We now make some specifications on the birth rate $\nu$ namely $\nu(t) \leq \frac{1}{t} ln(1+ \frac{1}{\sqrt{N}})$ for $t>0$ which ensures that $N(t) \leq N + \sqrt{N}$ for all $t>0$. Further $N(t) \in \{N,N+1,...,N+\sqrt{N}\}$ and $ \mathbb{E}[N(t)] = N e^{\nu t} \leq N + \sqrt{N}$ (Follows directly from negative binomial distribution).
So what I did is to model the logistic growth via choosing a time dependent birth rate $\nu$. Is this nonsense or does it make sense? I am really not sure.