Differential equations involving random variables First, consider a logistic decay model
$$\frac{dN}{dt} = -N(1-r+rN)$$
Where $r>0$. Depending on $r$, $N(t) \rightarrow 0$ or $N(t) \rightarrow 1-1/r$ are the possible behaviors. 
Second, consider a pure birth process with Poisson distribution $p_N(t)$ and birth rate $\lambda$: 
$$ p_N'(t) = (1-\lambda) p_{N}(t) + \lambda p_{N+1}(t)$$
The distribution function is
$$ p_N(t) = e^{-\lambda t} \frac{(\lambda t)^N}{N!}$$
Now my question: 
How can I mix these two concepts together? 
I want to model a population which is a logistic decay from the initial condition as the birth rate goes to zero, but it is a pure birth process as the death rate goes to zero. 
I'd like to obtain the probability distribution of a logistically decaying population with random birth. 
Any input is appreciated. Thanks!
 A: Recall the solution to the deterministic logistic equation:
$$N(t) = \frac{1}{1 + (\frac{1}{N_0} - 1)e^{-rt}}$$
for initial population size $N_0$. Because the birth rate $\lambda$ is (extraordinarily luckily) independent of the population size, a realization of this process will consist of chained intervals of these solutions for initial conditions $N_i$ over time intervals $W_i$ where the size of each time interval $W_i$ is distributed exponentially with parameter $\lambda$ and the initial conditions at each time interval $N_{i>0}$ are defined recursively as
$$N_{i+1} = \frac{1}{1 + (\frac{1}{N_i} - 1)e^{-rW_i}}+1$$
In this sense, the only source of randomness is $W_i$ which then feeds into the value of $N(t)$ through the definition of $N_{i+1}$. All of this is conditional on the fact that the logistic decay process "resets" at every birth event.
Finding information about the specific distribution this system follows is very difficult, since this is a continuous-time continuous-space system, but it can be easily found computationally.
Hope this helps!
