I know that the probability of extinction for a simple birth and death process is given by

$p_{0}(t) = \bigg(\frac{\mu-\mu e^{(\mu-\lambda)t}}{\lambda-\mu e^{(\mu-\lambda)t}}\bigg)^{N_0}$

where $\mu$ is the per capita death rate, $\lambda$ is the per capita birth rate and $N_0$ is the initial population. Here no assumptions on the upper limit of the population size is made.

I have 2 questions:

1) Does $p_0(t)$ change when there is an upper limit of the population size (say $N$)? This means that when the population is $N$, a death occurs with a probability of $\frac{\mu}{\lambda+\mu}$ and nothing happens with a probability of $\frac{\lambda}{\lambda+\mu}$. Hence $N$ is a carrying capacity, forming a reflecting boundary condition on the process.

2) If the answer above is yes, is there an analytical expression for $p_0(t)$ accounting for $N$?


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