# Thread: Birth and death process

1. ## Birth and death process

Hello guys

I consider a linear growth birth and death process for a population with immigration, this leads me to define the growth parameter as $\displaystyle g = i + k_1\cdot n$ and death parameter as $\displaystyle d = k_2\cdot n$ for $\displaystyle i>0, \: k_1 < k_2$ and $\displaystyle n$ is the current population.
I then have no problem doing the first couple of questions regarding the mean, variance and long run distributions. But, I then need to find an expression for the long run time that the the size of the population is less than some number $\displaystyle N$, which stumbles me as it does not regard probabilities directly.

Kind regards

2. ## Re: Birth and death process

It looks obvious to me that n(t+1), population in the next "year" will be $\displaystyle n(t+1)= i+ k_1n- k_2n= i- (k_2- k_1)n(t)$. Since you say that $\displaystyle k_1< k_2$, the death rate is greater than the birth rate, $\displaystyle k_2- k_1$ is positive. That is, n(t) is i, the initial population minus something. The population is decreasing and will never be larger than the initial population.

3. ## Re: Birth and death process

Yeah, you are right should have realized that. However, as a side note if I sum the individual components of the limiting distribution, $\displaystyle \pi_i$, up to some number $\displaystyle N$
say $\displaystyle t = \sum_{i=0}^N \pi_i$ can I interpret that as the long run percentage of time spend in the states $\displaystyle i \leq N$?