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Thread: Birth and death process

  1. #1
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    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
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  2. #2
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    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.
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  3. #3
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    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 $\displaystyle i \leq N$?
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