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Discrete Time Birth and Death Process

  1. Sep 24, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider a discrete time birth and death process in which the maximal population
    size is N = 6. Birth rates and death rates are directly proportional to the current
    size Xt of the population at time t (t = 0; 1; 2; : : :). If the maximal population size
    is reached, no more births can take place. That is bi = bi (i = 0; 1; : : : ; 5); b6 = 0
    di = di (i = 0; 1; : : : ; 6) At each time step, either one birth or one death or neither (but never both) will take place.

    Suppose the population starts with size 3 (i.e., X0 = 3). Let b = d = 0.02.
    Write down the transition probability matrix of the Markov chain.

    2. Relevant equations
    All probabilities in the matrix must be 0<=x<=1 and must sum up to 1 across the rows.
    Possible states of the birth and death process are the possible sizes of the population (current state = current population size).
    Also, the growth rate of the population through births is proportional to the population size with proportionality constant λ as long as the population remains below N (here N=6). The death rate is also proportional to the size of the population with proportionality constant μ as long as the population size remains strictly positive. That is bi = λi and di=μi for λ,μ >0 and 0<I<N

    3. The attempt at a solution

    I'm having a problem setting this up. It seems to me that if the population starts with 3, then the first 3 rows should all be 0 since if the population starts at 3, then the first 3 states shouldn't really matter initally. Here is what I have come up with.
    0 1 2 3 4 5 6
    0 0 0 0 0 0 0 0
    1 0 0 0 0 0 0 0
    2 0 0 0 0 0 0 0
    3 0 0 .06 .88 .06 0 0
    4 0 0 0 .08 .84 .08 0
    5 0 0 0 0 .10 .80 .10
    6 0 0 0 0 0 .12 .88

    For states 3-6, I have multiplied the current state * the birth or death rate to get the probability of going to the next or previous state.

    Please help.
     
    Last edited: Sep 24, 2013
  2. jcsd
  3. Sep 24, 2013 #2

    Ray Vickson

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    Have you really never seen one-step transition matrices before? If you had, you would know that the initial state is irrelevant: the matrix gives the one-step probability for going from state i to state j for any i,j pair. After you have obtained the matrix then---and only then---can you use it to compute subsequent state-probability vectors through time. After all, you want to know what happens if you start from state 1 or from state 4 or from state 6 or from state 2, etc. Starting from state 3 was just one of six possible examples.
     
  4. Sep 24, 2013 #3
    Ray, so are you suggesting that the birth rate and death rate does not change depending on the state I am in? You're saying that b=d=.02 for every state except for 0 and 6 where in 0 we can't have any deaths and in 6 we can't have any births?
     
  5. Sep 24, 2013 #4

    Ray Vickson

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    No, I am not suggesting that. The question stated very clearly that the birth-death probabilities are proportional to the state; that is, when in state i the probabilities of going next to i-1 or to i+1 are a linear functions of i. You wrote that!
     
  6. Sep 24, 2013 #5
    Okay, so for example in state 0 we can either stay in state 0 or move to state 1, but this actually means if there are no people than there will always be no people, correct? So state 0's row would have a 1 in the first entry and 0's everywhere else.. moving onto state 1 we would either go to state 0 (prob=1*.02=.02), stay in state 1(1-(b+d)=1-.04=.96), or move to state 2 (prob = 1*.02 = .02). Starting in state 2, we would go to state 1 (2*.02), stay in state 2 (1-.08=.92) or move to state 3(2*.02). Am I correct here?
     
  7. Sep 24, 2013 #6
    Saying it in a more aesthetically easier way, b0=0, b1=b=.02, b2=2b, b3=3b, b4=4b, b5=5b, b6=0 and d0=0, d1=d=.02, d2=2d, d3=3d, d4=4d, d5=5d, and d6=6d which would in turn make the following transition probability matrix...

    0 1 2 3 4 5 6
    0 1 0.................................
    1 .02 .96 .02 0.....................
    2 0 .04 .92 .04 0.............
    3 0 0 .06 .88 .12 0.......
    4 0 0 0 .08 .84 .08 0
    5 0 0 0 0 .10 .80 .10
    6 0 0 0 0 0 .12 .88
     
  8. Sep 25, 2013 #7

    Ray Vickson

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    Yes: state 0 is an absorbing state. Once the system hits state 0 it can never again leave it. That corresponds to population extinction.

    You seem to have the transition probability calculations down pat, now. Good work.
     
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