Discrete Time Birth and Death Process

[daneault23]

Homework Statement

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.

Homework 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

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.

 

[Ray Vickson]

Homework Statement

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.

Homework 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

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.

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.

 

[daneault23]

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?

 

[Ray Vickson]

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?

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!

 

[daneault23]

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!

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?

 

[daneault23]

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

 

[Ray Vickson]

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?

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.