Discrete Time Birth and Death Process

In summary, the transition probability matrix for this discrete time birth and death process with a maximal population size of 6 is as follows: 0 1 2 3 4 5 60 0 0 0 0 0 0 01 0 0 0 0 0 0 02 0 0 0 0 0 0 03 0 0 .06 .88 .06 0 04 0 0
  • #1
daneault23
32
0

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.
 
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  • #2
daneault23 said:

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.
 
  • #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?
 
  • #4
daneault23 said:
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!
 
  • #5
Ray Vickson said:
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?
 
  • #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
 
  • #7
daneault23 said:
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.
 

FAQ: Discrete Time Birth and Death Process

1. What is a Discrete Time Birth and Death Process?

A Discrete Time Birth and Death Process is a mathematical model used to describe the changes in a population over time. It involves the counting of individuals within a population at discrete intervals, with births and deaths occurring at each interval.

2. How is a Discrete Time Birth and Death Process different from a Continuous Time Birth and Death Process?

The main difference between a Discrete Time Birth and Death Process and a Continuous Time Birth and Death Process is the time variable. In a Discrete Time process, the changes in population occur at discrete intervals, while in a Continuous Time process, the changes occur continuously over time.

3. What are the assumptions made in a Discrete Time Birth and Death Process?

There are several assumptions made in a Discrete Time Birth and Death Process, including a constant population size, a constant birth rate, and a constant death rate. Additionally, it is assumed that births and deaths occur independently of each other, and the population is closed (no migration).

4. How is a Discrete Time Birth and Death Process used in real-world applications?

A Discrete Time Birth and Death Process can be used to model various real-world phenomena, such as the spread of infectious diseases, population growth, and the survival of species. It can also be used in finance and stock market analysis to model changes in prices over time.

5. What are the limitations of a Discrete Time Birth and Death Process?

One limitation of a Discrete Time Birth and Death Process is that it may not accurately reflect the complexities of real-world populations, as it makes several simplifying assumptions. Additionally, it may not be suitable for modeling populations with small numbers or those that experience significant fluctuations in birth and death rates.

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