Question

Hypothetical Human population matrix over a period of time of 20 year intervals.

Age 0-20 20-40 40-60 60-80   

0-20 [ .24 .98 0 0]

20-40 [ .77 0 .92 0]

40-60 [ .04 0 0 .57]

60-80 [ 0 0 0   0]

Complete the calculation to determine what the population distribution will be 200 years after the initial probability distribution shown in the example as P= [1000, 1000, 1000, 1000]. The formula is P*T^10 (there are 10 sets of 20 in 200)

After 200 years, # of people in the 0-20 range = _______________?

After 200 years, # of people in the 20-40 range = _______________?

After 200 years, # of people in the 40-60 range = _______________?

After 200 years, # of people in the 60-80 range = _______________?

Now, using the same population dynamics matrix, determine what the probability distribution will be after 320 yrars if the initial probability distribution is P= [1100, 1700, 1100, 1000] ? The formula is P*T^16 (there are 16 sets of 20 in 320)

After 320 years, # of people in the 0-20 range = _______________?

After 320 years, # of people in the 20-40 range = _______________?

After 320 years, # of people in the 40-60 range = _______________?

After 320 years, # of people in the 60-80 range = _______________?

Answer 1

Hypothetical Human population matrix over a period of time of 20-year intervals is given below:

0.24	0.98	0	0
0.77	0	0.92	0
0.04	0	0	0.57
0	0	0	0

which is denoted by "T" here.

Now our initial population vector was P = [1000,1000,1000,1000]

Now we need to find the answers for two questions.

After 200 years, what will be the population distribution matrix?

After 200 years, # of people in the 0-20 range = 1204.11

After 200 years, # of people in the 20-40 range = 1162.25

After 200 years, # of people in the 40-60 range = 1049.14

After 200 years, # of people in the 60-80 range = 589.90

Again, if we change the initial population then after 320 years, the population matrix will be changed to the following.

After 320 years, # of people in the 0-20 range = 1809.95

After 320 years, # of people in the 20-40 range = 1748.37

After 320 years, # of people in the 40-60 range = 1576.51

After 320 years, # of people in the 60-80 range = 887.75

Appendix:

The detailed calculation for obtaining the results:

a)

At first, we calculated the T^10 and then we multiplied P with T^10 to get the result

T^10 =

	A1	A2	A3	A4
1	0.6764	0.6085	0.6053	0.2964
2	0.5029	0.5274	0.4231	0.2794
3	0.0248	0.0263	0.0208	0.014
4	0	0	0	0

Now after the multiplication with P, the value becomes  

	C1	C2	C3	C4
1	1204.1	1162.3	1049.1	589.9

Hence the answer came.

b) The same rule, we first calculated the T^16 matrix then multiplied with P which is different from the first P.

T^16

	A1	A2	A3	A4
1	0.7301	0.6971	0.6389	0.3517
2	0.5738	0.5594	0.498	0.2855
3	0.0285	0.0278	0.0247	0.0142
4	0	0	0	0

Now after multiplication with the P in the 2nd question, we have the following matrix

P*T^16

	C1	C2	C3	C4
1	1810	1748.4	1576.5	887.75

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