Problem 4: Leslie Model The following matrix represents the reproductive cycle o
ID: 3042750 • Letter: P
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Problem 4: Leslie Model The following matrix represents the reproductive cycle of a lion population 0 4.6 0.8 L | 1.2 0 0.6 0 where lik is the average number of female cubs born to a single female lion during the time she is in age class k, and IU-1V-2, 3) is the fraction of female lions in age classj-1 that will survive and pass into class j. (a) What is the number of female lions in each class after 3, 6, 9 years if each class initially consist of 400 female lions? (b) For what initial distribution will the number of female lions in each class change by the same proportion? What is this rate of change?Explanation / Answer
From given matrix we can obtain the transition probability matrix P as follows:
The probability of lions that that will survive after 3 years and will pass to next class will be obtain by P3
So P3 =
0
0.978052
0.0219478
0
1
0
0.148148
0.851852
0
And number of lions out of 400 that will survive and will pass to next class will be obtained by multiplying P3 by 400, which is
At the end of 3 years
0
391
9
0
400
0
59
341
0
The probability of lions that that will survive after 6 years and will pass to next class will be obtain by P6
So P6 =
0.003252
0.996749
0
0
1
0
0
0.996749
0.003251
And number of lions out of 400 that will survive and will pass to next class will be obtained by multiplying P6 by 400, which is
At the end of 6 years
1
399
0
0
400
0
0
399
1
The probability of lions that that will survive after 9 years and will pass to next class will be obtain by P9
So P9 =
0
0.999929
0.0000714
0
1
0
0.000482
0.999518
0
And number of lions out of 400 that will survive and will pass to next class will be obtained by multiplying P9 by 400, which is
At the end of 9 years
0
400
0
0
400
0
0
400
0
Hence we see that after 9 years, distribution of lions will become stationary and hence the corresponding TPM can be taken as initial distribution.
0
0.978052
0.0219478
0
1
0
0.148148
0.851852
0
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