Mudassir Lodi
Homework for Lecture 6 of Dr. Z.’s Dynamical Models in Biology class
Email the answers (either as .pdf file or .txt file) to ShaloshBEkhad@gmail.com by 8:00pm Monday, Sept. 27, 2021. Subject: hw6 with an attachment hw6FirstLast.pdf and/or hw6FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post
1. In a certain population there are are only 4 age groups:
0-year-olds; 1-year-olds; 2-year-olds; 3-year-olds;
We have the following data
•	The probability of survival from 0-year old to 1-year old is 0.95
•	The probability of survival from 1-year old to 2-year old is 0.97
•	The probability of survival from 2-year old to 3-year old is 0.9
•	The fertility rate of 0-year olds is 0.1
•	The fertility rate of 1-year olds is 1.2
•	The fertility rate of 2-year olds is 0.9
•	The fertility rate of 3-year olds is 0.1
Set up a recurrence, in human language, for n0(t), the number of 0-year-olds at time t. Also express it in our Maple notation, as a list REC.
n0(t) = 0.1 n0(t-1) + 1.2(0.95) n0(t-2)
REC = [0.1, (1.2)(0.95)]
Use procedure GrowthCe from https://sites.math.rutgers.edu/~zeilberg/Bio21/M5.txt to find the growth constant of the population.
GrowthCe:=proc(REC) local x,i:
evalf([solve(1-add(REC[i]/x^i,i=1..nops(REC)))])[1]:
2.	Set up the Leslie Matrix for the population from the previous question. Verify that the largest eigenvalue agrees with the growth constant that you found in Problem 1.
[0.1	   1.2	0]
[0.95   0        0]
[0       0.91    0]
3.	In a mini-internet there are four websites, let’s call them S1,S2,S3,S4.
If a random surfer is currently at a given site, his or her
•	Probability of staying at site S1 is 0.5
•	Probability of staying at site S2 is 0.4
•	Probability of staying at site S3 is 0.3
•	Probability of staying at site S4 is 0.2
We also assume that, for each of the four sites , the probabilities of moving to another site are the same. In other words, for example, the probability of moving from site 1 to site 2, probability of moving from site 1 to site 3, probability of moving from site 1 to site 4 are the same, and similarly for the departures from each site. (Of course, it is not possible that the probability of moving from any site to a different site are all equal to each other). a: Set up the transition matrix P for this Markov chain.
S1         S2        S3          S4
0.50      0         0.16        0.39       S1
0.17      0.40      0.22        0.22       S2
0.17      0.27      0.30        0.27       S3
0.17      0.32      0.32        0.20       S4
b. By using the Linear Algebra package in Maple, find P1000, make sure that all the rows are identical (up to the floating point approximation done by Maple). What fraction of the surfers stay in each of the above web-sites? Use this to determine the the page-ranks of these four web-sites in this mini-internet. In other words rank them according to “popularity”.
Rank: S1, S3, S2, S4