Mudassir Lodi

Homework for Lecture 5 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. 20, 2021. Subject: hw5 with an attachment hw5FirstLast.pdf and/or hw5FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post

0. Convert the recurrence
	6a(n − 1) + a(n + 3) + 5a(n + 1) = 0	,
into canonical form where a(n) is expressed in terms of a(n − 1),a(n − 2),.... Use procedure RecToSeq(INI,REC,N) from today’s Maple code, https://sites.math.rutgers.edu/~zeilberg/Bio21/M5.txt to find a(1000) if a(0) = 1,a(1) = 2,a(2) = 4,a(3) = 11. 

1. Using the Maple code done in today’s class
http://sites.math.rutgers.edu/~zeilberg/Bio21/M4.txt find the growth constant in two different ways of the recurrence
10
	a(n) = Xa(n − i)	,
i=1
with arbitrary positive initial conditions.

GrowthC:=proc([6,1,5,0],[1,2,4,11],K) local L,a,b:
L:=RecToSeq(INI,REC,K):

2. Assuming that
•	A female older than 45 can no longer have babies
•	The youngest age a female can have babies is 16
•	the average fertility of a female between ages 16 and 30 is  
•	the average fertility of a female between ages 31 and 45 is  
•	the survival probability from one year to the next is always equal to  
Find the growth rate of the population, in two ways.
(i)	Using procedure LeslieMod (followed by using procedure GrowthCe).
(ii)	Using procedure LeslieMat (followed by finding the largest eigenvaue (in absolute value) of the Leslie matrix).

GrowthCe:=proc(REC) local x,i:
evalf([solve(1-add(REC[i]/x^i,i=1..nops(REC)))])[1]:

end:

3. Using today’s Maple code, Find the growth rate of Salmons given in Eq. [1.1] in Chapter 1, p.5, of the Ellner-Guckenheimer book DMB:
https://sites.math.rutgers.edu/~zeilberg/Bio21/dmb/dmb1.pdf

GrowthC:=proc(INI,REC,K) local L,a,b:
L:=RecToSeq(INI,REC,K):
 
4. Using the box at the bottom of page 16 of Leah Edelstein-Keshet’s book
https://sites.math.rutgers.edu/~zeilberg/Bio21/keshet/keshet1.pdf and today’s Maple code write a Maple procedure
PlantGseq(alpha,beta,gamma,sigma,INI,K)
that inputs the relevant botanical parameters and outputs the first K terms. Check whether the two tables in Table 1.1 (p. 18) are correct.

PlantGseq(0.5,0.25,2.0,0.8,INI,K)

5. Using the box at the bottom of page 16 of Leah Edelstein’s book
https://sites.math.rutgers.edu/~zeilberg/Bio21/keshet/keshet1.pdf and today’s Maple code write a procedure
PlantGseq(alpha,beta,gamma,sigma)
that inputs the relevant botanical parameters and outputs the growth constant of these plants.
Experiment with altering the parameters to get extinction, stability

PlantGseq(0.5,0.25,2.0,0.8)