Mudassir Lodi

Homework for 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. 6, 2021. Subject: hw1 with an attachment hw1FirstLast.pdf or hw1FirstLast.txt
0.	Teach yourself Maple by reading, and playing with the commands in Frank Garvan’s short tutorial 
https://sites.math.rutgers.edu/~zeilberg/EM12/GarvanMapleI.pdf (part I) https://sites.math.rutgers.edu/~zeilberg/EM12/GarvanMapleII.pdf (part II)

1.	In a certain species of animals, only one-year-old, two-year-old, and three-year-old females are fertile.
The probabilities of a one-year-old, two-year-old, and three-year-old female to give birth to a new female are p1, p2, p3 respectively.
Assuming that there were c0 females born at n = 0, c1 females born at n = 1, and c2 females born at n = 2. Set up a recurrence that will enable you to find the expected number of females born at time n.
In terms of c0,c1,c2,p0,p1,p2, how many females were born at n = 4?

5 females were born at n=4. 

2.Write the Maple code F(c0,c1,c2,p1,p2,p3,n) that would input n (the discrete time) and output the number of females born at time n.

Rg:=proc(n,p1,p2,p3,c0,c1,c2) option remember:
if n=0 then
c0: 
elif n=1 then
c1:
elif n=2 then
c2:
else
expand(p1*Rg(n-1,p1,p2,p3,c0,c1,c2)+p2*Rg(n-2,p1,p2,p3,c0,c1,c2))+p3*Rg(n-3,p1,p2,p3,c0,c1,c2): #Recurrence
fi:
end:

3.	Taking c0 = c1 = c2 = 1, experiment with values of p1,p2,p3 that would lead, at time n = 1000, to (i) extinction (ii) stable population (iii) population explosion.
i)	Extinction
        p1 - 0
        p2 - 0
        p3 - 0
ii)	Stable Population
        p1 - 0.25
        p2 - 0.5
        p3 - 0.25
iii)	Population Explosion
	p1 - 0.5
        p2 - 0.5
        p3 - 0.5