Mudassir Lodi
Homework for Lecture 15 of Dr. Z.’s Dynamical Models in Biology class
Email the answers (either as .pdf file and/or .txt file) to ShaloshBEkhad@gmail.com by 8:00pm Monday, Oct. 25,, 2021. Subject: hw15 with an attachment hw15FirstLast.pdf and/or hw15FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post
1. Read and understand the Maple code for procedure Dis1(F,y,y0,h,A) https://sites.math.rutgers.edu/~zeilberg/Bio21/M15.txt	.
2.
(i)	Let ai := the i-th digit of your RUID (if it is 0 make it 2).
Using Maple’s dsolve solve the following initial value problems
	 	,
	 	,
then use Maple’s plot command to plot these solutions from t = 0 to t = 5.
X’(t) = (1 – x(t))(1 – x(t))(8 – x(t)), x(0) = 1
X’(t) = (1 – x(t))(1 – x(t))(8 – x(t)), x(0) = 2
(ii)	Use Dis1, followed by Maple’s built-in command plot, to find the results of discretization, first with h = 0.1 and then with h = 0.01. Convince yourself that they look similar. Hint: The syntax for plotting a function f(t)=Expression(t) plot(Expression, t=a..b).
For example to plot the function x(t) = sin(t) from t = 0 to t = 10 you type
plot(sin(t),t=0..10);
To plot a list of points L (the output of Dis1) you use the same Maple command plot but the syntax is
plot(L);
Dis1:=proc(F,y,y0,h,A) local L,x,i:
L:=Orb(x+h*subs(y=x,F),x,y0,0,trunc(A/h)):

2.
(i)	Fully by hand convert the fourth-order recurrence
 
into a first-order system with four sequences x1(n),x2(n),x3(n),x4(n), where you renamed x(n), x1(n).
(ii)	Check that this agrees with the output of ToSys(k,z,f,INI). What do you enter for k, f, and INI? What is the output?
ToSys:=proc(k,z,f,INI) local i:
[f,seq(z[i-1],i=2..k)],INI:
end:

3.	Use procedure Orbk (see Help13();) to numerically find a stable fixed points (if it exists) of the second-order recurrence (with the given initial conditions)
	x(n) = (1 − x(n − 1))(1 − x(n − 2))	,	x(0) = 2.5	,	x(1) = 2.7	.
Confirm this rigorously by first using ToSys and then SFP2.
Orbk:=proc(k,z,f,INI,K1,K2) local L, i,newguy:
L:=INI:  #We start out with the list of initial values

if not (type(k, integer) and type(z,symbol) and type(INI,list) and nops(INI)=k and type(K1,integer) and type(K2,integer) and K1>0 and K2>K1) then #checking that the input is OK
 print(`bad input`):
 RETURN(FAIL):
fi:

4.	Carefully read section 3.6 of Leah Edelstein-Keshet book
https://sites.math.rutgers.edu/~zeilberg/Bio21/keshet/keshet3.pdf
Fill-in all the ”blanks” and fully derive the Hardy-Weinberg ”dynamical system”. But instead of lumping-up, for example “Father=AA and Mother=Aa” and “Mother=AA and Father=Aa” into one-category, keep the 9 categories separate.
You can make a 3 × 3 table, and in each of the boxes sub-divide it into three compartments AA, Aa, aa
and exrpess then in terms of u,v,w (of course sometimes you would get 0, for example if both parents are AA, then the number of aA as well as the number of aa is obviously 0).
By adding-up all the terms for AA, Aa, aa, each derive Eqs. ((53a), (53b), (53(c)) that formed the basis of HW3(u,v,w) and HW2(u,v).