Mudassir Lodi
Homework for Lecture 17 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, Nov. 1,, 2021. Subject: hw17 with an attachment hw17FirstLast.pdf and/or hw17FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post
1.
Carefully read, https://sites.math.rutgers.edu/~zeilberg/Bio21/att17.pdf . understand it. Then do the similar questions (with all the parts) for the following system
	x0(t) = 3x(t) − y(t)	,	y0(t) = 2x(t)	;	x(0) = 2	,	y(0) = 3
Except for (iii) this should be all by hand.
i)	X”(t) = 3x’(t) – y’(t)
X”(t) = 3x’(t) – 2x(t)
X”(t) – 3x’(t) – 2x(t)
R^2 – 3R – 2
ii)	X’(t) = 3 * x(t) – 1* y(t)
Y’(t) = 2 * x(t) – 0 * y(t)
[3 - λ      -1]
[2       0 – λ]

-3 λ + λ2
Λ(-3 + λ)
Λ = 0, 3
iii)	dsolve({diff(x(t), t) = 3x(t) - y(t), diff(y(t), t) = 2x(t), x(0) = 2, y(0) = 3}, {x(t), y(t)});

2.	You are welcome to use Maple as a calculator (including the Eigenvectors command (that includes the eigenvalues), becasue it is a random problem, with probably messy answer).
Let ai be the i-th digit of your RUID. Solve, in three different ways (as in the attendance pop quiz), the system
	x0(t) = a1 x(t) + a2y(t)	,	y0(t) = a3 x(t) − a5y(t)	;	x(0) = 1 ,	y(0) = 0
x’(t) = 2x(t) + y(t), y’(t) = x(t)
i)	X”(t) = 2X’(t) + X(t)
X”(t) – 2X’(t) + x(t)
R^2 – 2R + 1
R = 1
ii)	[2 - λ      1]
[1          - λ]

-2 λ + λ2 - 1
Λ2 - 2 λ - 1

Λ = -0.414, 2.414
iii)	dsolve({diff(x(t), t) = 2x(t) + y(t), diff(y(t), t) = x(t), x(0) = 1, y(0) = 0}, {x(t), y(t)});
3.	You either the (i) or (ii) (you pick) to solve the system (you are welcome to use Maple as a calculator)
	 	,
 
 
with the initial conditions
	x1(0) = 1	,	x2(0) = 2	,	x3(0) = −1	.
Check your answer with dsolve.
X1’(t) = x1(t) + x1(t) + x2(t) + x1(t) = 3x1(t) + x2(t)
4.	Careully read and understand the Maple code for procedure HW2g(u,v,M) in https://sites.math.rutgers.edu/~zeilberg/Bio21/M17.txt
(based on Anne Somalwar’s solution)
(i)	Using M=[[1,1,1],[1,1,1],[1,1,1]] and Orb2 with arbitrary initial point [a,b], rigorously prove the Hardy-Weinberg rule, that after only one generation things stabilize.
(ii)	By experimenting with ten different random M show that things stabilize in the long-run but sometimes some of the alleles die out.