Mudassir Lodi

Homework for Lecture 10 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, Oct. 11,, 2021. Subject: hw10 with an attachment hw10FirstLast.pdf and/or hw10FirstLast.txt Also please indicate (EITHER way) whether it is OK to post 
1. Explain why x = 0 is always a fixed point of the difference equation
	 	.
What is the other fixed point? For what values of c is x = 0 a stable fixed point? For what values of c is the other point stable?
f(x) = x/(x+c)
x = x/(x+c)
x/(x+c) – x = 0
x = 0 is a stable fixed point for c > 1 and c < -1. 

2. The origin (0,0) is obviously a fixed point of any transformation (that takes points in the plane to points in the plane) of the form
	(x,y) → (a11x + a12y	,	a21x + a22y)	,
(where a11,a12,a21,a22 are real numbers.
(Recall that these are called linear transformations).
For which of the following transformation is (0,0) a stable fixed point? Explain!
You should do it all by hand, but you are welcome to check your answers with Maple, to make sure. You must show your work. For each case, verify the stability (or non-stability) numerically (with Maple) by seeing what happens when you apply the transformation 1000 times to the point
(0.3,0.6).
(i)
	 	.
Eigenvalues: 9.748, -9.748. This is not a stable fixed point. 
(ii)
	 	.
Eigenvalues = 3/2, 2/3. This is a stable fixed point. 
(iii)
	 	.
Eigenvalues = ¾, -1/2. This is a stable fixed point.