Mudassir Lodi

Homework for Lecture 11 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: hw11 with an attachment hw11FirstLast.pdf and/or hw11FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post
1.	Read and understand the Maple code for the Maple procedure SFPe(f,x) . Use it prove that for k < 3 the difference equation
	xn = k xn−1(1 − xn−1)	,
has one stable fixed point. What is it?
The stable fixed point is x = ½. 

2.	(i) By experimenting with the Maple procedure Orb for f=k*x*(1-x) (in M9.txt, also in M11.txt), starting at k=3.1, and incremetning it by 0.1 each time, i.e. k = 3.1 k = 3.2, and later on (once you get close to the bifurcation point), by k = 0.01 etc., estimate the value of k where the second bifurcation point happens, i.e. when the discrete Logistic equation switches from an ultimate (long-run) period of 2 to an ultimate (long-run) period of 4 (the second period-doubling event (as you know from (1), the first period-doubling, from period 1 to period 2, occurs exactly at k = 3.
evalf(Orb(f, x, 0.4, 3, 3.41))
Second bifurcation point occurs at 3.41. 

ii (Optional, a little bit challenging, 5 brownie points),
by looking at the quartic function Comp(k*x*(1-x),x) (the composition of the kx(1 − x) with itself, i.e. the function that maps today’s value to the value at the day-after-tomorrow) and using SFPe find the EXACT value (up to the round-off error of Maple) of the number you found in (i).

3. Using SFPe(f,x) , for each of the three biological (single species) models given in section 3.1 of Professor Leah Edelstein-Keshet wonderful book
https://sites.math.rutgers.edu/~zeilberg/Bio21/keshet/keshet3.pdf
Do the following
•	Find all the fixed points (expressed as expressions in the parameters of the model)
•	For each of the above fixed point, expressed the condition (in human language, not in Maple), in terms of the parameters of the model, for it to be a stable fixed point.
Here is a simple example to illustrate what I mean. For the difference equation
	 	,
with only one parameter (c) the answer would be x = 0; It is a stable fixed point if and only if c > 1 or c < −1. x = 1 − c. It is a stable fixed point if and only if −1 < c < 1.
i) N = lambda/alpha (N^(1-b))

4. Read and understand the Maple code for
Orbk(k,z,f,INI,K1,K2).
(i)	Use it to numerically find the equilibrium point (if it exist, I am telling you there is at most one) of the second-order difference equation
 
using initial conditions x0 = 1.1,x1 = 5.3. Do it for all 16 choices of a = 1,2,3,4 ; b = 1,2,3,4.

(ii)	By hand, find the nice explicit expression, in terms of a and b, for the only equilibrium point.
Hint: In the long run, (by the definition of equilibrium ) all the values are the same so set xn = xn−1 = xn − 2 = z in the above recurrence, and solve for z (it is a simple equation). The root z = 0 should be ruled out, since if you plug-it-in you get division by 0.
Compare your answer to the numerical output of (i) (for those cases where there was a stable equilibrium, numerically).