Mudassir Lodi
Homework for Lecture 16 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: hw16 with an attachment hw16FirstLast.pdf and/or hw16FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post
0.	Pick or suggest a final project. You are welcome to pick your own team-mates (max. size: 3 students including the leader) by Nov. 1, 2021. Otherwise I will assign you a project and form random teams.
Team: Alan Ho, Mudassir Lodi, Max Mekhanikov
1.	(For everyone!, some people got close, but no one did it fully) Carefully read, https://sites.math.rutgers.edu/~zeilberg/Bio21/att16.pdf .
understand it. Then do the similar questions (with all the parts) for the following difference equation
(a)
	 	.
X1(n) = X1(n-1)(3/2 – X2(n-1))
X2(n) = X1(n-1)
z = z(5/3 – z) 
x = 0, 2/3 are the equilibrium points
(b)
	x(n) = x(n − 1)(2 − x(n − 2))	.
x1(n) = x1(n-1)(2 - x2(n-2))
x2(n) = x1(n-1)
z = z(2 – z)
x = 0, 1 are the equilibrium points
Confirm your results numerically by running
Orbk(k,z,f,INI,K1,K2) with K1 = 1000 and K2 = 1020.
(in the Maple code given here: https://sites.math.rutgers.edu/~zeilberg/Bio21/M15.txt .
2.	(A little bit challenging, but do your best, it is mandatory to attempt it). Consider the family of second-order difference equations, featuring the parameter a, assumed to be positive.
	x(n) = x(n − 1)(a − x(n − 2))	.
Find all the equilibrium points (0 is always one of them, but find the other one), and find conditions on a for the stability of x = 0 and conditions for a for the stability of the other equilibrium point.
X1(n) = x1(n-1)(a – x2(n-2))
X2(n) = x1(n-1)
Z = z(a – z)
0 = az – z - z^2
0 = z(a-1-z)
X = 0, a - 1
3.	(Optional challenge, 20 brownie-points). Consider the family of third-order difference equations, featuring the parameters a,b, assumed to be positive.
	x(n) = x(n − 1)(a − x(n − 2))(b − x(n − 3))	.
Find all the equilibrium points (0 is always one of them, but find the other ones, expressed in terms of a and b), and find conditions on a,b for the stability of x = 0 and conditions on (a,b) for the stability of the other equilibrium points.
Confession: I did not do it myself, and I have no idea how hard it is.
4.	For each of the following (autonomous) first-order ordinary differential equations
(i)	Find all the equilibrium points
(ii)	For each of them decide whether it is a stable equilibrium or not
(iii)	Confirm then by using
plot(Dis1(F,y,y0,h,A)) with h = 0.01 and A = 20.
(it is available from
https://sites.math.rutgers.edu/~zeilberg/Bio21/M15.txt
)
with h = 0.01
Worked Example: x0(t) = x(t)(3 − x(t))
Unerlying function : F(x) = x(3 − x) (recall that the format of a first-order autonomous differential equation is x0(t) = F(x(t))
(i): Equilibrium points: Solve x(3 − x) = 0, getting x = 0 and x = 3.
(ii): F0(x) = 3 − 2x.
When x = 0, F0(0) = 3, this is not negative, so it is unstable When x = 3, F0(0) = −3, this is negative, so it is stable.
(iii)
To invesigate numerically the status of x = 0 you do:
plot(Dis1(y*(3-y),y,0.01,0.01,20));
Note: for this simple case, Maple can solve it, so you can also use
plot(op(2,dsolve(diff(x(t),t)=x(t)*(3-x(t)), x(0)=0.01,x(t))),t=0..10); (but for the more complicated cases coming up,dsolve is too slow and unreliable).
The horizontal asymptote is the other equilibrium point x = 3! So x = 0 is indeed unstable.
To invesigate numerically the status of x = 3 you do:
plot(Dis1(y*(3-y),y,0.01,3.01,20));
(OR plot(op(2,dsolve(diff(x(t),t)=x(t)*(3-x(t)), x(0)=3.01,x(t))),t=0..10);) a: x0(t) = x(t)(3 − x(t))(5 − x(t)) b: x0(t) = x(t)2 (3 − x(t))(5 − x(t))(7 − x(t)) .
(You are welcome to use Maple for differentianting F(x) and for plugging-in).