Homework for Lecture 21 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, Dec. 1,, 2025. Subject: hw21 with an attachment hw21FirstLast.pdf and/or hw21FirstLast.txt

1. By hand solve the system

\frac{dx}{dt} = x − y , \frac{dy}{dt} = y − x , x(0) = 1 , y(0) = 1 .

Plot, by hand, the phase-plane diagram.

2. Now use Maple with the command

S:=dsolve({diff(x(t),t)=y(t),diff(y(t),t)=-x(t),x(0)=1,y(0)=0},{x(t),y(t)}); plot([subs(S,x(t)),subs(S,y(t)),t=0..10]);

did you get the same thing?

3. Use Maple to solve and then plot the phase-plane diagram for the system

\frac{dx}{dt} = a_{11}x + a_{12}y , \frac{dy}{dt} = a_{21}x + a_{22}y , x(0) = 1 , y(0) = 1 ,

for three randomly chosen matrices

[ \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ] .

4. Carefully read, and understand, the Maple code for the following procedures (type Help(ProcedureName); for instructions)

Lotka, Volterra, VolterraM

in the Maple package

https://sites.math.rutgers.edu/~zeilberg/Bio25/DMB.txt ,

For each of them, experiment with three randmom choices of parameters, and random initial conditions, using Dis (with h = 0.01), of each of the quantities in question.

Send me these nice plots. Confirm the numerics by using SEquP.