MAT 331 Homework Exercises. Week 8 (Nov 9, 99).


#26 (exp. 11/16)
Determine the eigenvalues of

A = $\displaystyle \left[\vphantom{
\begin{array}{cc}
1 & k \\
1 & 0
\end{array} }\right.$$\displaystyle \begin{array}{cc}
1 & k \\
1 & 0
\end{array}$ $\displaystyle \left.\vphantom{
\begin{array}{cc}
1 & k \\
1 & 0
\end{array} }\right]$.

Distinguish all the possible cases (positive, negative, complex eigenvalues) in k.

#27 (exp. 11/16)
For the system of differential equations of prob. #23,

$\displaystyle \left\{\vphantom{
\begin{array}{rcl}
\dot{x} &\!=\!& x^2 + y, \\
\dot{y} &\!=\!& x (y^2 - 1),
\end{array} }\right.$$\displaystyle \begin{array}{rcl}
\dot{x} &\!=\!& x^2 + y, \\
\dot{y} &\!=\!& x (y^2 - 1),
\end{array}$ $\displaystyle \left.\vphantom{
\begin{array}{rcl}
\dot{x} &\!=\!& x^2 + y, \\
\dot{y} &\!=\!& x (y^2 - 1),
\end{array} }\right.$

find the eigenvalues and eigenvectors of the Jacobian at the fixed points. [This is a give-away if you have done #23.]

#28 (exp. 11/16)
We will study the Volterra predator-prey equations: In a very simple ecosystem, at the time t (which is expressed, say, in years), there is a population of f (t) foxes and r(t) rabbits. The evolution of these quantities obeys the system

$\displaystyle \left\{\vphantom{
\begin{array}{rcl}
\dot{f}(t) &\!=\!& G_f\, f...
...t), \\
\dot{r}(t) &\!=\!& G_r\, r(t) - E\, f(t)\, r(t);
\end{array} }\right.$$\displaystyle \begin{array}{rcl}
\dot{f}(t) &\!=\!& G_f\, f(t) + E\, f(t)\, r(t), \\
\dot{r}(t) &\!=\!& G_r\, r(t) - E\, f(t)\, r(t);
\end{array}$ $\displaystyle \left.\vphantom{
\begin{array}{rcl}
\dot{f}(t) &\!=\!& G_f\, f(...
...t), \\
\dot{r}(t) &\!=\!& G_r\, r(t) - E\, f(t)\, r(t);
\end{array} }\right.$

where Gf and Gr are the growth rates for the foxes and the rabbits, respectively, in the absence of each other. E is the probability of a fatal encounter between a fox and a rabbit (normalized per number of foxes and rabbits).
First, write some words to explain why these equations make sense. Then, fix Gf = 0.4, Gr = 2.4 (it's notorius that rabbits have the tendency to reproduce quickly) and E = 0.01. For a few initial conditions of your choice, plot the trajectories in the (f, r)-plane (say, with 0$ \le$f$ \le$1000 and 0$ \le$r$ \le$1000). For the same initial conditions, plot the actual solutions too (i.e, f (t) against t, and r(t) against t). Write some comments as to what happens to the two species and why.
Finally, repeat the same procedure with Gf = - 1.1. Things change substantially. Again, what is the ``physical'' interpretation of this?

#29 (exp. 11/30)
Consider the equations of the glider with no drag term (R = 0). Use dsolve, type=numeric to solve them numerically with initial conditions $ \theta$(0) = 0, v(0) = 0.8. Then solve exactly the linearized system around the fixed point ($ \theta_{0}^{}$, v0) = (0, 1), with the same initial conditions. Graph the two functions for 0$ \le$t$ \le$5, and give a good estimate of their maximum difference. What happens if we take a larger t-range?

#30 (exp. 11/30)
Write a Maple procedure whose input is the pair of functions F1, F2. With regards to the equation $ {\frac{d}{dt}}$(x, y) = (F1(x, y), F2(x, y)), program the procedure to print a list of all the real fixed points, together with the eigenvalues of the Jacobian at each point, the eigenvectors if the eigenvalues are real, and the nature of the fixed point (e.g., saddle point, spiraling sink, etc.--for the trickier case of a double eigenvalue, printing "Double eigenvalue" will suffice).



 


Translated from LaTeX by MAT 331
1999-11-09