MAT331 homework problems

17.

(expires 18 March)     Consider the differential equation $\dot{\mathbf {z}}(t) = \mathbf {F}(\mathbf {z}(t))$, where the vector $\mathbf {z}(t) =(x(t),y(t))$ and the field $\mathbf {F}(x,y) = (-y,x-y)$. Plot a few solutions. What happens to them when $t \to +\infty$? Give a ``Maple-proof'' that this is a general fact for every solution. [A ``Maple-proof'' is an argument that is rigorous once we accept Maple results as incontrovertibly true.]

18.

(expires 18 March)    (No Maple.) For the equation $\dot{\mathbf {z}} = \mathbf {F}(\mathbf {z})$, $\mathbf {z} =(x,y)$, with the vector field

$$\mathbf {F}(x,y) = \left\langle -x(x^4+y^4)-y , x-y(x^4+y^4) \right\rangle,$$

prove that the origin is an attractor in the future, i.e., every solution verifies

$$\lim_{t \to +\infty} \mathbf {z}(t) = 0.$$

[You can ask around how to do this, but then you have to show clearly that you have understood it.]

19.

(expires 18 March)     We will study the Lotke-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

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

where $G_f$ and $G_r$ 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 $G_f = 0.4$, $G_r = 2.4$ (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 interpreting how the behaviour of the solutions relates to what happens to the two species.

Finally, repeat the same procedure with $G_f = -1.1$. Things change substantially. Again, what is the ``physical'' interpretation of this?