(a) Consider the system

$$\begin{aligned} &\frac{d x}{d t}=x(1-x)-x y \\ &\frac{d y}{d t}=\frac{1}{8} y(4 x-1) \end{aligned}$$

for $x(t) \geqslant 0, y(t) \geqslant 0$. Find the critical points, determine their type and explain, with the help of a diagram, the behaviour of solutions for large positive times $t$.

(b) Consider the system

$$\begin{aligned} &\frac{d x}{d t}=y+\left(1-x^{2}-y^{2}\right) x \\ &\frac{d y}{d t}=-x+\left(1-x^{2}-y^{2}\right) y \end{aligned}$$

for $(x(t), y(t)) \in \mathbb{R}^{2}$. Rewrite the system in polar coordinates by setting $x(t)=$ $r(t) \cos \theta(t)$ and $y(t)=r(t) \sin \theta(t)$, and hence describe the behaviour of solutions for large positive and large negative times.