(i) State and prove Dulac's Criterion for the non-existence of periodic orbits in $\mathbb{R}^{2}$. Hence show (choosing a weighting factor of the form $x^{\alpha} y^{\beta}$ ) that there are no periodic orbits of the equations

$$\dot{x}=x\left(2-6 x^{2}-5 y^{2}\right), \quad \dot{y}=y\left(-3+10 x^{2}+3 y^{2}\right)$$

(ii) State the Poincaré-Bendixson Theorem. A model of a chemical reaction (the Brusselator) is defined by the second order system

$$\dot{x}=a-x(1+b)+x^{2} y, \quad \dot{y}=b x-x^{2} y$$

where $a, b$ are positive parameters. Show that there is a unique fixed point. Show that, for a suitable choice of $p>0$, trajectories enter the closed region bounded by $x=p, y=b / p$, $x+y=a+b / p$ and $y=0$. Deduce that when $b>1+a^{2}$, the system has a periodic orbit.