Consider the predator$-$prey model

x rx

x x

x y

$=$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$+$

$1$

$2$

$12,$

y y

x

x

y

$=+$

$+$

$2$

$1$

where r $0$ and x y

$,$ p $0.$ Prove this system has no closed orbits by invoking

Dulac$$s criterion with the function gxy

x

x

$(,)=1+$ y B $1$

for a suitable choice of B