Let $$\left\{\begin{align} \dot x &= x- \frac{xy}{1+\alpha x}\\ \dot y &= -y + \frac{xy}{1+\alpha x}+\delta y^2 \end{align} \right.$$ be a predator-prey model.
Prove that the following polynomial system has the same orbits as the original model. $$\left\{\begin{align} \dot x &= x(1+\alpha x)-xy\\ \dot y &= -(y+\delta y^2)(1+\alpha x) +xy \end{align} \right.$$
Could someone provide a hint?
I have tried: Let $(x(t),y(t))$ be an orbit of the second system where $x(t), y(t) >0 \quad (\forall t)$. Then I should prove that this is also an orbit of the first system. A substitution leads to $(1+\alpha \cdot x(t) ) \dot x(t) = \dot x(t)$ which seems to imply $\dot x(t) = 0$, nonsense!
I guess I should translate/manipulate the orbit $(x(t),y(t))$, but how?
The vector fields in the two systems differ by a scalar multiple $1+\alpha x$.