I'm doing the following problem:
Consider the following system modelling the interaction between two species:
$$\begin{cases}\dot{x}=x(ax^n+by^n+c) \\ \dot{y}=y(dx^n+ey^n+f) \end{cases}$$
with $(a,b,c,d,e,f)\in\mathbb{R^6}$, $n\in\mathbb{N}$. Find a relation $\phi(a,b,c,d,e,f,n)$ such that:
$i)$ If $\phi(a,b,c,d,e,f,n)\neq 0$, then the system has no periodic orbits. Indication: Prove that you can reduce the problem to the first quadrant, and then use the following Dulac function: $D(x,y)=x^\alpha y^\beta$, with properly $\alpha$ and $\beta$.
I've proved that we can reduce the problem to the first quadrant by seeing that $x=0$ and $y=0$ are orbits of the system, so by uniqueness of solutions, we cannot have a periodic orbit crossing them.
Furthermore, we can change the variables to reduce it to the first quadrant.
Now we have to use the Bendixon-Dulac Theorem, with $B(x,y)$ the function they gave to us. Now the main thing that I don't know how to do is to obtain that relation $\phi(a,b,c,d,e,f,n)$. I don't know how to get there.
I found the divergence of the Bendixon Dulac theorem but there's a lot of terms and it's really confusing. Do we have to equal it to zero to find the terms $\alpha$ and $\beta$?
The Bendixson-Dulac needs application only inside each quadrant because $x=0$ and $y = 0$ are orbits.
Now with
$$ \dot x = u(x,y)\\ \dot y = v(x,y) $$
and defining $\phi(x,y) = x^{\alpha}y^{\beta}$ we have
$$ (\phi(x,y)u(x,y))_x + (\phi(x,y)v(x,y))_y = \phi(x,y) \left(x^n (a \alpha +a n+a+\beta d+d)+y^n (\alpha b+b+\beta e+e n+e)+(\alpha +1) c+(\beta +1) f\right) $$
Now choosing
$$ \left\{ \begin{array}{rcl} \alpha & = & \displaystyle -\frac{-a e n-a e+b d+d e n}{b d-a e}\\ \beta & = & \displaystyle-\frac{a b n-a e n-a e+b d}{b d-a e} \end{array} \right. $$
from
$$ \left\{ \begin{array}{rcl} a \alpha +a n+a+\beta d+d & = & 0\\ \alpha b+b+\beta e+e n+e & = & 0 \end{array} \right. $$
we get
$$ (\phi(x,y)u(x,y))_x + (\phi(x,y)v(x,y))_y = \phi(x,y)\left(\frac{n (a f (e-b)+c e (a-d))}{b d-a e} \right) $$
which is non null inside any quadrant, after careful check for the fraction numerator/denominator.