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**alexmahone** $\displaystyle \frac{dx}{dt}=a_1x-b_1x^2-c_1xy=x(a_1-b_1x-c_1y)$

$\displaystyle \frac{dy}{dt}=a_2y-b_2y^2-c_2xy=y(a_2-b_2y-c_2x)$

$\displaystyle a_1, a_2, b_1, b_2, c_1, c_2$ are positive constants.

Consider the nonzero critical point $\displaystyle (x_E, y_E)$, which is the solution of the system $\displaystyle a_1-b_1x-c_1y=0$ and $\displaystyle a_2-b_2x-c_2y=0$.

The stability of $\displaystyle (x_E, y_E)$ depends on whether $\displaystyle c_1c_2<b_1b_2$ or $\displaystyle c_1c_2>b_1b_2$. Could someone please direct me to a proof of this or guide me through it if it is not too computationally tedious?