Consider the inter-specific competition system
$$\frac{dx_1}{dt} = r_1 x_1 \left(1 - \frac{x_1 +\alpha_{12}x_2}{K_1} \right)$$ $$\frac{dx_2}{dt} = r_2 x_2 \left(1 - \frac{x_2 +\alpha_{21}x_1}{K_2} \right)$$
where all parameters are positive.
I need to determine the conditions for a coexistent steady state.
The nullclines are $x_2 = \frac{K_1 - x_1}{\alpha_{12}}$ or $x_1 = 0$ from $\dot{x_1} = 0$ and $x_2 = K_2 - \alpha_{21}x_1$ or $x_2 = 0$ from $\dot{x_2} = 0$.
Apparently, the conditions for coexistence are $\alpha_{12}K_2 < K_1$ and $\alpha_{21}K_1 < K_2$
Now I have plotted the phase portrait. I can see where the first condition ($\alpha_{12}K_2 < K_1$) comes from by comparing $y$ intercepts, but where does the second condition ($\alpha_{21}K_1 < K_2$) come from?
Note: I can get the conditions by using the Jacobean, but I would like to see where the second one comes from by simply considering the nullcines.
One way the coexistence conditions come up is by solving for $x_1$ and $x_2$. Then, under the assumptions $x_1,x_2\ge 0, \alpha_{12}\alpha_{21}<1$, the coexistence conditions follow.