If $b_1b_2-c_1c_2=0$ then find the condition for Co-existence in two species Competition model
The two-species competition model is,
$$ \begin{align} \frac{du}{dt}&=(a_1-b_1u-c_1v)u\\ \frac{dv}{dt}&=(a_2-b_2v-c_2u)v \end{align} $$
One equilibrium point is $(u,v)=\left(\frac{a_1b_2-a_2c_1}{b_1b_2-c_1c_2},\frac{a_2b_1-a_1c_2}{b_1b_2-c_1c_2}\right)$ where both species exist. Now, I didn't understand from this information how could I get the condition. The answer mention those conditions:
- If $a_1c_2=a_2c_1$ then the two species can co-exist
- If $a_1c_2>a_2c_1$ then $v\rightarrow0$ as $t\rightarrow\infty$
- If $a_1c_2<a_2c_1$ then $u\rightarrow0$ as $t\rightarrow\infty$
How do they get those conditions? Any help will be appreciated. TIA
For $b_1b_2-c_1c_2\neq0$, I manage to get some conditions but for $b_1b_2-c_1c_2=0$ I didn't manage to get anything.
