Given the following model
$$\begin{cases} \frac{du}{d\tau} = \beta u \big(1- \frac{u}{v}\big)\\ \frac{dv}{d\tau} = v(1-v) - \alpha uv \end{cases}$$
where $\alpha, \beta \geq 0$.
I'm trying to calculate and determine the stability of the steady states. The only one I can find though is $(u,v) = (0,1)$.
I've tried letting $\dot{u} = \dot{v} = 0$ but it doesn't make things easier.
This differential system is found in population dynamics.
You will find a general setting in the excellent document "Modeling Population Dynamics" (https://staff.fnwi.uva.nl/a.m.deroos/downloads/pdf_readers/syllabus.pdf): see equations (7.2) page 137.