Hi this is the system i need to work on: $$ \left\{\begin{array}{rcl} \dot x&=&x(1-x)-axy\\ \dot y&=&y(1-y)-bxy \end{array} \right. $$ I found four stationary points: $(0,0)$;$(1,0)$;$(0,1)$;$((a-1)/(ab-1),(b-1)/(ab-1))$
I found that $(0,0)$ is stable with linearization, but for the other three there are values of a and b for which i need to find a Lyapunov function and i have no idea where to start. The values being $a=1$ or $b=1$. Could you please give me any ideas? Also what happens if I choose values for $a$ and $b$ like, for example, $2$ and $1/2$. The last point isn't defined anymore, does it just disappear from the phase portrait or what? Anyway I'm quite confused, can anyone answer this questions please? Thanks in advance