I want to solve exercise 3 here:
In the competitive system $$x'=x(k-ax-by), \quad y'=y(m-cx-dy), \quad k,m,a,b,c,d>0 $$ if the lines $ax+by=k$ and $cx+dy=m$ do not intersect in the first quadrangle $x,y>0$ find the limit set.
I'm not sure what is meant by the limit set, but I believe it's either the $\alpha$- or $\omega$- limit set as defined here. The critical points of the system are $$(0,0),(0,m/d),(k/a,0) $$ as well as the intersection of the two lines, it it exists. Hence there are no critical points in the domain $D=\{(x,y):x,y>0\}$. Moreover, the axes $x=0$ and $y=0$ are invriant sets of the flow. This means that if the initial data is $(x_0,y_0) \in D$, the solution will remain in $D$ for all time $t$. I've tried solving the system numerically with various parameter values, but couldn't see a pattern. My question is: what is the limit set given the above?
Thank you!