find the nullclines
I don't know how to find the nullclines in the system (see picture attached) according to my professor (0,0) is one of them. I'm so confuse looking for it.
1 Answer
$\begingroup$The nullclines (null meaning zero, cline meaning slope) of the system $$x'=f(x,y),\quad y'=g(x,y)$$ occur when $f(x,y)=0$ or $g(x,y)=0$. You have found these curves to be $$y=ax,\quad y={x\over b(1+x)}.$$
This means that for any point $P$ on the curve $y=ax$, that $x'\big|_P=0$. That is, on the curve $y=ax$, $x$ is not changing with respect to $t$.
Similarly, for any point $Q$ on the curve $y={x\over b(1+x)}$, that $y'\big|_Q=0$, so on the curve $y={x\over b(1+x)}$, $y$ is not changing with respect to $t$.
Finally, where these two curves intersect, namely, $$ax={x\over b(1+x)},$$ we get the points where neither $x$ not $y$ is changing with respect to $t$. These are called the fixed points of the system. Here, the fixed points occur at $$x=0,\ y=0 \quad\text{and}\quad x={1-ab\over ab},\ y={1-ab\over b}.$$
As for sketching the phase portrait, take a look at this and this.
$\endgroup$
