Hint
One way to determine the direction vectors is to just test points $(p_1,p_2)$ in each region and nullcline segment, directly calculating $\diff{p_1}{t}$ and $\diff{p_2}{t}$ to determine the directions.
Another way is to just reason about how the signs of $\diff{p_1}{t}$ and $\diff{p_2}{t}$ must change when crossing a nullcline. Here's one way to reason about the signs of $\diff{p_1}{t}$ and $\diff{p_2}{t}$ in the different regions. (All answer blanks in this section are optional.)
At the $p_1$-nullcline, the sign of $\diff{p_1}{t}$ must be
. Generally, as one crosses the $p_1$-nullcline, the sign of $\diff{p_1}{t}$ will
. Similarly, for the the sign of $\diff{p_2}{t}$ when crossing the $p_2$-nullcline.
We begin by deducing what the sign of $\diff{p_1}{t}$ must be on either side of the diagonal portion of the $p_1$-nullcline. (Since the vertical/horizontal lines of the nullclines border the biologically plausible quadrant of the nullclines, we don't have to worry about crossing those nullcline segments.)
The one piece of $p_1$-nullcline that is a diagonal line is
. In the region below this line, we know that $p_1 + \frac{1}{2}p_2 \lt$
. Since we are in the biologically plausible quadrant, we know that $p_1 \gt$
. Therefore, below the diagonal line of the $p_1$-nullcline, we can conclude that $\diff{p_1}{t}$ is
. (Think about the signs of the factors of the expression for $\diff{p_1}{t}$.)
This means that direction vectors below this diagonal line must point
(although they may also point either up or down depending on the sign of $\diff{p_1}{t}$).
On the other hand, above the diagonal line of the $p_1$-nullcline, the sign of $\diff{p_1}{t}$ must switch to being
so the direction vectors switch to pointing
.
Here is similar reasoning for the sign of $\diff{p_2}{t}$ on either side of the diagonal line of the $p_2$-nullcline.
The one piece of $p_2$-nullcline that is a diagonal line is
. In the region below this line, we know that $p_2 + \frac{1}{2}p_1 \lt$
. Since we are in the biologically plausible quadrant, we know that $p_2 \gt$
. Therefore, below the diagonal line of the $p_2$-nullcline, we can conclude that $\diff{p_2}{t}$ is
. This means that direction vectors below this diagonal line must point
(although they may also point either left or right depending on the sign of $\diff{p_2}{t}$).
On the other hand, above the diagonal line of the $p_2$-nullcline, the sign of $\diff{p_2}{t}$ must switch to being
so the direction vectors switch to pointing
.
All four regions of the phase plane are defined as being above or below these two diagonal lines, so the above information is enough to deduce the direction of the vectors in each region.
The same logic will work for determining the direction vectors on the nullcline. For example, for direction vectors on the $p_1$-nullcline, all you need to know is whether or not you are above or below the diagonal line of the $p_2$-nullcline. The above logic will then give you the sign of the $\diff{p_2}{t}$ in order to tell you if the direction vector should point straight
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